From 19bde89b84619defc2a822c3bf96bdca9c97eee7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Wed, 28 Jun 2023 12:16:10 +0200
Subject: Reorganize backends/lean/Base

---
 backends/lean/Base/Diverge/Elab.lean | 182 +++++++++++++++++++++++++++++++++++
 1 file changed, 182 insertions(+)
 create mode 100644 backends/lean/Base/Diverge/Elab.lean

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
new file mode 100644
index 00000000..313c5a79
--- /dev/null
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -0,0 +1,182 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Base.Diverge.Base
+import Base.Diverge.ElabBase
+
+namespace Diverge
+
+/- Automating the generation of the encoding and the proofs so as to use nice
+   syntactic sugar. -/
+
+syntax (name := divergentDef)
+  declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command
+
+open Lean Elab Term Meta Primitives
+
+initialize registerTraceClass `Diverge.divRecursion (inherited := true)
+
+set_option trace.Diverge.divRecursion true
+
+/- The following was copied from the `wfRecursion` function. -/
+
+open WF in
+def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
+  let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
+  logInfo ("divRecursion: defs: " ++ msg)
+
+  -- CHANGE HERE This function should add definitions with these names/types/values ^^
+  -- Temporarily add the predefinitions as axioms
+  for preDef in preDefs do
+    addAsAxiom preDef
+
+  -- TODO: what is this?
+  for preDef in preDefs do
+    applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
+
+  -- Process the definitions
+  addAndCompilePartialRec preDefs
+
+-- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
+-- This is the only part where we make actual changes and hook into the equation compiler.
+-- (I've removed all the well-founded stuff to make it easier to read though.)
+
+open private ensureNoUnassignedMVarsAtPreDef betaReduceLetRecApps partitionPreDefs
+  addAndCompilePartial addAsAxioms from Lean.Elab.PreDefinition.Main
+
+def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLCtx {} {} do
+  for preDef in preDefs do
+    trace[Elab.definition.body] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+  let preDefs ← preDefs.mapM ensureNoUnassignedMVarsAtPreDef
+  let preDefs ← betaReduceLetRecApps preDefs
+  let cliques := partitionPreDefs preDefs
+  let mut hasErrors := false
+  for preDefs in cliques do
+    trace[Elab.definition.scc] "{preDefs.map (·.declName)}"
+    try
+      logInfo "calling divRecursion"
+      withRef (preDefs[0]!.ref) do
+        divRecursion preDefs
+      logInfo "divRecursion succeeded"
+    catch ex =>
+      -- If it failed, we 
+      logInfo "divRecursion failed"
+      hasErrors := true
+      logException ex
+      let s ← saveState
+      try
+        if preDefs.all fun preDef => preDef.kind == DefKind.def ||
+           preDefs.all fun preDef => preDef.kind == DefKind.abbrev then
+          -- try to add as partial definition
+          try
+            addAndCompilePartial preDefs (useSorry := true)
+          catch _ =>
+            -- Compilation failed try again just as axiom
+            s.restore
+            addAsAxioms preDefs
+        else return ()
+      catch _ => s.restore
+
+-- The following two functions are copy&pasted from Lean.Elab.MutualDef
+
+open private elabHeaders levelMVarToParamHeaders getAllUserLevelNames withFunLocalDecls elabFunValues
+  instantiateMVarsAtHeader instantiateMVarsAtLetRecToLift checkLetRecsToLiftTypes withUsed from Lean.Elab.MutualDef
+
+def Term.elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM Unit := do
+    let scopeLevelNames ← getLevelNames
+    let headers ← elabHeaders views
+    let headers ← levelMVarToParamHeaders views headers
+    let allUserLevelNames := getAllUserLevelNames headers
+    withFunLocalDecls headers fun funFVars => do
+      for view in views, funFVar in funFVars do
+        addLocalVarInfo view.declId funFVar
+      let values ←
+        try
+          let values ← elabFunValues headers
+          Term.synthesizeSyntheticMVarsNoPostponing
+          values.mapM (instantiateMVars ·)
+        catch ex =>
+          logException ex
+          headers.mapM fun header => mkSorry header.type (synthetic := true)
+      let headers ← headers.mapM instantiateMVarsAtHeader
+      let letRecsToLift ← getLetRecsToLift
+      let letRecsToLift ← letRecsToLift.mapM instantiateMVarsAtLetRecToLift
+      checkLetRecsToLiftTypes funFVars letRecsToLift
+      withUsed vars headers values letRecsToLift fun vars => do
+        let preDefs ← MutualClosure.main vars headers funFVars values letRecsToLift
+        for preDef in preDefs do
+          trace[Elab.definition] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+        let preDefs ← withLevelNames allUserLevelNames <| levelMVarToParamPreDecls preDefs
+        let preDefs ← instantiateMVarsAtPreDecls preDefs
+        let preDefs ← fixLevelParams preDefs scopeLevelNames allUserLevelNames
+        for preDef in preDefs do
+          trace[Elab.definition] "after eraseAuxDiscr, {preDef.declName} : {preDef.type} :=\n{preDef.value}"
+        checkForHiddenUnivLevels allUserLevelNames preDefs
+        addPreDefinitions preDefs
+
+open Command in
+def Command.elabMutualDef (ds : Array Syntax) : CommandElabM Unit := do
+  let views ← ds.mapM fun d => do
+    let `($mods:declModifiers divergent def $id:declId $sig:optDeclSig $val:declVal) := d
+      | throwUnsupportedSyntax
+    let modifiers ← elabModifiers mods
+    let (binders, type) := expandOptDeclSig sig
+    let deriving? := none
+    pure { ref := d, kind := DefKind.def, modifiers,
+           declId := id, binders, type? := type, value := val, deriving? }
+  runTermElabM fun vars => Term.elabMutualDef vars views
+
+-- Special command so that we don't fall back to the built-in mutual when we produce an error.
+local syntax "_divergent" Parser.Command.mutual : command
+elab_rules : command | `(_divergent mutual $decls* end) => Command.elabMutualDef decls
+
+macro_rules
+  | `(mutual $decls* end) => do
+    unless !decls.isEmpty && decls.all (·.1.getKind == ``divergentDef) do
+      Macro.throwUnsupported
+    `(command| _divergent mutual $decls* end)
+
+open private setDeclIdName from Lean.Elab.Declaration
+elab_rules : command
+  | `($mods:declModifiers divergent%$tk def $id:declId $sig:optDeclSig $val:declVal) => do
+    let (name, _) := expandDeclIdCore id
+    if (`_root_).isPrefixOf name then throwUnsupportedSyntax
+    let view := extractMacroScopes name
+    let .str ns shortName := view.name | throwUnsupportedSyntax
+    let shortName' := { view with name := shortName }.review
+    let cmd ← `(mutual $mods:declModifiers divergent%$tk def $(⟨setDeclIdName id shortName'⟩):declId $sig:optDeclSig $val:declVal end)
+    if ns matches .anonymous then
+      Command.elabCommand cmd
+    else
+      Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
+
+mutual
+  divergent def is_even (i : Int) : Result Bool :=
+    if i = 0 then return true else return (← is_odd (i - 1))
+
+  divergent def is_odd (i : Int) : Result Bool :=
+    if i = 0 then return false else return (← is_even (i - 1))
+end
+
+example (i : Int) : is_even i = .ret (i % 2 = 0) ∧ is_odd i = .ret (i % 2 ≠ 0) := by
+  induction i
+  unfold is_even
+  sorry    
+
+divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+  match ls with
+  | [] => .fail .panic
+  | x :: ls =>
+    if i = 0 then return x
+    else return (← list_nth ls (i - 1))
+
+mutual
+  divergent def foo (i : Int) : Result Nat :=
+    if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10
+
+  divergent def bar (i : Int) : Result Nat :=
+    if i > 20 then foo (i / 20) else .ret 42
+end
+
+end Diverge
-- 
cgit v1.3.1


From a6de153f3bfda7feb27d16fcdf2131d37f99c7a3 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 29 Jun 2023 11:22:32 +0200
Subject: Start working on Elab.lean

---
 backends/lean/Base/Diverge/Base.lean     |   3 +
 backends/lean/Base/Diverge/Elab.lean     | 138 ++++++++++++++++++++++++++++---
 backends/lean/Base/Diverge/ElabBase.lean |  75 ++++++++++++++++-
 3 files changed, 203 insertions(+), 13 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 0f92e682..2e60f6e8 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -4,6 +4,9 @@ import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 import Mathlib.Tactic.Linarith
 
+-- For debugging
+import Base.Diverge.ElabBase
+
 /-
 TODO:
 - we want an easier to use cases:
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 313c5a79..22e0039f 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -15,16 +15,53 @@ syntax (name := divergentDef)
 
 open Lean Elab Term Meta Primitives
 
-initialize registerTraceClass `Diverge.divRecursion (inherited := true)
-
-set_option trace.Diverge.divRecursion true
+set_option trace.Diverge.def true
 
 /- The following was copied from the `wfRecursion` function. -/
 
 open WF in
+
+
+
+-- Replace the recursive calls by a call to the continuation
+-- def replace_rec_calls
+
+#check Lean.Meta.forallTelescope
+#check Expr
+#check withRef
+#check MonadRef.withRef
+#check Nat
+#check Array
+#check Lean.Meta.inferType
+#check Nat
+#check Int
+
+#check (0, 1)
+#check Prod
+#check ()
+#check Unit
+#check Sigma
+
+--  print_decl is_even_body
+#check instOfNatNat
+#check OfNat.ofNat -- @OfNat.ofNat ℕ 2 ...
+#check OfNat.ofNat -- @OfNat.ofNat (Fin 2) 1 ...
+#check Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat
+
+
+-- TODO: is there already such a utility somewhere?
+-- TODO: change to mkSigmas
+def mkProds (tys : List Expr) : MetaM Expr :=
+  match tys with
+  | [] => do return (Expr.const ``PUnit.unit [])
+  | [ty] => do return ty
+  | ty :: tys => do
+    let pty ← mkProds tys
+    mkAppM ``Prod.mk #[ty, pty]
+
 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
-  logInfo ("divRecursion: defs: " ++ msg)
+  trace[Diverge.def] ("divRecursion: defs: " ++ msg)
 
   -- CHANGE HERE This function should add definitions with these names/types/values ^^
   -- Temporarily add the predefinitions as axioms
@@ -35,6 +72,85 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   for preDef in preDefs do
     applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
 
+  -- Retrieve the name of the first definition, that we will use as the namespace
+  -- for the definitions common to the group
+  let def0 := preDefs[0]!
+  let grName := def0.declName
+  trace[Diverge.def] "group name: {grName}"
+
+  /- Compute the type of the continuation.
+
+     We do the following
+     - we make sure all the definitions have the same universe parameters
+       (we can make this more general later)
+     - we group all the type parameters together, make sure all the
+       definitions have the same type parameters, and enforce
+       a uniform polymorphism (we can also lift this later).
+       This would require generalizing a bit our indexed fixed point to
+       make the output type parametric in the input.
+     - we group all the non-type parameters: we parameterize the continuation
+       by those
+  -/
+  let grLvlParams := def0.levelParams
+  trace[Diverge.def] "def0 type: {def0.type}"
+
+  -- Small utility: compute the list of type parameters
+  let getTypeParams (ty: Expr) : MetaM (List Expr × List Expr × Expr) :=
+    Lean.Meta.forallTelescope ty fun tys out_ty => do
+      trace[Diverge.def] "types: {tys}"
+/-      let (_, params) ← StateT.run (do
+        for x in tys do
+          let ty ← Lean.Meta.inferType x
+          match ty with
+          | .sort _ => do
+            let st ← StateT.get
+            StateT.set (ty :: st)
+          | _ => do break
+        ) ([] : List Expr)
+      let params := params.reverse
+      trace[Diverge.def] "  type parameters {params}"
+      return params -/
+      let rec get_params (ls : List Expr) : MetaM (List Expr × List Expr) :=
+        match ls with
+        | x :: tl => do
+          let ty ← Lean.Meta.inferType x
+          match ty with
+          | .sort _ => do
+            let (ty_params, params) ← get_params tl
+            return (x :: ty_params, params)
+          | _ => do return ([], ls)
+        | _ => do return ([], [])
+      let (ty_params, params) ← get_params tys.toList
+      trace[Diverge.def] "  parameters: {ty_params}; {params}"
+      return (ty_params, params, out_ty)
+  let (grTyParams, _, _) ← do
+    getTypeParams def0.type
+
+  -- Compute the input types and the output types
+  let all_tys ← preDefs.mapM fun preDef => do
+    let (tyParams, params, ret_ty) ← getTypeParams preDef.type
+    -- TODO: this is not complete, there are more checks to perform
+    if tyParams.length ≠ grTyParams.length then
+      throwError "Non-uniform polymorphism"
+    return (params, ret_ty)
+
+  -- TODO: I think there are issues with the free variables
+  let (input_tys, output_tys) := List.unzip all_tys.toList
+  let input_tys : List Expr ← liftM (List.mapM mkProds input_tys)
+
+  trace[Diverge.def] "  in/out tys: {input_tys}; {output_tys}"
+
+  -- Compute the names set
+  let names := preDefs.map PreDefinition.declName
+  let names := HashSet.empty.insertMany names
+
+  --
+  for preDef in preDefs do
+    trace[Diverge.def] "about to explore: {preDef.declName}"
+    explore_term "" preDef.value
+
+  -- Compute the bodies
+
   -- Process the definitions
   addAndCompilePartialRec preDefs
 
@@ -47,21 +163,21 @@ open private ensureNoUnassignedMVarsAtPreDef betaReduceLetRecApps partitionPreDe
 
 def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLCtx {} {} do
   for preDef in preDefs do
-    trace[Elab.definition.body] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+    trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
   let preDefs ← preDefs.mapM ensureNoUnassignedMVarsAtPreDef
   let preDefs ← betaReduceLetRecApps preDefs
   let cliques := partitionPreDefs preDefs
   let mut hasErrors := false
   for preDefs in cliques do
-    trace[Elab.definition.scc] "{preDefs.map (·.declName)}"
+    trace[Diverge.elab] "{preDefs.map (·.declName)}"
     try
-      logInfo "calling divRecursion"
+      trace[Diverge.elab] "calling divRecursion"
       withRef (preDefs[0]!.ref) do
         divRecursion preDefs
-      logInfo "divRecursion succeeded"
+      trace[Diverge.elab] "divRecursion succeeded"
     catch ex =>
       -- If it failed, we 
-      logInfo "divRecursion failed"
+      trace[Diverge.elab] "divRecursion failed"
       hasErrors := true
       logException ex
       let s ← saveState
@@ -106,12 +222,12 @@ def Term.elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM U
       withUsed vars headers values letRecsToLift fun vars => do
         let preDefs ← MutualClosure.main vars headers funFVars values letRecsToLift
         for preDef in preDefs do
-          trace[Elab.definition] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+          trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
         let preDefs ← withLevelNames allUserLevelNames <| levelMVarToParamPreDecls preDefs
         let preDefs ← instantiateMVarsAtPreDecls preDefs
         let preDefs ← fixLevelParams preDefs scopeLevelNames allUserLevelNames
         for preDef in preDefs do
-          trace[Elab.definition] "after eraseAuxDiscr, {preDef.declName} : {preDef.type} :=\n{preDef.value}"
+          trace[Diverge.elab] "after eraseAuxDiscr, {preDef.declName} : {preDef.type} :=\n{preDef.value}"
         checkForHiddenUnivLevels allUserLevelNames preDefs
         addPreDefinitions preDefs
 
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index e693dce2..84b73a30 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -2,8 +2,79 @@ import Lean
 
 namespace Diverge
 
-open Lean
+open Lean Elab Term Meta
 
-initialize registerTraceClass `Diverge.divRecursion (inherited := true)
+initialize registerTraceClass `Diverge.elab (inherited := true)
+initialize registerTraceClass `Diverge.def (inherited := true)
+
+-- TODO: move
+-- TODO: small helper
+def explore_term (incr : String) (e : Expr) : TermElabM Unit :=
+  match e with
+  | .bvar _ => do logInfo m!"{incr}bvar: {e}"; return ()
+  | .fvar _ => do logInfo m!"{incr}fvar: {e}"; return ()
+  | .mvar _ => do logInfo m!"{incr}mvar: {e}"; return ()
+  | .sort _ => do logInfo m!"{incr}sort: {e}"; return ()
+  | .const _ _ => do logInfo m!"{incr}const: {e}"; return ()
+  | .app fn arg => do
+    logInfo m!"{incr}app: {e}"
+    explore_term (incr ++ "  ") fn
+    explore_term (incr ++ "  ") arg
+  | .lam _bName bTy body _binfo => do
+    logInfo m!"{incr}lam: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .forallE _bName bTy body _bInfo => do
+    logInfo m!"{incr}forallE: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .letE _dName ty val body _nonDep => do
+    logInfo m!"{incr}letE: {e}"
+    explore_term (incr ++ "  ") ty
+    explore_term (incr ++ "  ") val
+    explore_term (incr ++ "  ") body
+  | .lit _ => do logInfo m!"{incr}lit: {e}"; return ()
+  | .mdata _ e => do
+    logInfo m!"{incr}mdata: {e}"
+    explore_term (incr ++ "  ") e
+  | .proj _ _ struct => do
+    logInfo m!"{incr}proj: {e}"
+    explore_term (incr ++ "  ") struct
+
+def explore_decl (n : Name) : TermElabM Unit := do
+  logInfo m!"Name: {n}"
+  let env ← getEnv
+  let decl := env.constants.find! n
+  match decl with
+  | .defnInfo val =>
+     logInfo m!"About to explore defn: {decl.name}"
+     logInfo m!"# Type:"
+     explore_term "" val.type
+     logInfo m!"# Value:"
+     explore_term "" val.value
+  | .axiomInfo _  => throwError m!"axiom: {n}"
+  | .thmInfo _    => throwError m!"thm: {n}"
+  | .opaqueInfo _ => throwError m!"opaque: {n}"
+  | .quotInfo _   => throwError m!"quot: {n}"
+  | .inductInfo _ => throwError m!"induct: {n}"
+  | .ctorInfo _   => throwError m!"ctor: {n}"
+  | .recInfo _    => throwError m!"rec: {n}"
+
+syntax (name := printDecl) "print_decl " ident : command
+
+open Lean.Elab.Command
+
+@[command_elab printDecl] def elabPrintDecl : CommandElab := fun stx => do
+  liftTermElabM do
+    let id := stx[1]
+    addCompletionInfo <| CompletionInfo.id id id.getId (danglingDot := false) {} none
+    let cs ← resolveGlobalConstWithInfos id
+    explore_decl cs[0]!
+
+private def test1 : Nat := 0
+private def test2 (x : Nat) : Nat := x
+
+print_decl test1
+print_decl test2
 
 end Diverge
-- 
cgit v1.3.1


From fdc8693772ecb1978873018c790061854f00a015 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 29 Jun 2023 23:15:20 +0200
Subject: Write function to compute the input/output types

---
 backends/lean/Base/Diverge/Base.lean     |   3 +-
 backends/lean/Base/Diverge/Elab.lean     | 154 ++++++++++++++++++++++++-------
 backends/lean/Base/Diverge/ElabBase.lean |   1 +
 3 files changed, 126 insertions(+), 32 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 630c0bf6..22b59bd0 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -560,6 +560,7 @@ namespace FixI
     kk_ty id a b → kk_ty id a b
 
   def in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
+  -- TODO: remove?
   @[simp] def mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
     in_out_ty :=
     Sigma.mk in_ty out_ty
@@ -1143,7 +1144,7 @@ namespace Ex6
         if i = 0 then .ret hd
         else list_nth tl (i - 1)
     := by
-    have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
+    have Heq := is_valid_fix_fixed_eq list_nth_body_is_valid
     simp [list_nth]
     conv => lhs; rw [Heq]
 
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 22e0039f..116c5d8b 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -13,7 +13,7 @@ namespace Diverge
 syntax (name := divergentDef)
   declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command
 
-open Lean Elab Term Meta Primitives
+open Lean Elab Term Meta Primitives Lean.Meta
 
 set_option trace.Diverge.def true
 
@@ -21,27 +21,9 @@ set_option trace.Diverge.def true
 
 open WF in
 
-
-
 -- Replace the recursive calls by a call to the continuation
 -- def replace_rec_calls
 
-#check Lean.Meta.forallTelescope
-#check Expr
-#check withRef
-#check MonadRef.withRef
-#check Nat
-#check Array
-#check Lean.Meta.inferType
-#check Nat
-#check Int
-
-#check (0, 1)
-#check Prod
-#check ()
-#check Unit
-#check Sigma
-
 --  print_decl is_even_body
 #check instOfNatNat
 #check OfNat.ofNat -- @OfNat.ofNat ℕ 2 ...
@@ -59,6 +41,100 @@ def mkProds (tys : List Expr) : MetaM Expr :=
     let pty ← mkProds tys
     mkAppM ``Prod.mk #[ty, pty]
 
+/- Generate the input type of a function body, which is a sigma type (i.e., a
+   dependent tuple) which groups all its inputs.
+
+   Example:
+   - xl = [(a:Type), (ls:List a), (i:Int)]
+
+   Generates:
+   `(a:Type) × (ls:List a) × (i:Int)`
+
+ -/
+def mkSigmasTypesOfTypes (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
+    return (Expr.const ``PUnit.unit [])
+  | [x] => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
+    let ty ← Lean.Meta.inferType x
+    return ty
+  | x :: xl => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
+    let alpha ← Lean.Meta.inferType x
+    let sty ← mkSigmasTypesOfTypes xl
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]: alpha={alpha}, sty={sty}"
+    let beta ← mkLambdaFVars #[x] sty
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
+    mkAppOptM ``Sigma #[some alpha, some beta]
+
+def mk_indexed_name (index : Nat) : Name := .num (.str .anonymous "_uniq") index
+
+/- Generate the out_ty of the body of a function, which from an input (a sigma
+   type generated by `mkSigmasTypesOfTypes`) gives the output type of the function.
+
+   Example:
+   - xl = `[a:Type, ls:List a, i:Int]`
+   - out_ty = `a`
+   - index = 0      -- For naming purposes: we use it to numerotate the "scrutinee" variables
+
+   Generates:
+   ```
+   match scrut0 with
+   | Sigma.mk x scrut1 =>
+     match scrut1 with
+     | Sigma.mk ls i =>
+       a
+   ```
+-/
+def mkSigmasOutType (xl : List Expr) (out_ty : Expr) (index : Nat := 0) : MetaM Expr :=
+  match xl with
+  | [] => do
+    -- This would be unexpected
+    throwError "mkSigmasOutType: empyt list of input parameters"
+  | [x] => do
+    -- In the explanations above: inner match case
+    trace[Diverge.def.sigmas] "mkSigmasOutType: [{x}]"
+    mkLambdaFVars #[x] out_ty
+  | fst :: xl => do
+    -- In the explanations above: outer match case
+    -- Remark: for the naming purposes, we use the same convention as for the
+    -- fields and parameters in `Sigma.casesOn and `Sigma.mk
+    trace[Diverge.def.sigmas] "mkSigmasOutType: [{fst}::{xl}]"
+    let alpha ← Lean.Meta.inferType fst
+    let snd_ty ← mkSigmasTypesOfTypes xl
+    let beta ← mkLambdaFVars #[fst] snd_ty
+    let snd ← mkSigmasOutType xl out_ty (index + 1)
+    let scrut_ty ← mkSigmasTypesOfTypes (fst :: xl)
+    withLocalDeclD (mk_indexed_name index) scrut_ty fun scrut => do
+    let mk ← mkLambdaFVars #[fst] snd
+    trace[Diverge.def.sigmas] "mkSigmasOutType: scrut: ({scrut}) : ({← inferType scrut})"
+    let motive ← mkLambdaFVars #[scrut] (← inferType out_ty)
+    trace[Diverge.def.sigmas] "mkSigmasOutType:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
+    let out ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+    let out ← mkLambdaFVars #[scrut] out
+    trace[Diverge.def.sigmas] "mkSigmasOutType: out: {out}"
+    return out
+
+/- Small tests for list_nth: give a model of what `mkSigmasOutType` should generate -/
+private def list_nth_out_ty2 (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
+  @Sigma.casesOn (List a)
+                 (fun (_ls : List a) => Int)
+                 (fun (_scrut1:@Sigma (List a) (fun (_ls : List a) => Int)) => Type)
+                 scrut1
+                 (fun (_ls : List a) (_i : Int) => Diverge.Primitives.Result a)
+
+private def list_nth_out_ty1 (scrut0 : @Sigma (Type) (fun (a:Type) =>
+                      @Sigma (List a) (fun (_ls : List a) => Int))) :=
+  @Sigma.casesOn (Type)
+                 (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))
+                 (fun (_scrut0:@Sigma (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))) => Type)
+                 scrut0
+                 (fun (a : Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) =>
+                  list_nth_out_ty2 a scrut1)
+/- -/
+
 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
   trace[Diverge.def] ("divRecursion: defs: " ++ msg)
@@ -94,7 +170,23 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let grLvlParams := def0.levelParams
   trace[Diverge.def] "def0 type: {def0.type}"
 
-  -- Small utility: compute the list of type parameters
+  -- Compute the list of pairs: (input type × output type)
+  let inOutTys : Array (Expr × Expr) ←
+      preDefs.mapM (fun preDef => do
+        -- Check the universe parameters - TODO: I'm not sure what the best thing
+        -- to do is. In practice, all the type parameters should be in Type 0, so
+        -- we shouldn't have universe issues.
+        if preDef.levelParams ≠ grLvlParams then
+          throwError "Non-uniform polymorphism in the universes"
+        forallTelescope preDef.type (fun in_tys out_ty => do
+          let in_ty ← liftM (mkSigmasTypesOfTypes in_tys.toList)
+          let out_ty ← liftM (mkSigmasOutType in_tys.toList out_ty)
+          return (in_ty, out_ty)
+        )
+      )
+  trace[Diverge.def] "inOutTys: {inOutTys}"
+
+/-  -- Small utility: compute the list of type parameters
   let getTypeParams (ty: Expr) : MetaM (List Expr × List Expr × Expr) :=
     Lean.Meta.forallTelescope ty fun tys out_ty => do
       trace[Diverge.def] "types: {tys}"
@@ -138,16 +230,16 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let (input_tys, output_tys) := List.unzip all_tys.toList
   let input_tys : List Expr ← liftM (List.mapM mkProds input_tys)
 
-  trace[Diverge.def] "  in/out tys: {input_tys}; {output_tys}"
+  trace[Diverge.def] "  in/out tys: {input_tys}; {output_tys}" -/
 
   -- Compute the names set
   let names := preDefs.map PreDefinition.declName
   let names := HashSet.empty.insertMany names
 
   --
-  for preDef in preDefs do
-    trace[Diverge.def] "about to explore: {preDef.declName}"
-    explore_term "" preDef.value
+  -- for preDef in preDefs do
+  --  trace[Diverge.def] "about to explore: {preDef.declName}"
+  --  explore_term "" preDef.value
 
   -- Compute the bodies
 
@@ -267,6 +359,13 @@ elab_rules : command
     else
       Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
 
+divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+  match ls with
+  | [] => .fail .panic
+  | x :: ls =>
+    if i = 0 then return x
+    else return (← list_nth ls (i - 1))
+
 mutual
   divergent def is_even (i : Int) : Result Bool :=
     if i = 0 then return true else return (← is_odd (i - 1))
@@ -280,13 +379,6 @@ example (i : Int) : is_even i = .ret (i % 2 = 0) ∧ is_odd i = .ret (i % 2 ≠
   unfold is_even
   sorry    
 
-divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
-  match ls with
-  | [] => .fail .panic
-  | x :: ls =>
-    if i = 0 then return x
-    else return (← list_nth ls (i - 1))
-
 mutual
   divergent def foo (i : Int) : Result Nat :=
     if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index 84b73a30..441b25f0 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -5,6 +5,7 @@ namespace Diverge
 open Lean Elab Term Meta
 
 initialize registerTraceClass `Diverge.elab (inherited := true)
+initialize registerTraceClass `Diverge.def.sigmas (inherited := true)
 initialize registerTraceClass `Diverge.def (inherited := true)
 
 -- TODO: move
-- 
cgit v1.3.1


From 1c9331ce92b68b9a83c601212149a6c24591708f Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Fri, 30 Jun 2023 15:53:39 +0200
Subject: Generate the fixed-point bodies in Elab.lean

---
 backends/lean/Base/Diverge/Base.lean     |   8 +-
 backends/lean/Base/Diverge/Elab.lean     | 451 +++++++++++++++++++++++--------
 backends/lean/Base/Diverge/ElabBase.lean |  47 +++-
 3 files changed, 391 insertions(+), 115 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index 22b59bd0..aa0539ba 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -554,14 +554,14 @@ namespace FixI
   /- Some utilities to define the mutually recursive functions -/
 
   -- TODO: use more
-  @[simp] def kk_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
+  abbrev kk_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
     (i:id) → (x:a i) → Result (b i x)
-  @[simp] def k_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
+  abbrev k_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
     kk_ty id a b → kk_ty id a b
 
-  def in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
+  abbrev in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
   -- TODO: remove?
-  @[simp] def mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
+  abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
     in_out_ty :=
     Sigma.mk in_ty out_ty
 
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 116c5d8b..f7de7518 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -16,31 +16,62 @@ syntax (name := divergentDef)
 open Lean Elab Term Meta Primitives Lean.Meta
 
 set_option trace.Diverge.def true
+-- set_option trace.Diverge.def.sigmas true
 
 /- The following was copied from the `wfRecursion` function. -/
 
 open WF in
 
--- Replace the recursive calls by a call to the continuation
--- def replace_rec_calls
+def mkList (xl : List Expr) (ty : Expr) : MetaM Expr :=
+  match xl with
+  | [] =>
+    mkAppOptM ``List.nil #[some ty]
+  | x :: tl => do
+    let tl ← mkList tl ty
+    mkAppOptM ``List.cons #[some ty, some x, some tl]
 
---  print_decl is_even_body
-#check instOfNatNat
-#check OfNat.ofNat -- @OfNat.ofNat ℕ 2 ...
-#check OfNat.ofNat -- @OfNat.ofNat (Fin 2) 1 ...
-#check Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat
+def mkProd (x y : Expr) : MetaM Expr :=
+  mkAppM ``Prod.mk #[x, y]
 
+def mkInOutTy (x y : Expr) : MetaM Expr :=
+  mkAppM ``FixI.mk_in_out_ty #[x, y]
 
 -- TODO: is there already such a utility somewhere?
 -- TODO: change to mkSigmas
 def mkProds (tys : List Expr) : MetaM Expr :=
   match tys with
-  | [] => do return (Expr.const ``PUnit.unit [])
-  | [ty] => do return ty
+  | [] => do pure (Expr.const ``PUnit.unit [])
+  | [ty] => do pure ty
   | ty :: tys => do
     let pty ← mkProds tys
     mkAppM ``Prod.mk #[ty, pty]
 
+-- Return the `a` in `Return a`
+def get_result_ty (ty : Expr) : MetaM Expr :=
+  ty.withApp fun f args => do
+  if ¬ f.isConstOf ``Result ∨ args.size ≠ 1 then
+    throwError "Invalid argument to get_result_ty: {ty}"
+  else
+    pure (args.get! 0)
+
+-- Group a list of expressions into a dependent tuple
+def mkSigmas (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.sigmas] "mkSigmas: []"
+    pure (Expr.const ``PUnit.unit [])
+  | [x] => do
+    trace[Diverge.def.sigmas] "mkSigmas: [{x}]"
+    pure x
+  | fst :: xl => do
+    trace[Diverge.def.sigmas] "mkSigmas: [{fst}::{xl}]"
+    let alpha ← Lean.Meta.inferType fst
+    let snd ← mkSigmas xl
+    let snd_ty ← inferType snd
+    let beta ← mkLambdaFVars #[fst] snd_ty
+    trace[Diverge.def.sigmas] "mkSigmas:\n{alpha}\n{beta}\n{fst}\n{snd}"
+    mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
+
 /- Generate the input type of a function body, which is a sigma type (i.e., a
    dependent tuple) which groups all its inputs.
 
@@ -55,11 +86,11 @@ def mkSigmasTypesOfTypes (xl : List Expr) : MetaM Expr :=
   match xl with
   | [] => do
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
-    return (Expr.const ``PUnit.unit [])
+    pure (Expr.const ``PUnit.unit [])
   | [x] => do
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
     let ty ← Lean.Meta.inferType x
-    return ty
+    pure ty
   | x :: xl => do
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
     let alpha ← Lean.Meta.inferType x
@@ -71,15 +102,26 @@ def mkSigmasTypesOfTypes (xl : List Expr) : MetaM Expr :=
 
 def mk_indexed_name (index : Nat) : Name := .num (.str .anonymous "_uniq") index
 
-/- Generate the out_ty of the body of a function, which from an input (a sigma
-   type generated by `mkSigmasTypesOfTypes`) gives the output type of the function.
+/- Given a list of values `[x0:ty0, ..., xn:ty1]` where every `xi` might use the previous
+   `xj` (j < i) and a value `out` which uses `x0`, ..., `xn`, generate the following
+   expression:
+   ```
+   fun x:((x0:ty0) × ... × (xn:tyn) => -- **Dependent** tuple
+   match x with
+   | (x0, ..., xn) => out
+   ```
+   
+   The `index` parameter is used for naming purposes: we use it to numerotate the
+   bound variables that we introduce.
 
    Example:
+   ========
+   More precisely:
    - xl = `[a:Type, ls:List a, i:Int]`
-   - out_ty = `a`
-   - index = 0      -- For naming purposes: we use it to numerotate the "scrutinee" variables
+   - out = `a`
+   - index = 0
 
-   Generates:
+   generates:
    ```
    match scrut0 with
    | Sigma.mk x scrut1 =>
@@ -88,36 +130,47 @@ def mk_indexed_name (index : Nat) : Name := .num (.str .anonymous "_uniq") index
        a
    ```
 -/
-def mkSigmasOutType (xl : List Expr) (out_ty : Expr) (index : Nat := 0) : MetaM Expr :=
+partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr :=
   match xl with
   | [] => do
     -- This would be unexpected
-    throwError "mkSigmasOutType: empyt list of input parameters"
+    throwError "mkSigmasMatch: empyt list of input parameters"
   | [x] => do
     -- In the explanations above: inner match case
-    trace[Diverge.def.sigmas] "mkSigmasOutType: [{x}]"
-    mkLambdaFVars #[x] out_ty
+    trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]"
+    mkLambdaFVars #[x] out
   | fst :: xl => do
     -- In the explanations above: outer match case
     -- Remark: for the naming purposes, we use the same convention as for the
     -- fields and parameters in `Sigma.casesOn and `Sigma.mk
-    trace[Diverge.def.sigmas] "mkSigmasOutType: [{fst}::{xl}]"
+    trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]"
     let alpha ← Lean.Meta.inferType fst
     let snd_ty ← mkSigmasTypesOfTypes xl
     let beta ← mkLambdaFVars #[fst] snd_ty
-    let snd ← mkSigmasOutType xl out_ty (index + 1)
+    let snd ← mkSigmasMatch xl out (index + 1)
     let scrut_ty ← mkSigmasTypesOfTypes (fst :: xl)
     withLocalDeclD (mk_indexed_name index) scrut_ty fun scrut => do
     let mk ← mkLambdaFVars #[fst] snd
-    trace[Diverge.def.sigmas] "mkSigmasOutType: scrut: ({scrut}) : ({← inferType scrut})"
-    let motive ← mkLambdaFVars #[scrut] (← inferType out_ty)
-    trace[Diverge.def.sigmas] "mkSigmasOutType:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
-    let out ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
-    let out ← mkLambdaFVars #[scrut] out
-    trace[Diverge.def.sigmas] "mkSigmasOutType: out: {out}"
-    return out
-
-/- Small tests for list_nth: give a model of what `mkSigmasOutType` should generate -/
+    trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})"
+    -- TODO: make the computation of the motive more efficient
+    let motive ← do
+      let out_ty ← inferType out
+      match out_ty  with
+      | .sort _ | .lit _ | .const .. =>
+        -- The type of the motive doesn't depend on the scrutinee
+        mkLambdaFVars #[scrut] out_ty
+      | _ =>
+        -- The type of the motive *may* depend on the scrutinee
+        -- TODO: make this more efficient (we could change the output type of
+        -- mkSigmasMatch
+        mkSigmasMatch (fst :: xl) out_ty
+    trace[Diverge.def.sigmas] "mkSigmasMatch:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
+    let sm ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+    let sm ← mkLambdaFVars #[scrut] sm
+    trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}"
+    pure sm
+
+/- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/
 private def list_nth_out_ty2 (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
   @Sigma.casesOn (List a)
                  (fun (_ls : List a) => Int)
@@ -135,14 +188,199 @@ private def list_nth_out_ty1 (scrut0 : @Sigma (Type) (fun (a:Type) =>
                   list_nth_out_ty2 a scrut1)
 /- -/
 
+-- TODO: move
+-- TODO: we can use Array.mapIdx
+@[specialize] def mapiAux (i : Nat) (f : Nat → α → β) : List α → List β
+  | []    => []
+  | a::as => f i a :: mapiAux (i+1) f as
+
+@[specialize] def mapi (f : Nat → α → β) : List α → List β := mapiAux 0 f
+
+#check Array.map
+-- Return the expression: `Fin n`
+-- TODO: use more
+def mkFin (n : Nat) : Expr :=
+  mkAppN (.const ``Fin []) #[.lit (.natVal n)]
+
+-- Return the expression: `i : Fin n`
+def mkFinVal (n i : Nat) : MetaM Expr := do
+  let n_lit : Expr := .lit (.natVal (n - 1))
+  let i_lit : Expr := .lit (.natVal i)
+  -- We could use `trySynthInstance`, but as we know the instance that we are
+  -- going to use, we can save the lookup
+  let ofNat ← mkAppOptM ``Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat #[n_lit, i_lit]
+  mkAppOptM ``OfNat.ofNat #[none, none, ofNat]
+
+-- TODO: remove?
+def mkFinValOld (n i : Nat) : MetaM Expr := do
+  let finTy := mkFin n
+  let ofNat ← mkAppM ``OfNat #[finTy, .lit (.natVal i)]
+  match ← trySynthInstance ofNat with
+  | LOption.some x =>
+    mkAppOptM ``OfNat.ofNat #[none, none, x]
+  | _ => throwError "mkFinVal: could not synthesize an instance of {ofNat} "
+
+/- Generate and declare as individual definitions the bodies for the individual funcions:
+   - replace the recursive calls with calls to the continutation `k`
+   - make those bodies take one single dependent tuple as input
+
+   We name the declarations: "[original_name].body".
+   We return the new declarations.
+ -/
+def mkDeclareUnaryBodies (grLvlParams : List Name) (k_var : Expr)
+  (preDefs : Array PreDefinition) :
+  MetaM (Array Expr) := do
+  let grSize := preDefs.size
+
+  -- Compute the map from name to index - the continuation has an indexed type:
+  -- we use the index (a finite number of type `Fin`) to control the function
+  -- we call at the recursive call
+  let nameToId : HashMap Name Nat :=
+    let namesIds := mapi (fun i d => (d.declName, i)) preDefs.toList
+    HashMap.ofList namesIds
+
+  trace[Diverge.def.genBody] "nameToId: {nameToId.toList}"
+
+  -- Auxiliary function to explore the function bodies and replace the
+  -- recursive calls
+  let visit_e (e : Expr) : MetaM Expr := do
+    trace[Diverge.def.genBody] "visiting expression: {e}"
+    match e with
+    | .app .. => do
+      e.withApp fun f args => do
+        trace[Diverge.def.genBody] "this is an app: {f} {args}"
+        -- Check if this is a recursive call
+        if f.isConst then
+          let name := f.constName!
+          match nameToId.find? name with
+          | none => pure e
+          | some id =>
+            -- This is a recursive call: replace it
+            -- Compute the index
+            let i ← mkFinVal grSize id
+            -- Put the arguments in one big dependent tuple
+            let args ← mkSigmas args.toList
+            mkAppM' k_var #[i, args]
+        else
+          -- Not a recursive call: do nothing
+          pure e
+     | .const name _ =>
+       -- Sanity check: we eliminated all the recursive calls
+       if (nameToId.find? name).isSome then
+         throwError "mkUnaryBodies: a recursive call was not eliminated"
+       else pure e
+     | _ => pure e
+
+  -- Explore the bodies
+  preDefs.mapM fun preDef => do
+    -- Replace the recursive calls
+    let body ← mapVisit visit_e preDef.value
+
+    -- Change the type
+    lambdaLetTelescope body fun args body => do
+    let body ← mkSigmasMatch args.toList body 0
+
+    -- Add the declaration
+    let value ← mkLambdaFVars #[k_var] body
+    let name := preDef.declName.append "body"
+    let levelParams := grLvlParams
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := levelParams
+      type := ← inferType value -- TODO: change the type
+      value := value
+      hints := ReducibilityHints.regular (getMaxHeight (← getEnv) value + 1)
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+    trace[Diverge.def] "individual body of {preDef.declName}: {body}"
+    -- Return the constant
+    let body := Lean.mkConst name (levelParams.map .param)
+    -- let body ← mkAppM' body #[k_var]
+    trace[Diverge.def] "individual body (after decl): {body}"
+    pure body
+
+-- Generate a unique function body from the bodies of the mutually recursive group,
+-- and add it as a declaration in the context
+def mkDeclareMutualBody (grName : Name) (grLvlParams : List Name)
+  (i_var k_var : Expr)
+  (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
+  (bodies : Array Expr) : MetaM Expr := do
+  -- Generate the body
+  let grSize := bodies.size
+  let finTypeExpr := mkFin grSize
+  -- TODO: not very clean
+  let inOutTyType ← do
+    let (x, y) := inOutTys.get! 0
+    inferType (← mkInOutTy x y)
+  let rec mkFuns (inOutTys : List (Expr × Expr)) (bl : List Expr) : MetaM Expr :=
+    match inOutTys, bl with
+    | [], [] =>
+      mkAppOptM ``FixI.Funs.Nil #[finTypeExpr, in_ty, out_ty]
+    | (ity, oty) :: inOutTys, b :: bl => do
+      -- Retrieving ity and oty - this is not very clean
+      let inOutTysExpr ← mkList (← inOutTys.mapM (λ (x, y) => mkInOutTy x y)) inOutTyType
+      let fl ← mkFuns inOutTys bl
+      mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
+    | _, _ => throwError "mkDeclareMutualBody: `tys` and `bodies` don't have the same length"
+  let bodyFuns ← mkFuns inOutTys bodies.toList
+  -- Wrap in `get_fun`
+  let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, k_var]
+  -- Add the index `i` and the continuation `k` as a variables
+  let body ← mkLambdaFVars #[k_var, i_var] body
+  trace[Diverge.def] "mkDeclareMutualBody: body: {body}"
+  -- Add the declaration
+  let name := grName.append "mutrec_body"
+  let levelParams := grLvlParams
+  let decl := Declaration.defnDecl {
+    name := name
+    levelParams := levelParams
+    type := ← inferType body
+    value := body
+    hints := ReducibilityHints.regular (getMaxHeight (← getEnv) body + 1)
+    safety := .safe
+    all := [name]
+  }
+  addDecl decl
+  -- Return the constant
+  pure (Lean.mkConst name (levelParams.map .param))
+
+-- Generate the final definions by using the mutual body and the fixed point operator.
+def mkDeclareFixDefs (mutBody : Expr) (preDefs : Array PreDefinition) :
+  TermElabM Unit := do
+  let grSize := preDefs.size
+  let _ ← preDefs.mapIdxM fun idx preDef => do
+    lambdaLetTelescope preDef.value fun xs _ => do
+    -- Create the index
+    let idx ← mkFinVal grSize idx.val
+    -- Group the inputs into a dependent tuple
+    let input ← mkSigmas xs.toList
+    -- Apply the fixed point
+    let fixedBody ← mkAppM ``FixI.fix #[mutBody, idx, input]
+    let fixedBody ← mkLambdaFVars xs fixedBody
+    -- Create the declaration
+    let name := preDef.declName
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := preDef.levelParams
+      type := preDef.type
+      value := fixedBody
+      hints := ReducibilityHints.regular (getMaxHeight (← getEnv) fixedBody + 1)
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+  pure ()
+
 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
   trace[Diverge.def] ("divRecursion: defs: " ++ msg)
 
   -- CHANGE HERE This function should add definitions with these names/types/values ^^
   -- Temporarily add the predefinitions as axioms
-  for preDef in preDefs do
-    addAsAxiom preDef
+  -- for preDef in preDefs do
+  --  addAsAxiom preDef
 
   -- TODO: what is this?
   for preDef in preDefs do
@@ -154,25 +392,14 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let grName := def0.declName
   trace[Diverge.def] "group name: {grName}"
 
-  /- Compute the type of the continuation.
-
-     We do the following
-     - we make sure all the definitions have the same universe parameters
-       (we can make this more general later)
-     - we group all the type parameters together, make sure all the
-       definitions have the same type parameters, and enforce
-       a uniform polymorphism (we can also lift this later).
-       This would require generalizing a bit our indexed fixed point to
-       make the output type parametric in the input.
-     - we group all the non-type parameters: we parameterize the continuation
-       by those
-  -/
+  /- # Compute the input/output types of the continuation `k`. -/
   let grLvlParams := def0.levelParams
-  trace[Diverge.def] "def0 type: {def0.type}"
+  trace[Diverge.def] "def0 universe levels: {def0.levelParams}"
 
-  -- Compute the list of pairs: (input type × output type)
+  -- We first compute the list of pairs: (input type × output type)
   let inOutTys : Array (Expr × Expr) ←
       preDefs.mapM (fun preDef => do
+        withRef preDef.ref do -- is the withRef useful?
         -- Check the universe parameters - TODO: I'm not sure what the best thing
         -- to do is. In practice, all the type parameters should be in Type 0, so
         -- we shouldn't have universe issues.
@@ -180,68 +407,74 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
           throwError "Non-uniform polymorphism in the universes"
         forallTelescope preDef.type (fun in_tys out_ty => do
           let in_ty ← liftM (mkSigmasTypesOfTypes in_tys.toList)
-          let out_ty ← liftM (mkSigmasOutType in_tys.toList out_ty)
-          return (in_ty, out_ty)
+          -- Retrieve the type in the "Result"
+          let out_ty ← get_result_ty out_ty
+          let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
+          pure (in_ty, out_ty)
         )
       )
   trace[Diverge.def] "inOutTys: {inOutTys}"
-
-/-  -- Small utility: compute the list of type parameters
-  let getTypeParams (ty: Expr) : MetaM (List Expr × List Expr × Expr) :=
-    Lean.Meta.forallTelescope ty fun tys out_ty => do
-      trace[Diverge.def] "types: {tys}"
-/-      let (_, params) ← StateT.run (do
-        for x in tys do
-          let ty ← Lean.Meta.inferType x
-          match ty with
-          | .sort _ => do
-            let st ← StateT.get
-            StateT.set (ty :: st)
-          | _ => do break
-        ) ([] : List Expr)
-      let params := params.reverse
-      trace[Diverge.def] "  type parameters {params}"
-      return params -/
-      let rec get_params (ls : List Expr) : MetaM (List Expr × List Expr) :=
-        match ls with
-        | x :: tl => do
-          let ty ← Lean.Meta.inferType x
-          match ty with
-          | .sort _ => do
-            let (ty_params, params) ← get_params tl
-            return (x :: ty_params, params)
-          | _ => do return ([], ls)
-        | _ => do return ([], [])
-      let (ty_params, params) ← get_params tys.toList
-      trace[Diverge.def] "  parameters: {ty_params}; {params}"
-      return (ty_params, params, out_ty)
-  let (grTyParams, _, _) ← do
-    getTypeParams def0.type
-
-  -- Compute the input types and the output types
-  let all_tys ← preDefs.mapM fun preDef => do
-    let (tyParams, params, ret_ty) ← getTypeParams preDef.type
-    -- TODO: this is not complete, there are more checks to perform
-    if tyParams.length ≠ grTyParams.length then
-      throwError "Non-uniform polymorphism"
-    return (params, ret_ty)
-
-  -- TODO: I think there are issues with the free variables
-  let (input_tys, output_tys) := List.unzip all_tys.toList
-  let input_tys : List Expr ← liftM (List.mapM mkProds input_tys)
-
-  trace[Diverge.def] "  in/out tys: {input_tys}; {output_tys}" -/
-
-  -- Compute the names set
-  let names := preDefs.map PreDefinition.declName
-  let names := HashSet.empty.insertMany names
-
-  --
-  -- for preDef in preDefs do
-  --  trace[Diverge.def] "about to explore: {preDef.declName}"
-  --  explore_term "" preDef.value
-
-  -- Compute the bodies
+  -- Turn the list of input/output type pairs into an expresion
+  let inOutTysExpr ← inOutTys.mapM (λ (x, y) => mkInOutTy x y)
+  let inOutTysExpr ← mkList inOutTysExpr.toList (← inferType (inOutTysExpr.get! 0))
+
+  -- From the list of pairs of input/output types, actually compute the
+  -- type of the continuation `k`.
+  -- We first introduce the index `i : Fin n` where `n` is the number of
+  -- functions in the group.
+  let i_var_ty := mkFin preDefs.size
+  withLocalDeclD (.num (.str .anonymous "i") 0) i_var_ty fun i_var => do
+  let in_out_ty ← mkAppM ``List.get #[inOutTysExpr, i_var]
+  trace[Diverge.def] "in_out_ty := {in_out_ty} : {← inferType in_out_ty}"
+  -- Add an auxiliary definition for `in_out_ty`
+  let in_out_ty ← do
+    let value ← mkLambdaFVars #[i_var] in_out_ty
+    let name := grName.append "in_out_ty"
+    let levelParams := grLvlParams
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := levelParams
+      type := ← inferType value
+      value := value
+      hints := .abbrev
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+    -- Return the constant
+    let in_out_ty := Lean.mkConst name (levelParams.map .param)
+    mkAppM' in_out_ty #[i_var]
+  trace[Diverge.def] "in_out_ty (after decl) := {in_out_ty} : {← inferType in_out_ty}"
+  let in_ty ← mkAppM ``Sigma.fst #[in_out_ty]
+  trace[Diverge.def] "in_ty: {in_ty}"
+  withLocalDeclD (.num (.str .anonymous "x") 1) in_ty fun input => do
+  let out_ty ← mkAppM' (← mkAppM ``Sigma.snd #[in_out_ty]) #[input]
+  trace[Diverge.def] "out_ty: {out_ty}"
+
+  -- Introduce the continuation `k`
+  let in_ty ← mkLambdaFVars #[i_var] in_ty
+  let out_ty ← mkLambdaFVars #[i_var, input] out_ty
+  let k_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty] --
+  trace[Diverge.def] "k_var_ty: {k_var_ty}"
+  withLocalDeclD (.num (.str .anonymous "k") 2) k_var_ty fun k_var => do
+  trace[Diverge.def] "k_var: {k_var}"
+
+  -- Replace the recursive calls in all the function bodies by calls to the
+  -- continuation `k` and and generate for those bodies declarations
+  let bodies ← mkDeclareUnaryBodies grLvlParams k_var preDefs
+  -- Generate the mutually recursive body
+  let body ← mkDeclareMutualBody grName grLvlParams i_var k_var in_ty out_ty inOutTys.toList bodies
+  trace[Diverge.def] "mut rec body (after decl): {body}"
+
+  -- Prove that the mut rec body satisfies the validity criteria required by
+  -- our fixed-point
+  -- TODO
+
+  -- Generate the final definitions
+  let defs ← mkDeclareFixDefs body preDefs
+
+  -- Prove the unfolding equations
+  -- TODO
 
   -- Process the definitions
   addAndCompilePartialRec preDefs
@@ -366,6 +599,10 @@ divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
     if i = 0 then return x
     else return (← list_nth ls (i - 1))
 
+#print list_nth.in_out_ty
+#check list_nth.body
+#print list_nth
+
 mutual
   divergent def is_even (i : Int) : Result Bool :=
     if i = 0 then return true else return (← is_odd (i - 1))
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index 441b25f0..82f79f94 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -4,13 +4,14 @@ namespace Diverge
 
 open Lean Elab Term Meta
 
-initialize registerTraceClass `Diverge.elab (inherited := true)
-initialize registerTraceClass `Diverge.def.sigmas (inherited := true)
-initialize registerTraceClass `Diverge.def (inherited := true)
+initialize registerTraceClass `Diverge.elab
+initialize registerTraceClass `Diverge.def
+initialize registerTraceClass `Diverge.def.sigmas
+initialize registerTraceClass `Diverge.def.genBody
 
 -- TODO: move
 -- TODO: small helper
-def explore_term (incr : String) (e : Expr) : TermElabM Unit :=
+def explore_term (incr : String) (e : Expr) : MetaM Unit :=
   match e with
   | .bvar _ => do logInfo m!"{incr}bvar: {e}"; return ()
   | .fvar _ => do logInfo m!"{incr}fvar: {e}"; return ()
@@ -78,4 +79,42 @@ private def test2 (x : Nat) : Nat := x
 print_decl test1
 print_decl test2
 
+-- We adapted this from AbstractNestedProofs.visit
+-- A map visitor function for expressions
+partial def mapVisit (k : Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+  let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
+    let localInstances ← getLocalInstances
+    let mut lctx ← getLCtx
+    for x in xs do
+      let xFVarId := x.fvarId!
+      let localDecl ← xFVarId.getDecl
+      let type      ← mapVisit k localDecl.type
+      let localDecl := localDecl.setType type
+      let localDecl ← match localDecl.value? with
+         | some value => let value ← mapVisit k value; pure <| localDecl.setValue value
+         | none       => pure localDecl
+      lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
+    withLCtx lctx localInstances k2
+  -- TODO: use a cache? (Lean.checkCache)
+  -- Explore
+  let e ← k e
+  match e with
+  | .bvar _
+  | .fvar _
+  | .mvar _
+  | .sort _
+  | .lit _
+  | .const _ _ => pure e
+  | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (mapVisit k))
+  | .lam .. =>
+    lambdaLetTelescope e fun xs b =>
+      mapVisitBinders xs do mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
+  | .forallE .. => do
+    forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← mapVisit k b)
+  | .letE .. => do
+    lambdaLetTelescope e fun xs b => mapVisitBinders xs do
+      mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
+  | .mdata _ b => return e.updateMData! (← mapVisit k b)
+  | .proj _ _ b => return e.updateProj! (← mapVisit k b)
+
 end Diverge
-- 
cgit v1.3.1


From 37e5d5501e024869037bf0ea1559229a8be62da7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 3 Jul 2023 16:24:44 +0200
Subject: Generate the proofs of validity in Elab.lean

---
 backends/lean/Base/Diverge/Base.lean     |  76 +++++-
 backends/lean/Base/Diverge/Elab.lean     | 403 ++++++++++++++++++++++++++++---
 backends/lean/Base/Diverge/ElabBase.lean |   1 +
 3 files changed, 446 insertions(+), 34 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index aa0539ba..89365d25 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -434,6 +434,23 @@ namespace Fix
     is_valid_p k (λ k => k x) := by
     simp_all [is_valid_p, is_mono_p_rec, is_cont_p_rec]
 
+  theorem is_valid_p_ite
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 e2 : ((x:a) → Result (b x)) → Result c}
+    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+    is_valid_p k (ite cond e1 e2) := by
+    split <;> assumption
+
+  theorem is_valid_p_dite
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 : cond → ((x:a) → Result (b x)) → Result c}
+    {e2 : Not cond → ((x:a) → Result (b x)) → Result c}
+    (he1: ∀ x, is_valid_p k (e1 x)) (he2 : ∀ x, is_valid_p k (e2 x)) :
+    is_valid_p k (dite cond e1 e2) := by
+    split <;> simp [*]
+
   -- Lean is good at unification: we can write a very general version
   -- (in particular, it will manage to figure out `g` and `h` when we
   -- apply the lemma)
@@ -680,6 +697,24 @@ namespace FixI
     is_valid_p k (λ k => k i x) := by
     simp [is_valid_p, k_to_gen, e_to_gen, kk_to_gen, kk_of_gen]
 
+  theorem is_valid_p_ite
+    (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 e2 : ((i:id) → (x:a i) → Result (b i x)) → Result c}
+    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+    is_valid_p k (λ k => ite cond (e1 k) (e2 k)) := by
+    split <;> assumption
+
+  theorem is_valid_p_dite
+    (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 : ((i:id) → (x:a i) → Result (b i x)) → cond → Result c}
+    {e2 : ((i:id) → (x:a i) → Result (b i x)) → Not cond → Result c}
+    (he1: ∀ x, is_valid_p k (λ k => e1 k x))
+    (he2 : ∀ x, is_valid_p k (λ k => e2 k x)) :
+    is_valid_p k (λ k => dite cond (e1 k) (e2 k)) := by
+    split <;> simp [*]
+
   theorem is_valid_p_bind
     {{k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
     {{g : ((i:id) → (x:a i) → Result (b i x)) → Result c}}
@@ -699,6 +734,9 @@ namespace FixI
     | .Nil => True
     | .Cons f fl => (∀ x, FixI.is_valid_p k (λ k => f k x)) ∧ fl.is_valid_p k
 
+  theorem Funs.is_valid_p_Nil (k : k_ty id a b) :
+    Funs.is_valid_p k Funs.Nil := by simp [Funs.is_valid_p]
+
   def Funs.is_valid_p_is_valid_p_aux
     {k : k_ty id a b}
     {tys : List in_out_ty}
@@ -1116,7 +1154,7 @@ namespace Ex6
   def body (k : (i : Fin 1) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 1) :
     (x : input_ty i) → Result (output_ty i x) := get_fun bodies i k
 
-  theorem list_nth_body_is_valid: is_valid body := by
+  theorem body_is_valid: is_valid body := by
     -- Split the proof into proofs of validity of the individual bodies
     rw [is_valid]
     simp only [body]
@@ -1131,6 +1169,20 @@ namespace Ex6
     split <;> simp
     split <;> simp
 
+  -- Writing the proof terms explicitly
+  theorem list_nth_body_is_valid' (k : k_ty (Fin 1) input_ty output_ty)
+    (x : (a : Type u) × List a × Int) : is_valid_p k (fun k => list_nth_body k x) :=
+    let ⟨ a, ls, i ⟩ := x
+    match ls with
+    | [] => is_valid_p_same k (.fail .panic)
+    | hd :: tl =>
+      is_valid_p_ite k (Eq i 0) (is_valid_p_same k (.ret hd)) (is_valid_p_rec k 0 ⟨a, tl, i-1⟩)
+
+  theorem body_is_valid' : is_valid body :=
+    fun k =>
+    Funs.is_valid_p_is_valid_p tys k bodies
+      (And.intro (list_nth_body_is_valid' k) (Funs.is_valid_p_Nil k))
+
   noncomputable
   def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
     fix body 0 ⟨ a, ls , i ⟩
@@ -1144,8 +1196,28 @@ namespace Ex6
         if i = 0 then .ret hd
         else list_nth tl (i - 1)
     := by
-    have Heq := is_valid_fix_fixed_eq list_nth_body_is_valid
+    have Heq := is_valid_fix_fixed_eq body_is_valid
     simp [list_nth]
     conv => lhs; rw [Heq]
 
+  -- Write the proof term explicitly: the generation of the proof term (without tactics)
+  -- is automatable, and the proof term is actually a lot simpler and smaller when we
+  -- don't use tactics.
+  theorem list_nth_eq'.{u} {a : Type u} (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    :=
+    -- Use the fixed-point equation
+    have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+    -- Add the index
+    have Heqi := congr_fun Heq 0
+    -- Add the input
+    have Heqix := congr_fun Heqi { fst := a, snd := (ls, i) }
+    -- Done
+    Heqix
+
 end Ex6
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index f7de7518..cf40ea8f 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -16,6 +16,7 @@ syntax (name := divergentDef)
 open Lean Elab Term Meta Primitives Lean.Meta
 
 set_option trace.Diverge.def true
+set_option trace.Diverge.def.valid true
 -- set_option trace.Diverge.def.sigmas true
 
 /- The following was copied from the `wfRecursion` function. -/
@@ -196,7 +197,6 @@ private def list_nth_out_ty1 (scrut0 : @Sigma (Type) (fun (a:Type) =>
 
 @[specialize] def mapi (f : Nat → α → β) : List α → List β := mapiAux 0 f
 
-#check Array.map
 -- Return the expression: `Fin n`
 -- TODO: use more
 def mkFin (n : Nat) : Expr :=
@@ -227,7 +227,7 @@ def mkFinValOld (n i : Nat) : MetaM Expr := do
    We name the declarations: "[original_name].body".
    We return the new declarations.
  -/
-def mkDeclareUnaryBodies (grLvlParams : List Name) (k_var : Expr)
+def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
   (preDefs : Array PreDefinition) :
   MetaM (Array Expr) := do
   let grSize := preDefs.size
@@ -260,7 +260,7 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (k_var : Expr)
             let i ← mkFinVal grSize id
             -- Put the arguments in one big dependent tuple
             let args ← mkSigmas args.toList
-            mkAppM' k_var #[i, args]
+            mkAppM' kk_var #[i, args]
         else
           -- Not a recursive call: do nothing
           pure e
@@ -281,8 +281,8 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (k_var : Expr)
     let body ← mkSigmasMatch args.toList body 0
 
     -- Add the declaration
-    let value ← mkLambdaFVars #[k_var] body
-    let name := preDef.declName.append "body"
+    let value ← mkLambdaFVars #[kk_var] body
+    let name := preDef.declName.append "sbody"
     let levelParams := grLvlParams
     let decl := Declaration.defnDecl {
       name := name
@@ -297,16 +297,17 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (k_var : Expr)
     trace[Diverge.def] "individual body of {preDef.declName}: {body}"
     -- Return the constant
     let body := Lean.mkConst name (levelParams.map .param)
-    -- let body ← mkAppM' body #[k_var]
+    -- let body ← mkAppM' body #[kk_var]
     trace[Diverge.def] "individual body (after decl): {body}"
     pure body
 
 -- Generate a unique function body from the bodies of the mutually recursive group,
--- and add it as a declaration in the context
-def mkDeclareMutualBody (grName : Name) (grLvlParams : List Name)
-  (i_var k_var : Expr)
+-- and add it as a declaration in the context.
+-- We return the list of bodies (of type `Funs ...`) and the mutually recursive body.
+def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
+  (kk_var i_var : Expr)
   (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
-  (bodies : Array Expr) : MetaM Expr := do
+  (bodies : Array Expr) : MetaM (Expr × Expr) := do
   -- Generate the body
   let grSize := bodies.size
   let finTypeExpr := mkFin grSize
@@ -323,15 +324,15 @@ def mkDeclareMutualBody (grName : Name) (grLvlParams : List Name)
       let inOutTysExpr ← mkList (← inOutTys.mapM (λ (x, y) => mkInOutTy x y)) inOutTyType
       let fl ← mkFuns inOutTys bl
       mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
-    | _, _ => throwError "mkDeclareMutualBody: `tys` and `bodies` don't have the same length"
+    | _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length"
   let bodyFuns ← mkFuns inOutTys bodies.toList
   -- Wrap in `get_fun`
-  let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, k_var]
+  let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, kk_var]
   -- Add the index `i` and the continuation `k` as a variables
-  let body ← mkLambdaFVars #[k_var, i_var] body
-  trace[Diverge.def] "mkDeclareMutualBody: body: {body}"
+  let body ← mkLambdaFVars #[kk_var, i_var] body
+  trace[Diverge.def] "mkDeclareMutRecBody: body: {body}"
   -- Add the declaration
-  let name := grName.append "mutrec_body"
+  let name := grName.append "mut_rec_body"
   let levelParams := grLvlParams
   let decl := Declaration.defnDecl {
     name := name
@@ -344,10 +345,348 @@ def mkDeclareMutualBody (grName : Name) (grLvlParams : List Name)
   }
   addDecl decl
   -- Return the constant
-  pure (Lean.mkConst name (levelParams.map .param))
+  pure (bodyFuns, Lean.mkConst name (levelParams.map .param))
+
+def isCasesExpr (e : Expr) : MetaM Bool := do
+  let e := e.getAppFn
+  if e.isConst then
+    return isCasesOnRecursor (← getEnv) e.constName
+  else return false
+
+structure MatchInfo where
+  matcherName       : Name
+  matcherLevels     : Array Level
+  params            : Array Expr
+  motive            : Expr
+  scruts            : Array Expr
+  branchesNumParams : Array Nat
+  branches          : Array Expr
+
+instance : ToMessageData MatchInfo where
+  -- This is not a very clean formatting, but we don't need more
+  toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}"
+
+-- An expression which doesn't use the continuation kk is valid
+def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveNoKExprIsValid: {e}"
+  let eIsValid ← mkAppM ``FixI.is_valid_p_same #[k_var, e]
+  trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}"
+  pure eIsValid
+
+mutual
+
+partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveValid: {e}"
+  match e with
+  | .bvar _
+  | .fvar _
+  | .mvar _
+  | .sort _
+  | .lit _
+  | .const _ _ => throwError "Unimplemented"
+  | .lam .. => throwError "Unimplemented"
+  | .forallE .. => throwError "Unreachable" -- Shouldn't get there
+  | .letE .. => throwError "TODO"
+    -- lambdaLetTelescope e fun xs b => mapVisitBinders xs do
+    --  mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
+  | .mdata _ b => proveExprIsValid k_var kk_var b
+  | .proj _ _ _ =>
+    -- The projection shouldn't use the continuation
+    proveNoKExprIsValid k_var e
+  | .app .. =>
+    e.withApp fun f args => do
+      -- There are several cases: first, check if this is a match/if
+      -- The expression is a (dependent) if then else
+      let isIte := e.isIte
+      if isIte || e.isDIte then do
+        e.withApp fun f args => do
+        trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
+        if args.size ≠ 5 then
+           throwError "Wrong number of parameters for {f}: {args}"
+        let cond := args.get! 1
+        let dec := args.get! 2
+        -- Prove that the branches are valid
+        let br0 := args.get! 3
+        let br1 := args.get! 4
+        let proveBranchValid (br : Expr) : MetaM Expr :=
+          if isIte then proveExprIsValid k_var kk_var br
+          else do
+            -- There is a lambda -- TODO: how do we remove exacly *one* lambda?
+            lambdaLetTelescope br fun xs br => do
+            let x := xs.get! 0
+            let xs := xs.extract 1 xs.size
+            let br ← mkLambdaFVars xs br
+            let brValid ← proveExprIsValid k_var kk_var br
+            mkLambdaFVars #[x] brValid
+        let br0Valid ← proveBranchValid br0
+        let br1Valid ← proveBranchValid br1
+        let const := if isIte then ``FixI.is_valid_p_ite else ``FixI.is_valid_p_dite
+        let eIsValid ← mkAppOptM const #[none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid]
+        trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
+        pure eIsValid
+      -- The expression is a match (this case is for when the elaborator
+      -- introduces auxiliary definitions to hide the match behind syntactic
+      -- sugar)
+      else if let some me := ← matchMatcherApp? e then do
+        trace[Diverge.def.valid]
+          "matcherApp:
+           - params: {me.params}
+           - motive: {me.motive}
+           - discrs: {me.discrs}
+           - altNumParams: {me.altNumParams}
+           - alts: {me.alts}
+           - remaining: {me.remaining}"
+        -- matchMatcherApp has already done the work for us
+        if me.remaining.size ≠ 0 then
+          throwError "MatcherApp: non empty remaining array: {me.remaining}"
+        let me : MatchInfo := {
+          matcherName := me.matcherName
+          matcherLevels := me.matcherLevels
+          params := me.params
+          motive := me.motive
+          scruts := me.discrs
+          branchesNumParams := me.altNumParams
+          branches := me.alts
+        }
+        proveMatchIsValid k_var kk_var me
+      -- The expression is a raw match (this case is for when the expression
+      -- is a direct call to the primitive `casesOn` function, without
+      -- syntactic sugar)
+      else if ← isCasesExpr f then do
+        trace[Diverge.def.valid] "rawMatch: {e}"
+        -- The casesOn definition is always of the following shape:
+        -- input parameters (implicit parameters), then motive (implicit), 
+        -- scrutinee (explicit), branches (explicit).
+        let matcherName := f.constName!
+        let matcherLevels := f.constLevels!.toArray
+        -- Find the first explicit parameter: this is the scrutinee
+        forallTelescope (← inferType f) fun xs _ => do
+        let rec findFirstExplicit (i : Nat) : MetaM Nat := do
+          if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
+          else
+            let x := xs.get! i
+            let xFVarId := x.fvarId!
+            let localDecl ← xFVarId.getDecl
+            match localDecl.binderInfo with
+            | .default => pure i
+            | _ => findFirstExplicit (i + 1)
+        let scrutIdx ← findFirstExplicit 0
+        -- Split the arguments
+        let params := args.extract 0 (scrutIdx - 1)
+        let motive := args.get! (scrutIdx - 1)
+        let scrut := args.get! scrutIdx
+        let branches := args.extract (scrutIdx + 1) args.size
+        -- Compute the number of parameters for the branches: for this we use
+        -- the type of the uninstantiated casesOn constant
+        let branchesNumParams : Array Nat ← do
+          let env ← getEnv
+          let decl := env.constants.find! matcherName
+          let ty := decl.type
+          forallTelescope ty fun xs _ => do
+          let xs := xs.extract (scrutIdx + 1) xs.size
+          xs.mapM fun x => do
+          let xty ← inferType x
+          forallTelescope xty fun ys _ => do
+          pure ys.size
+        let me : MatchInfo := {
+          matcherName,
+          matcherLevels,
+          params,
+          motive,
+          scruts := #[scrut],
+          branchesNumParams,
+          branches,
+        }
+        proveMatchIsValid k_var kk_var me
+      -- Monadic let-binding
+      else if f.isConstOf ``Bind.bind then do
+        trace[Diverge.def.valid] "bind:\n{args}"
+        let x := args.get! 4
+        let y := args.get! 5
+        -- Prove that the subexpressions are valid
+        let xValid ← proveExprIsValid k_var kk_var x
+        trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
+        let yValid ← do
+          -- This is a lambda expression -- TODO: how do we remove exacly *one* lambda?
+          lambdaLetTelescope y fun xs y => do
+          let x := xs.get! 0
+          let xs := xs.extract 1 xs.size
+          let y ← mkLambdaFVars xs y
+          trace[Diverge.def.valid] "bind: y: {y}"
+          let yValid ← proveExprIsValid k_var kk_var y
+          trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
+          trace[Diverge.def.valid] "bind: yValid: x: {x}"
+          let yValid ← mkLambdaFVars #[x] yValid
+          trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
+          pure yValid
+        -- Put everything together
+        trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
+        mkAppM ``FixI.is_valid_p_bind #[xValid, yValid]
+      -- Recursive call
+      else if f.isFVarOf kk_var.fvarId! then do
+        trace[Diverge.def.valid] "rec: args: \n{args}"
+        if args.size ≠ 2 then throwError "Recursive call with invalid number of parameters: {args}"
+        let i_arg := args.get! 0
+        let x_arg := args.get! 1
+        let eIsValid ← mkAppM ``FixI.is_valid_p_rec #[k_var, i_arg, x_arg]
+        trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
+        pure eIsValid
+      else do
+        -- Remaining case: normal application.
+        -- It shouldn't use the continuation
+        proveNoKExprIsValid k_var e
+
+partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveMatchIsValid: {me}"
+  -- Prove the validity of the branch expressions
+  let branchesValid:Array Expr ← me.branches.mapIdxM fun idx br => do
+    -- Go inside the lambdas - note that we have to be careful: some of the
+    -- binders might come from the match, and some of the binders might come
+    -- from the fact that the expression in the match is a lambda expression:
+    -- we use the branchesNumParams field for this reason
+    lambdaLetTelescope br fun xs br => do
+    let numParams := me.branchesNumParams.get! idx
+    let xs_beg := xs.extract 0 numParams
+    let xs_end := xs.extract numParams xs.size
+    let br ← mkLambdaFVars xs_end br
+    -- Prove that the branch expression is valid
+    let brValid ← proveExprIsValid k_var kk_var br
+    -- Reconstruct the lambda expression
+    mkLambdaFVars xs_beg brValid
+  trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
+  -- Put together: compute the motive.
+  -- It must be of the shape:
+  -- ```
+  -- λ scrut => is_valid_p k (λ k => match scrut with ...)
+  -- ```
+  let validMotive : Expr ← do
+    -- The motive is a function of the scrutinees (i.e., a lambda expression):
+    -- introduce binders for the scrutinees
+    let declInfos := me.scruts.mapIdx fun idx scrut =>
+      let name : Name :=  (.num (.str .anonymous "scrut") idx)
+      let ty := λ (_ : Array Expr) => inferType scrut
+      (name, ty)
+    withLocalDeclsD declInfos fun scrutVars => do
+    -- Create a match expression but where the scrutinees have been replaced
+    -- by variables
+    let params : Array (Option Expr) := me.params.map some
+    let motive : Option Expr := some me.motive
+    let scruts : Array (Option Expr) := scrutVars.map some
+    let branches : Array (Option Expr) := me.branches.map some
+    let args := params ++ [motive] ++ scruts ++ branches
+    let matchE ← mkAppOptM me.matcherName args
+    -- let matchE ← mkLambdaFVars scrutVars (← mkAppOptM me.matcherName args)
+    -- Wrap in the `is_valid_p` predicate
+    let matchE ← mkLambdaFVars #[kk_var] matchE
+    let validMotive ← mkAppM ``FixI.is_valid_p #[k_var, matchE]
+    -- Abstract away the scrutinee variables
+    mkLambdaFVars scrutVars validMotive
+  trace[Diverge.def.valid] "valid motive: {validMotive}"
+  -- Put together
+  let valid ← do
+    let params : Array (Option Expr) := me.params.map (λ _ => none)
+    let motive := some validMotive
+    let scruts := me.scruts.map some
+    let branches := branchesValid.map some
+    let args := params ++ [motive] ++ scruts ++ branches
+    mkAppOptM me.matcherName args
+  trace[Diverge.def.valid] "proveMatchIsValid:\n{valid}:\n{← inferType valid}"
+  pure valid
+
+end
+
+-- Prove that a single body (in the mutually recursive group) is valid
+partial def proveSingleBodyIsValid (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
+  MetaM Expr := do
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: bodyConst: {bodyConst}"
+  -- Lookup the definition (`bodyConst` is the definition of the body, we want
+  -- to retrieve the value itself to dive inside)
+  let name := bodyConst.constName!
+  let env ← getEnv
+  let body := (env.constants.find! name).value!
+  trace[Diverge.def.valid] "body: {body}"
+  lambdaLetTelescope body fun xs body => do
+  assert! xs.size = 2
+  let kk_var := xs.get! 0
+  let x_var := xs.get! 1
+  -- State the type of the theorem to prove
+  let thmTy ← mkAppM ``FixI.is_valid_p
+    #[k_var, ← mkLambdaFVars #[kk_var] (← mkAppM' bodyConst #[kk_var, x_var])]
+  trace[Diverge.def.valid] "thmTy: {thmTy}"
+  -- Prove that the body is valid
+  let proof ← proveExprIsValid k_var kk_var body
+  let proof ← mkLambdaFVars #[k_var, x_var] proof
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: proof:\n{proof}:\n{← inferType proof}"
+  -- The target type (we don't have to do this: this is simply a sanity check,
+  -- and this allows a nicer debugging output)
+  let thmTy ← do
+    let body ← mkAppM' bodyConst #[kk_var, x_var]
+    let body ← mkLambdaFVars #[kk_var] body
+    let ty ← mkAppM ``FixI.is_valid_p #[k_var, body]
+    mkForallFVars #[k_var, x_var] ty
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}"
+  -- Save the theorem
+  let name := preDef.declName ++ "sbody_is_valid"
+  let decl := Declaration.thmDecl {
+    name
+    levelParams := preDef.levelParams
+    type := thmTy
+    value := proof
+    all := [name]
+  }
+  addDecl decl
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: added thm: {name}"
+  -- Return the theorem
+  pure (Expr.const name (preDef.levelParams.map .param))
+
+partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
+  (k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do
+  -- Create the big "and" expression, which groups the validity proof of the individual bodies
+  let rec mkValidConj (i : Nat) : MetaM Expr := do
+    if i = bodiesValid.size then
+      -- We reached the end
+      mkAppM ``FixI.Funs.is_valid_p_Nil #[k_var]
+    else do
+      -- We haven't reached the end: introduce a conjunction
+      let valid := bodiesValid.get! i
+      let valid ← mkAppM' valid #[k_var]
+      mkAppM ``And.intro #[valid, ← mkValidConj (i + 1)]
+  let andExpr ← mkValidConj 0
+  -- Wrap in the `is_valid_p_is_valid_p` theorem, and abstract the continuation
+  let isValid ← mkAppM ``FixI.Funs.is_valid_p_is_valid_p #[inOutTys, k_var, bodyFuns, andExpr]
+  mkLambdaFVars #[k_var] isValid
+
+-- Prove that the mut rec body is valid
+-- TODO: maybe this function should introduce k_var itself
+def proveMutRecIsValid
+  (grName : Name) (grLvlParams : List Name)
+  (inOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
+  (k_var : Expr) (preDefs : Array PreDefinition)
+  (bodies : Array Expr) : MetaM Expr := do
+  -- First prove that the individual bodies are valid
+  let bodiesValid ←
+    bodies.mapIdxM fun idx body => do
+      let preDef := preDefs.get! idx
+      proveSingleBodyIsValid k_var preDef body
+  -- Then prove that the mut rec body is valid
+  let isValid ← proveFunsBodyIsValid inOutTys bodyFuns k_var bodiesValid
+  -- Save the theorem
+  let thmTy ← mkAppM ``FixI.is_valid #[mutRecBodyConst]
+  let name := grName ++ "mut_rec_body_is_valid"
+  let decl := Declaration.thmDecl {
+    name
+    levelParams := grLvlParams
+    type := thmTy
+    value := isValid
+    all := [name]
+  }
+  addDecl decl
+  trace[Diverge.def.valid] "proveFunsBodyIsValid: added thm: {name}:\n{thmTy}"
+  -- Return the theorem
+  pure (Expr.const name (grLvlParams.map .param))
 
 -- Generate the final definions by using the mutual body and the fixed point operator.
-def mkDeclareFixDefs (mutBody : Expr) (preDefs : Array PreDefinition) :
+def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
   TermElabM Unit := do
   let grSize := preDefs.size
   let _ ← preDefs.mapIdxM fun idx preDef => do
@@ -357,7 +696,7 @@ def mkDeclareFixDefs (mutBody : Expr) (preDefs : Array PreDefinition) :
     -- Group the inputs into a dependent tuple
     let input ← mkSigmas xs.toList
     -- Apply the fixed point
-    let fixedBody ← mkAppM ``FixI.fix #[mutBody, idx, input]
+    let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
     let fixedBody ← mkLambdaFVars xs fixedBody
     -- Create the declaration
     let name := preDef.declName
@@ -454,24 +793,26 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   -- Introduce the continuation `k`
   let in_ty ← mkLambdaFVars #[i_var] in_ty
   let out_ty ← mkLambdaFVars #[i_var, input] out_ty
-  let k_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty] --
-  trace[Diverge.def] "k_var_ty: {k_var_ty}"
-  withLocalDeclD (.num (.str .anonymous "k") 2) k_var_ty fun k_var => do
-  trace[Diverge.def] "k_var: {k_var}"
+  let kk_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty]
+  trace[Diverge.def] "kk_var_ty: {kk_var_ty}"
+  withLocalDeclD (.num (.str .anonymous "kk") 2) kk_var_ty fun kk_var => do
+  trace[Diverge.def] "kk_var: {kk_var}"
 
   -- Replace the recursive calls in all the function bodies by calls to the
   -- continuation `k` and and generate for those bodies declarations
-  let bodies ← mkDeclareUnaryBodies grLvlParams k_var preDefs
+  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var preDefs
   -- Generate the mutually recursive body
-  let body ← mkDeclareMutualBody grName grLvlParams i_var k_var in_ty out_ty inOutTys.toList bodies
-  trace[Diverge.def] "mut rec body (after decl): {body}"
+  let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var in_ty out_ty inOutTys.toList bodies
+  trace[Diverge.def] "mut rec body (after decl): {mutRecBody}"
 
   -- Prove that the mut rec body satisfies the validity criteria required by
   -- our fixed-point
-  -- TODO
+  let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
+  withLocalDeclD (.num (.str .anonymous "k") 3) k_var_ty fun k_var => do
+  let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
 
   -- Generate the final definitions
-  let defs ← mkDeclareFixDefs body preDefs
+  let defs ← mkDeclareFixDefs mutRecBody preDefs
 
   -- Prove the unfolding equations
   -- TODO
@@ -496,13 +837,10 @@ def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLC
   for preDefs in cliques do
     trace[Diverge.elab] "{preDefs.map (·.declName)}"
     try
-      trace[Diverge.elab] "calling divRecursion"
       withRef (preDefs[0]!.ref) do
         divRecursion preDefs
-      trace[Diverge.elab] "divRecursion succeeded"
     catch ex =>
-      -- If it failed, we 
-      trace[Diverge.elab] "divRecursion failed"
+      -- If it failed, we add the functions as partial functions
       hasErrors := true
       logException ex
       let s ← saveState
@@ -600,7 +938,8 @@ divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
     else return (← list_nth ls (i - 1))
 
 #print list_nth.in_out_ty
-#check list_nth.body
+#check list_nth.sbody
+#check list_nth.mut_rec_body
 #print list_nth
 
 mutual
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index 82f79f94..281dbd6c 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -8,6 +8,7 @@ initialize registerTraceClass `Diverge.elab
 initialize registerTraceClass `Diverge.def
 initialize registerTraceClass `Diverge.def.sigmas
 initialize registerTraceClass `Diverge.def.genBody
+initialize registerTraceClass `Diverge.def.valid
 
 -- TODO: move
 -- TODO: small helper
-- 
cgit v1.3.1


From 7ceab6a725e5bd17c05bfd381753e453b15afaf7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 3 Jul 2023 16:46:59 +0200
Subject: Add a missing case in the validity proofs

---
 backends/lean/Base/Diverge/Elab.lean | 25 +++++++++++++++++++------
 1 file changed, 19 insertions(+), 6 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index cf40ea8f..063480a2 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -378,17 +378,22 @@ mutual
 partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
   trace[Diverge.def.valid] "proveValid: {e}"
   match e with
+  | .const _ _ => throwError "Unimplemented" -- Shouldn't get there?
   | .bvar _
   | .fvar _
-  | .mvar _
-  | .sort _
   | .lit _
-  | .const _ _ => throwError "Unimplemented"
+  | .mvar _
+  | .sort _ => throwError "Unreachable"
   | .lam .. => throwError "Unimplemented"
   | .forallE .. => throwError "Unreachable" -- Shouldn't get there
-  | .letE .. => throwError "TODO"
-    -- lambdaLetTelescope e fun xs b => mapVisitBinders xs do
-    --  mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
+  | .letE dName dTy dValue body _nonDep => do
+    -- Introduce a local declaration for the let-binding
+    withLetDecl dName dTy dValue fun decl => do
+    let isValid ← proveExprIsValid k_var kk_var body
+    -- Add the let-binding around (rem.: the let-binding should be
+    -- *inside* the `is_valid_p`, not outside, but because it reduces
+    -- in the end it doesn't matter)
+    mkLetFVars #[decl] isValid
   | .mdata _ b => proveExprIsValid k_var kk_var b
   | .proj _ _ _ =>
     -- The projection shouldn't use the continuation
@@ -963,4 +968,12 @@ mutual
     if i > 20 then foo (i / 20) else .ret 42
 end
 
+-- Testing dependent branching and let-bindings
+-- TODO: why the linter warning?
+divergent def is_non_zero (i : Int) : Result Bool :=
+  if _h:i = 0 then return false
+  else
+    let b := true
+    return b
+
 end Diverge
-- 
cgit v1.3.1


From 9214484c471ad931924865855687f9a2ffe255dd Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 3 Jul 2023 18:02:52 +0200
Subject: Automate the proofs of the unfolding theorems for Diverge

---
 backends/lean/Base/Diverge/Elab.lean     | 107 +++++++++++++++++++++++++------
 backends/lean/Base/Diverge/ElabBase.lean |   1 +
 2 files changed, 89 insertions(+), 19 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 063480a2..91c51a31 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -16,8 +16,9 @@ syntax (name := divergentDef)
 open Lean Elab Term Meta Primitives Lean.Meta
 
 set_option trace.Diverge.def true
-set_option trace.Diverge.def.valid true
+-- set_option trace.Diverge.def.valid true
 -- set_option trace.Diverge.def.sigmas true
+set_option trace.Diverge.def.unfold true
 
 /- The following was copied from the `wfRecursion` function. -/
 
@@ -390,9 +391,10 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
     -- Introduce a local declaration for the let-binding
     withLetDecl dName dTy dValue fun decl => do
     let isValid ← proveExprIsValid k_var kk_var body
-    -- Add the let-binding around (rem.: the let-binding should be
-    -- *inside* the `is_valid_p`, not outside, but because it reduces
-    -- in the end it doesn't matter)
+    -- Add the let-binding around.
+    -- Rem.: the let-binding should be *inside* the `is_valid_p`, not outside,
+    -- but because it reduces in the end it doesn't matter. More precisely:
+    -- `P (let x := v in y)` and `let x := v in P y` reduce to the same expression.
     mkLetFVars #[decl] isValid
   | .mdata _ b => proveExprIsValid k_var kk_var b
   | .proj _ _ _ =>
@@ -692,9 +694,9 @@ def proveMutRecIsValid
 
 -- Generate the final definions by using the mutual body and the fixed point operator.
 def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
-  TermElabM Unit := do
+  TermElabM (Array Name) := do
   let grSize := preDefs.size
-  let _ ← preDefs.mapIdxM fun idx preDef => do
+  let defs ← preDefs.mapIdxM fun idx preDef => do
     lambdaLetTelescope preDef.value fun xs _ => do
     -- Create the index
     let idx ← mkFinVal grSize idx.val
@@ -715,7 +717,58 @@ def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
       all := [name]
     }
     addDecl decl
-  pure ()
+    pure name
+  pure defs
+
+-- Prove the equations that we will use as unfolding theorems
+partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinition)
+  (decls : Array Name) : MetaM Unit := do
+  let grSize := preDefs.size
+  let proveIdx (i : Nat) : MetaM Unit := do
+    let preDef := preDefs.get! i
+    let defName := decls.get! i
+    -- Retrieve the arguments
+    lambdaLetTelescope preDef.value fun xs body => do
+    trace[Diverge.def.unfold] "proveUnfoldingThms: xs: {xs}"
+    trace[Diverge.def.unfold] "proveUnfoldingThms: body: {body}"
+    -- The theorem statement
+    let thmTy ← do
+      -- The equation: the declaration gives the lhs, the pre-def gives the rhs
+      let lhs ← mkAppOptM defName (xs.map some)
+      let rhs := body
+      let eq ← mkAppM ``Eq #[lhs, rhs]
+      mkForallFVars xs eq
+    trace[Diverge.def.unfold] "proveUnfoldingThms: thm statement: {thmTy}"
+    -- The proof
+    -- Use the fixed-point equation
+    let proof ← mkAppM ``FixI.is_valid_fix_fixed_eq #[isValidThm]
+    -- Add the index
+    let idx ← mkFinVal grSize i
+    let proof ← mkAppM ``congr_fun #[proof, idx]
+    -- Add the input argument
+    let arg ← mkSigmas xs.toList
+    let proof ← mkAppM ``congr_fun #[proof, arg]
+    -- Abstract the arguments away
+    let proof ← mkLambdaFVars xs proof
+    trace[Diverge.def.unfold] "proveUnfoldingThms: proof: {proof}:\n{← inferType proof}"
+    -- Declare the theorem
+    let name := preDef.declName ++ "unfold"
+    let decl := Declaration.thmDecl {
+      name
+      levelParams := preDef.levelParams
+      type := thmTy
+      value := proof
+      all := [name]
+    }
+    addDecl decl
+    trace[Diverge.def.unfold] "proveUnfoldingThms: added thm: {name}:\n{thmTy}"
+  let rec prove (i : Nat) : MetaM Unit := do
+    if i = preDefs.size then pure ()
+    else do
+      proveIdx i
+      prove (i + 1)
+  --
+  prove 0
 
 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
@@ -817,12 +870,12 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
 
   -- Generate the final definitions
-  let defs ← mkDeclareFixDefs mutRecBody preDefs
+  let decls ← mkDeclareFixDefs mutRecBody preDefs
 
-  -- Prove the unfolding equations
-  -- TODO
+  -- Prove the unfolding theorems
+  proveUnfoldingThms isValidThm preDefs decls
 
-  -- Process the definitions
+  -- Process the definitions - TODO
   addAndCompilePartialRec preDefs
 
 -- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
@@ -942,10 +995,23 @@ divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
     if i = 0 then return x
     else return (← list_nth ls (i - 1))
 
-#print list_nth.in_out_ty
-#check list_nth.sbody
-#check list_nth.mut_rec_body
-#print list_nth
+example {a: Type} (ls : List a) :
+  ∀ (i : Int),
+  0 ≤ i → i < ls.length →
+  ∃ x, list_nth ls i = .ret x := by
+  induction ls
+  . intro i hpos h; simp at h; linarith
+  . rename_i hd tl ih
+    intro i hpos h
+    rw [list_nth.unfold]; simp
+    split <;> simp [*]
+    . tauto
+    . -- TODO: we shouldn't have to do that
+      have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
+      simp at h
+      have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
+      simp [ih]
+      tauto
 
 mutual
   divergent def is_even (i : Int) : Result Bool :=
@@ -955,10 +1021,8 @@ mutual
     if i = 0 then return false else return (← is_even (i - 1))
 end
 
-example (i : Int) : is_even i = .ret (i % 2 = 0) ∧ is_odd i = .ret (i % 2 ≠ 0) := by
-  induction i
-  unfold is_even
-  sorry    
+#print is_even.unfold
+#print is_odd.unfold
 
 mutual
   divergent def foo (i : Int) : Result Nat :=
@@ -968,6 +1032,9 @@ mutual
     if i > 20 then foo (i / 20) else .ret 42
 end
 
+#print foo.unfold
+#print bar.unfold
+
 -- Testing dependent branching and let-bindings
 -- TODO: why the linter warning?
 divergent def is_non_zero (i : Int) : Result Bool :=
@@ -976,4 +1043,6 @@ divergent def is_non_zero (i : Int) : Result Bool :=
     let b := true
     return b
 
+#print is_non_zero.unfold
+
 end Diverge
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index 281dbd6c..fd95291e 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -9,6 +9,7 @@ initialize registerTraceClass `Diverge.def
 initialize registerTraceClass `Diverge.def.sigmas
 initialize registerTraceClass `Diverge.def.genBody
 initialize registerTraceClass `Diverge.def.valid
+initialize registerTraceClass `Diverge.def.unfold
 
 -- TODO: move
 -- TODO: small helper
-- 
cgit v1.3.1


From 75fae6384716f24fe137283d4a41836782b9aec7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 3 Jul 2023 19:26:27 +0200
Subject: Cleanup a bit Diverge/Elab.lean

---
 backends/lean/Base/Diverge/Elab.lean | 366 +++++++++++++++++++----------------
 1 file changed, 197 insertions(+), 169 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 91c51a31..cc580265 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -15,39 +15,16 @@ syntax (name := divergentDef)
 
 open Lean Elab Term Meta Primitives Lean.Meta
 
-set_option trace.Diverge.def true
--- set_option trace.Diverge.def.valid true
--- set_option trace.Diverge.def.sigmas true
-set_option trace.Diverge.def.unfold true
-
 /- The following was copied from the `wfRecursion` function. -/
 
 open WF in
 
-def mkList (xl : List Expr) (ty : Expr) : MetaM Expr :=
-  match xl with
-  | [] =>
-    mkAppOptM ``List.nil #[some ty]
-  | x :: tl => do
-    let tl ← mkList tl ty
-    mkAppOptM ``List.cons #[some ty, some x, some tl]
-
 def mkProd (x y : Expr) : MetaM Expr :=
   mkAppM ``Prod.mk #[x, y]
 
 def mkInOutTy (x y : Expr) : MetaM Expr :=
   mkAppM ``FixI.mk_in_out_ty #[x, y]
 
--- TODO: is there already such a utility somewhere?
--- TODO: change to mkSigmas
-def mkProds (tys : List Expr) : MetaM Expr :=
-  match tys with
-  | [] => do pure (Expr.const ``PUnit.unit [])
-  | [ty] => do pure ty
-  | ty :: tys => do
-    let pty ← mkProds tys
-    mkAppM ``Prod.mk #[ty, pty]
-
 -- Return the `a` in `Return a`
 def get_result_ty (ty : Expr) : MetaM Expr :=
   ty.withApp fun f args => do
@@ -56,26 +33,31 @@ def get_result_ty (ty : Expr) : MetaM Expr :=
   else
     pure (args.get! 0)
 
--- Group a list of expressions into a dependent tuple
-def mkSigmas (xl : List Expr) : MetaM Expr :=
+/- Group a list of expressions into a dependent tuple.
+
+   Example:
+   xl = [`a : Type`, `ls : List a`]
+   returns:
+   `⟨ (a:Type), (ls: List a) ⟩`
+ -/
+def mkSigmasVal (xl : List Expr) : MetaM Expr :=
   match xl with
   | [] => do
-    trace[Diverge.def.sigmas] "mkSigmas: []"
+    trace[Diverge.def.sigmas] "mkSigmasVal: []"
     pure (Expr.const ``PUnit.unit [])
   | [x] => do
-    trace[Diverge.def.sigmas] "mkSigmas: [{x}]"
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
     pure x
   | fst :: xl => do
-    trace[Diverge.def.sigmas] "mkSigmas: [{fst}::{xl}]"
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{fst}::{xl}]"
     let alpha ← Lean.Meta.inferType fst
-    let snd ← mkSigmas xl
+    let snd ← mkSigmasVal xl
     let snd_ty ← inferType snd
     let beta ← mkLambdaFVars #[fst] snd_ty
-    trace[Diverge.def.sigmas] "mkSigmas:\n{alpha}\n{beta}\n{fst}\n{snd}"
+    trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
     mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
 
-/- Generate the input type of a function body, which is a sigma type (i.e., a
-   dependent tuple) which groups all its inputs.
+/- Generate a Sigma type from a list of expressions.
 
    Example:
    - xl = [(a:Type), (ls:List a), (i:Int)]
@@ -84,7 +66,7 @@ def mkSigmas (xl : List Expr) : MetaM Expr :=
    `(a:Type) × (ls:List a) × (i:Int)`
 
  -/
-def mkSigmasTypesOfTypes (xl : List Expr) : MetaM Expr :=
+def mkSigmasType (xl : List Expr) : MetaM Expr :=
   match xl with
   | [] => do
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
@@ -96,15 +78,16 @@ def mkSigmasTypesOfTypes (xl : List Expr) : MetaM Expr :=
   | x :: xl => do
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
     let alpha ← Lean.Meta.inferType x
-    let sty ← mkSigmasTypesOfTypes xl
+    let sty ← mkSigmasType xl
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]: alpha={alpha}, sty={sty}"
     let beta ← mkLambdaFVars #[x] sty
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
     mkAppOptM ``Sigma #[some alpha, some beta]
 
-def mk_indexed_name (index : Nat) : Name := .num (.str .anonymous "_uniq") index
+def mkAnonymous (s : String) (i : Nat) : Name :=
+  .num (.str .anonymous s) i
 
-/- Given a list of values `[x0:ty0, ..., xn:ty1]` where every `xi` might use the previous
+/- Given a list of values `[x0:ty0, ..., xn:ty1]`, where every `xi` might use the previous
    `xj` (j < i) and a value `out` which uses `x0`, ..., `xn`, generate the following
    expression:
    ```
@@ -112,20 +95,22 @@ def mk_indexed_name (index : Nat) : Name := .num (.str .anonymous "_uniq") index
    match x with
    | (x0, ..., xn) => out
    ```
-   
+
    The `index` parameter is used for naming purposes: we use it to numerotate the
    bound variables that we introduce.
 
+   We use this function to currify functions (the function bodies given to the
+   fixed-point operator must be unary functions).
+
    Example:
    ========
-   More precisely:
    - xl = `[a:Type, ls:List a, i:Int]`
    - out = `a`
    - index = 0
 
-   generates:
+   generates (getting rid of most of the syntactic sugar):
    ```
-   match scrut0 with
+   λ scrut0 => match scrut0 with
    | Sigma.mk x scrut1 =>
      match scrut1 with
      | Sigma.mk ls i =>
@@ -138,21 +123,30 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
     -- This would be unexpected
     throwError "mkSigmasMatch: empyt list of input parameters"
   | [x] => do
-    -- In the explanations above: inner match case
+    -- In the example given for the explanations: this is the inner match case
     trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]"
     mkLambdaFVars #[x] out
   | fst :: xl => do
-    -- In the explanations above: outer match case
+    -- In the example given for the explanations: this is the outer match case
     -- Remark: for the naming purposes, we use the same convention as for the
-    -- fields and parameters in `Sigma.casesOn and `Sigma.mk
+    -- fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
+    -- those definitions might help)
+    --
+    -- We want to build the match expression:
+    -- ```
+    -- λ scrut =>
+    -- match scrut with
+    -- | Sigma.mk x ...  -- the hole is given by a recursive call on the tail
+    -- ```
     trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]"
     let alpha ← Lean.Meta.inferType fst
-    let snd_ty ← mkSigmasTypesOfTypes xl
+    let snd_ty ← mkSigmasType xl
     let beta ← mkLambdaFVars #[fst] snd_ty
     let snd ← mkSigmasMatch xl out (index + 1)
-    let scrut_ty ← mkSigmasTypesOfTypes (fst :: xl)
-    withLocalDeclD (mk_indexed_name index) scrut_ty fun scrut => do
     let mk ← mkLambdaFVars #[fst] snd
+    -- Introduce the "scrut" variable
+    let scrut_ty ← mkSigmasType (fst :: xl)
+    withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
     trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})"
     -- TODO: make the computation of the motive more efficient
     let motive ← do
@@ -166,38 +160,32 @@ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : Met
         -- TODO: make this more efficient (we could change the output type of
         -- mkSigmasMatch
         mkSigmasMatch (fst :: xl) out_ty
+    -- The final expression: putting everything together
     trace[Diverge.def.sigmas] "mkSigmasMatch:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
     let sm ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+    -- Abstracting the "scrut" variable
     let sm ← mkLambdaFVars #[scrut] sm
     trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}"
     pure sm
 
 /- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/
-private def list_nth_out_ty2 (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
+private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
   @Sigma.casesOn (List a)
                  (fun (_ls : List a) => Int)
                  (fun (_scrut1:@Sigma (List a) (fun (_ls : List a) => Int)) => Type)
                  scrut1
                  (fun (_ls : List a) (_i : Int) => Diverge.Primitives.Result a)
 
-private def list_nth_out_ty1 (scrut0 : @Sigma (Type) (fun (a:Type) =>
+private def list_nth_out_ty_outer (scrut0 : @Sigma (Type) (fun (a:Type) =>
                       @Sigma (List a) (fun (_ls : List a) => Int))) :=
   @Sigma.casesOn (Type)
                  (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))
                  (fun (_scrut0:@Sigma (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))) => Type)
                  scrut0
                  (fun (a : Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) =>
-                  list_nth_out_ty2 a scrut1)
+                  list_nth_out_ty_inner a scrut1)
 /- -/
 
--- TODO: move
--- TODO: we can use Array.mapIdx
-@[specialize] def mapiAux (i : Nat) (f : Nat → α → β) : List α → List β
-  | []    => []
-  | a::as => f i a :: mapiAux (i+1) f as
-
-@[specialize] def mapi (f : Nat → α → β) : List α → List β := mapiAux 0 f
-
 -- Return the expression: `Fin n`
 -- TODO: use more
 def mkFin (n : Nat) : Expr :=
@@ -212,15 +200,6 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
   let ofNat ← mkAppOptM ``Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat #[n_lit, i_lit]
   mkAppOptM ``OfNat.ofNat #[none, none, ofNat]
 
--- TODO: remove?
-def mkFinValOld (n i : Nat) : MetaM Expr := do
-  let finTy := mkFin n
-  let ofNat ← mkAppM ``OfNat #[finTy, .lit (.natVal i)]
-  match ← trySynthInstance ofNat with
-  | LOption.some x =>
-    mkAppOptM ``OfNat.ofNat #[none, none, x]
-  | _ => throwError "mkFinVal: could not synthesize an instance of {ofNat} "
-
 /- Generate and declare as individual definitions the bodies for the individual funcions:
    - replace the recursive calls with calls to the continutation `k`
    - make those bodies take one single dependent tuple as input
@@ -234,11 +213,11 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
   let grSize := preDefs.size
 
   -- Compute the map from name to index - the continuation has an indexed type:
-  -- we use the index (a finite number of type `Fin`) to control the function
-  -- we call at the recursive call
+  -- we use the index (a finite number of type `Fin`) to control which function
+  -- we call at the recursive call site.
   let nameToId : HashMap Name Nat :=
-    let namesIds := mapi (fun i d => (d.declName, i)) preDefs.toList
-    HashMap.ofList namesIds
+    let namesIds := preDefs.mapIdx (fun i d => (d.declName, i.val))
+    HashMap.ofList namesIds.toList
 
   trace[Diverge.def.genBody] "nameToId: {nameToId.toList}"
 
@@ -260,7 +239,7 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
             -- Compute the index
             let i ← mkFinVal grSize id
             -- Put the arguments in one big dependent tuple
-            let args ← mkSigmas args.toList
+            let args ← mkSigmasVal args.toList
             mkAppM' kk_var #[i, args]
         else
           -- Not a recursive call: do nothing
@@ -277,13 +256,14 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
     -- Replace the recursive calls
     let body ← mapVisit visit_e preDef.value
 
-    -- Change the type
+    -- Currify the function by grouping the arguments into a dependent tuple
+    -- (over which we match to retrieve the individual arguments).
     lambdaLetTelescope body fun args body => do
     let body ← mkSigmasMatch args.toList body 0
 
     -- Add the declaration
     let value ← mkLambdaFVars #[kk_var] body
-    let name := preDef.declName.append "sbody"
+    let name := preDef.declName.append "body"
     let levelParams := grLvlParams
     let decl := Declaration.defnDecl {
       name := name
@@ -304,7 +284,7 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
 
 -- Generate a unique function body from the bodies of the mutually recursive group,
 -- and add it as a declaration in the context.
--- We return the list of bodies (of type `Funs ...`) and the mutually recursive body.
+-- We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
 def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
   (kk_var i_var : Expr)
   (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
@@ -322,7 +302,7 @@ def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
       mkAppOptM ``FixI.Funs.Nil #[finTypeExpr, in_ty, out_ty]
     | (ity, oty) :: inOutTys, b :: bl => do
       -- Retrieving ity and oty - this is not very clean
-      let inOutTysExpr ← mkList (← inOutTys.mapM (λ (x, y) => mkInOutTy x y)) inOutTyType
+      let inOutTysExpr ← mkListLit inOutTyType (← inOutTys.mapM (λ (x, y) => mkInOutTy x y))
       let fl ← mkFuns inOutTys bl
       mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
     | _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length"
@@ -345,7 +325,7 @@ def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
     all := [name]
   }
   addDecl decl
-  -- Return the constant
+  -- Return the bodies and the constant
   pure (bodyFuns, Lean.mkConst name (levelParams.map .param))
 
 def isCasesExpr (e : Expr) : MetaM Bool := do
@@ -367,7 +347,8 @@ instance : ToMessageData MatchInfo where
   -- This is not a very clean formatting, but we don't need more
   toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}"
 
--- An expression which doesn't use the continuation kk is valid
+-- Small helper: prove that an expression which doesn't use the continuation `kk`
+-- is valid, and return the proof.
 def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
   trace[Diverge.def.valid] "proveNoKExprIsValid: {e}"
   let eIsValid ← mkAppM ``FixI.is_valid_p_same #[k_var, e]
@@ -376,6 +357,14 @@ def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
 
 mutual
 
+/- Prove that an expression is valid, and return the proof.
+
+   More precisely, if `e` is an expression which potentially uses the continution
+   `kk`, return an expression of type:
+   ```
+   is_valid_p k (λ kk => e)
+   ```
+ -/
 partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
   trace[Diverge.def.valid] "proveValid: {e}"
   match e with
@@ -403,7 +392,9 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
   | .app .. =>
     e.withApp fun f args => do
       -- There are several cases: first, check if this is a match/if
-      -- The expression is a (dependent) if then else
+      -- Check if the expression is a (dependent) if then else.
+      -- We treat the if then else expressions differently from the other matches,
+      -- and have dedicated theorems for them.
       let isIte := e.isIte
       if isIte || e.isDIte then do
         e.withApp fun f args => do
@@ -431,9 +422,9 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         let eIsValid ← mkAppOptM const #[none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid]
         trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
         pure eIsValid
-      -- The expression is a match (this case is for when the elaborator
+      -- Check if the expression is a match (this case is for when the elaborator
       -- introduces auxiliary definitions to hide the match behind syntactic
-      -- sugar)
+      -- sugar):
       else if let some me := ← matchMatcherApp? e then do
         trace[Diverge.def.valid]
           "matcherApp:
@@ -443,7 +434,8 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
            - altNumParams: {me.altNumParams}
            - alts: {me.alts}
            - remaining: {me.remaining}"
-        -- matchMatcherApp has already done the work for us
+        -- matchMatcherApp does all the work for us: we simply need to gather
+        -- the information and call the auxiliary helper `proveMatchIsValid`
         if me.remaining.size ≠ 0 then
           throwError "MatcherApp: non empty remaining array: {me.remaining}"
         let me : MatchInfo := {
@@ -456,14 +448,21 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
           branches := me.alts
         }
         proveMatchIsValid k_var kk_var me
-      -- The expression is a raw match (this case is for when the expression
-      -- is a direct call to the primitive `casesOn` function, without
-      -- syntactic sugar)
+      -- Check if the expression is a raw match (this case is for when the expression
+      -- is a direct call to the primitive `casesOn` function, without syntactic sugar).
+      -- We have to check this case because functions like `mkSigmasMatch`, which we
+      -- use to currify function bodies, introduce such raw matches.
       else if ← isCasesExpr f then do
         trace[Diverge.def.valid] "rawMatch: {e}"
+        -- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
+        --
         -- The casesOn definition is always of the following shape:
-        -- input parameters (implicit parameters), then motive (implicit), 
-        -- scrutinee (explicit), branches (explicit).
+        -- - input parameters (implicit parameters)
+        -- - motive (implicit), -- the motive gives the return type of the match
+        -- - scrutinee (explicit)
+        -- - branches (explicit).
+        -- In particular, we notice that the scrutinee is the first *explicit*
+        -- parameter - this is how we spot it.
         let matcherName := f.constName!
         let matcherLevels := f.constLevels!.toArray
         -- Find the first explicit parameter: this is the scrutinee
@@ -484,7 +483,9 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         let scrut := args.get! scrutIdx
         let branches := args.extract (scrutIdx + 1) args.size
         -- Compute the number of parameters for the branches: for this we use
-        -- the type of the uninstantiated casesOn constant
+        -- the type of the uninstantiated casesOn constant (we can't just
+        -- destruct the lambdas in the branch expressions because the result
+        -- of a match might be a lambda expression).
         let branchesNumParams : Array Nat ← do
           let env ← getEnv
           let decl := env.constants.find! matcherName
@@ -505,9 +506,11 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
           branches,
         }
         proveMatchIsValid k_var kk_var me
-      -- Monadic let-binding
+      -- Check if this is a monadic let-binding
       else if f.isConstOf ``Bind.bind then do
         trace[Diverge.def.valid] "bind:\n{args}"
+        -- We simply need to prove that the subexpressions are valid, and call
+        -- the appropriate lemma.
         let x := args.get! 4
         let y := args.get! 5
         -- Prove that the subexpressions are valid
@@ -529,7 +532,7 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         -- Put everything together
         trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
         mkAppM ``FixI.is_valid_p_bind #[xValid, yValid]
-      -- Recursive call
+      -- Check if this is a recursive call, i.e., a call to the continuation `kk`
       else if f.isFVarOf kk_var.fvarId! then do
         trace[Diverge.def.valid] "rec: args: \n{args}"
         if args.size ≠ 2 then throwError "Recursive call with invalid number of parameters: {args}"
@@ -540,9 +543,10 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         pure eIsValid
       else do
         -- Remaining case: normal application.
-        -- It shouldn't use the continuation
+        -- It shouldn't use the continuation.
         proveNoKExprIsValid k_var e
 
+-- Prove that a match expression is valid.
 partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do
   trace[Diverge.def.valid] "proveMatchIsValid: {me}"
   -- Prove the validity of the branch expressions
@@ -561,16 +565,18 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     -- Reconstruct the lambda expression
     mkLambdaFVars xs_beg brValid
   trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
-  -- Put together: compute the motive.
-  -- It must be of the shape:
+  -- Compute the motive, which has the following shape:
   -- ```
   -- λ scrut => is_valid_p k (λ k => match scrut with ...)
+  --                                 ^^^^^^^^^^^^^^^^^^^^
+  --                         this is the original match expression, with the
+  --                        the difference that the scrutinee(s) is a variable
   -- ```
   let validMotive : Expr ← do
     -- The motive is a function of the scrutinees (i.e., a lambda expression):
     -- introduce binders for the scrutinees
     let declInfos := me.scruts.mapIdx fun idx scrut =>
-      let name : Name :=  (.num (.str .anonymous "scrut") idx)
+      let name : Name := mkAnonymous "scrut" idx
       let ty := λ (_ : Array Expr) => inferType scrut
       (name, ty)
     withLocalDeclsD declInfos fun scrutVars => do
@@ -582,7 +588,6 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     let branches : Array (Option Expr) := me.branches.map some
     let args := params ++ [motive] ++ scruts ++ branches
     let matchE ← mkAppOptM me.matcherName args
-    -- let matchE ← mkLambdaFVars scrutVars (← mkAppOptM me.matcherName args)
     -- Wrap in the `is_valid_p` predicate
     let matchE ← mkLambdaFVars #[kk_var] matchE
     let validMotive ← mkAppM ``FixI.is_valid_p #[k_var, matchE]
@@ -591,6 +596,7 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
   trace[Diverge.def.valid] "valid motive: {validMotive}"
   -- Put together
   let valid ← do
+    -- We let Lean infer the parameters
     let params : Array (Option Expr) := me.params.map (λ _ => none)
     let motive := some validMotive
     let scruts := me.scruts.map some
@@ -602,12 +608,16 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
 
 end
 
--- Prove that a single body (in the mutually recursive group) is valid
-partial def proveSingleBodyIsValid (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
+-- Prove that a single body (in the mutually recursive group) is valid.
+--
+-- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+-- we prove that `is_even.body` and `is_odd.body` are valid.
+partial def proveSingleBodyIsValid
+  (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
   MetaM Expr := do
   trace[Diverge.def.valid] "proveSingleBodyIsValid: bodyConst: {bodyConst}"
-  -- Lookup the definition (`bodyConst` is the definition of the body, we want
-  -- to retrieve the value itself to dive inside)
+  -- Lookup the definition (`bodyConst` is a const, we want to retrieve its
+  -- definition to dive inside)
   let name := bodyConst.constName!
   let env ← getEnv
   let body := (env.constants.find! name).value!
@@ -633,7 +643,7 @@ partial def proveSingleBodyIsValid (k_var : Expr) (preDef : PreDefinition) (body
     mkForallFVars #[k_var, x_var] ty
   trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}"
   -- Save the theorem
-  let name := preDef.declName ++ "sbody_is_valid"
+  let name := preDef.declName ++ "body_is_valid"
   let decl := Declaration.thmDecl {
     name
     levelParams := preDef.levelParams
@@ -646,6 +656,11 @@ partial def proveSingleBodyIsValid (k_var : Expr) (preDef : PreDefinition) (body
   -- Return the theorem
   pure (Expr.const name (preDef.levelParams.map .param))
 
+-- Prove that the list of bodies are valid.
+--
+-- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+-- we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
+-- valid.
 partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
   (k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do
   -- Create the big "and" expression, which groups the validity proof of the individual bodies
@@ -663,7 +678,13 @@ partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
   let isValid ← mkAppM ``FixI.Funs.is_valid_p_is_valid_p #[inOutTys, k_var, bodyFuns, andExpr]
   mkLambdaFVars #[k_var] isValid
 
--- Prove that the mut rec body is valid
+-- Prove that the mut rec body (i.e., the unary body which groups the bodies
+-- of all the functions in the mutually recursive group and on which we will
+-- apply the fixed-point operator) is valid.
+--
+-- We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
+-- which we return.
+--
 -- TODO: maybe this function should introduce k_var itself
 def proveMutRecIsValid
   (grName : Name) (grLvlParams : List Name)
@@ -693,6 +714,12 @@ def proveMutRecIsValid
   pure (Expr.const name (grLvlParams.map .param))
 
 -- Generate the final definions by using the mutual body and the fixed point operator.
+--
+-- For instance:
+-- ```
+-- def is_even (i : Int) : Result Bool := mut_rec_body 0 i
+-- def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
+-- ```
 def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
   TermElabM (Array Name) := do
   let grSize := preDefs.size
@@ -701,7 +728,7 @@ def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
     -- Create the index
     let idx ← mkFinVal grSize idx.val
     -- Group the inputs into a dependent tuple
-    let input ← mkSigmas xs.toList
+    let input ← mkSigmasVal xs.toList
     -- Apply the fixed point
     let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
     let fixedBody ← mkLambdaFVars xs fixedBody
@@ -746,7 +773,7 @@ partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinitio
     let idx ← mkFinVal grSize i
     let proof ← mkAppM ``congr_fun #[proof, idx]
     -- Add the input argument
-    let arg ← mkSigmas xs.toList
+    let arg ← mkSigmasVal xs.toList
     let proof ← mkAppM ``congr_fun #[proof, arg]
     -- Abstract the arguments away
     let proof ← mkLambdaFVars xs proof
@@ -774,11 +801,6 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
   trace[Diverge.def] ("divRecursion: defs: " ++ msg)
 
-  -- CHANGE HERE This function should add definitions with these names/types/values ^^
-  -- Temporarily add the predefinitions as axioms
-  -- for preDef in preDefs do
-  --  addAsAxiom preDef
-
   -- TODO: what is this?
   for preDef in preDefs do
     applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
@@ -803,7 +825,7 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
         if preDef.levelParams ≠ grLvlParams then
           throwError "Non-uniform polymorphism in the universes"
         forallTelescope preDef.type (fun in_tys out_ty => do
-          let in_ty ← liftM (mkSigmasTypesOfTypes in_tys.toList)
+          let in_ty ← liftM (mkSigmasType in_tys.toList)
           -- Retrieve the type in the "Result"
           let out_ty ← get_result_ty out_ty
           let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
@@ -813,14 +835,14 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   trace[Diverge.def] "inOutTys: {inOutTys}"
   -- Turn the list of input/output type pairs into an expresion
   let inOutTysExpr ← inOutTys.mapM (λ (x, y) => mkInOutTy x y)
-  let inOutTysExpr ← mkList inOutTysExpr.toList (← inferType (inOutTysExpr.get! 0))
+  let inOutTysExpr ← mkListLit (← inferType (inOutTysExpr.get! 0)) inOutTysExpr.toList
 
   -- From the list of pairs of input/output types, actually compute the
   -- type of the continuation `k`.
   -- We first introduce the index `i : Fin n` where `n` is the number of
   -- functions in the group.
   let i_var_ty := mkFin preDefs.size
-  withLocalDeclD (.num (.str .anonymous "i") 0) i_var_ty fun i_var => do
+  withLocalDeclD (mkAnonymous "i" 0) i_var_ty fun i_var => do
   let in_out_ty ← mkAppM ``List.get #[inOutTysExpr, i_var]
   trace[Diverge.def] "in_out_ty := {in_out_ty} : {← inferType in_out_ty}"
   -- Add an auxiliary definition for `in_out_ty`
@@ -844,7 +866,7 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   trace[Diverge.def] "in_out_ty (after decl) := {in_out_ty} : {← inferType in_out_ty}"
   let in_ty ← mkAppM ``Sigma.fst #[in_out_ty]
   trace[Diverge.def] "in_ty: {in_ty}"
-  withLocalDeclD (.num (.str .anonymous "x") 1) in_ty fun input => do
+  withLocalDeclD (mkAnonymous "x" 1) in_ty fun input => do
   let out_ty ← mkAppM' (← mkAppM ``Sigma.snd #[in_out_ty]) #[input]
   trace[Diverge.def] "out_ty: {out_ty}"
 
@@ -853,7 +875,7 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let out_ty ← mkLambdaFVars #[i_var, input] out_ty
   let kk_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty]
   trace[Diverge.def] "kk_var_ty: {kk_var_ty}"
-  withLocalDeclD (.num (.str .anonymous "kk") 2) kk_var_ty fun kk_var => do
+  withLocalDeclD (mkAnonymous "kk" 2) kk_var_ty fun kk_var => do
   trace[Diverge.def] "kk_var: {kk_var}"
 
   -- Replace the recursive calls in all the function bodies by calls to the
@@ -866,7 +888,7 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   -- Prove that the mut rec body satisfies the validity criteria required by
   -- our fixed-point
   let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
-  withLocalDeclD (.num (.str .anonymous "k") 3) k_var_ty fun k_var => do
+  withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do
   let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
 
   -- Generate the final definitions
@@ -915,7 +937,7 @@ def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLC
         else return ()
       catch _ => s.restore
 
--- The following two functions are copy&pasted from Lean.Elab.MutualDef
+-- The following two functions are copy-pasted from Lean.Elab.MutualDef
 
 open private elabHeaders levelMVarToParamHeaders getAllUserLevelNames withFunLocalDecls elabFunValues
   instantiateMVarsAtHeader instantiateMVarsAtLetRecToLift checkLetRecsToLiftTypes withUsed from Lean.Elab.MutualDef
@@ -988,61 +1010,67 @@ elab_rules : command
     else
       Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
 
-divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
-  match ls with
-  | [] => .fail .panic
-  | x :: ls =>
-    if i = 0 then return x
-    else return (← list_nth ls (i - 1))
-
-example {a: Type} (ls : List a) :
-  ∀ (i : Int),
-  0 ≤ i → i < ls.length →
-  ∃ x, list_nth ls i = .ret x := by
-  induction ls
-  . intro i hpos h; simp at h; linarith
-  . rename_i hd tl ih
-    intro i hpos h
-    rw [list_nth.unfold]; simp
-    split <;> simp [*]
-    . tauto
-    . -- TODO: we shouldn't have to do that
-      have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
-      simp at h
-      have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
-      simp [ih]
-      tauto
-
-mutual
-  divergent def is_even (i : Int) : Result Bool :=
-    if i = 0 then return true else return (← is_odd (i - 1))
-
-  divergent def is_odd (i : Int) : Result Bool :=
-    if i = 0 then return false else return (← is_even (i - 1))
-end
-
-#print is_even.unfold
-#print is_odd.unfold
-
-mutual
-  divergent def foo (i : Int) : Result Nat :=
-    if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10
-
-  divergent def bar (i : Int) : Result Nat :=
-    if i > 20 then foo (i / 20) else .ret 42
-end
-
-#print foo.unfold
-#print bar.unfold
-
--- Testing dependent branching and let-bindings
--- TODO: why the linter warning?
-divergent def is_non_zero (i : Int) : Result Bool :=
-  if _h:i = 0 then return false
-  else
-    let b := true
-    return b
+namespace Tests
+  /- Some examples of partial functions -/
+
+  divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+    match ls with
+    | [] => .fail .panic
+    | x :: ls =>
+      if i = 0 then return x
+      else return (← list_nth ls (i - 1))
+
+  #check list_nth.unfold
+
+  example {a: Type} (ls : List a) :
+    ∀ (i : Int),
+    0 ≤ i → i < ls.length →
+    ∃ x, list_nth ls i = .ret x := by
+    induction ls
+    . intro i hpos h; simp at h; linarith
+    . rename_i hd tl ih
+      intro i hpos h
+      rw [list_nth.unfold]; simp
+      split <;> simp [*]
+      . tauto
+      . -- TODO: we shouldn't have to do that
+        have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
+        simp at h
+        have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
+        simp [ih]
+        tauto
+
+  mutual
+    divergent def is_even (i : Int) : Result Bool :=
+      if i = 0 then return true else return (← is_odd (i - 1))
+
+    divergent def is_odd (i : Int) : Result Bool :=
+      if i = 0 then return false else return (← is_even (i - 1))
+  end
+
+  #check is_even.unfold
+  #check is_odd.unfold
+
+  mutual
+    divergent def foo (i : Int) : Result Nat :=
+      if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10
+
+    divergent def bar (i : Int) : Result Nat :=
+      if i > 20 then foo (i / 20) else .ret 42
+  end
+
+  #check foo.unfold
+  #check bar.unfold
+
+  -- Testing dependent branching and let-bindings
+  -- TODO: why the linter warning?
+  divergent def is_non_zero (i : Int) : Result Bool :=
+    if _h:i = 0 then return false
+    else
+      let b := true
+      return b
 
-#print is_non_zero.unfold
+  #check is_non_zero.unfold
+end Tests
 
 end Diverge
-- 
cgit v1.3.1


From 4fd17e4bb91eb46d4704643dfbfbbf0874837b07 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Tue, 4 Jul 2023 12:49:37 +0200
Subject: Make Diverge use Primitives

---
 backends/lean/Base/Diverge/Base.lean     | 65 +-------------------------------
 backends/lean/Base/Diverge/Elab.lean     |  2 +-
 backends/lean/Base/Diverge/ElabBase.lean |  8 ++--
 backends/lean/Base/Primitives.lean       | 25 ++----------
 4 files changed, 10 insertions(+), 90 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
index a8503107..e22eb914 100644
--- a/backends/lean/Base/Diverge/Base.lean
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -4,8 +4,7 @@ import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 import Mathlib.Tactic.Linarith
 
--- For debugging
-import Base.Diverge.ElabBase
+import Base.Primitives
 
 /-
 TODO:
@@ -35,68 +34,6 @@ set_option profiler.threshold 100
 
 namespace Diverge
 
-namespace Primitives
-/-! # Copy-pasting from Primitives to make the file self-contained -/
-
-inductive Error where
-   | assertionFailure: Error
-   | integerOverflow: Error
-   | divisionByZero: Error
-   | arrayOutOfBounds: Error
-   | maximumSizeExceeded: Error
-   | panic: Error
-deriving Repr, BEq
-
-open Error
-
-inductive Result (α : Type u) where
-  | ret (v: α): Result α
-  | fail (e: Error): Result α
-  | div
-deriving Repr, BEq
-
-open Result
-
-instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
-  Inhabited.mk (fail panic)
-
-instance Result_Nonempty (α : Type u) : Nonempty (Result α) :=
-  Nonempty.intro div
-
-def bind {α : Type u} {β : Type v} (x: Result α) (f: α -> Result β) : Result β :=
-  match x with
-  | ret v  => f v 
-  | fail v => fail v
-  | div => div
-
-@[simp] theorem bind_ret (x : α) (f : α → Result β) : bind (.ret x) f = f x := by simp [bind]
-@[simp] theorem bind_fail (x : Error) (f : α → Result β) : bind (.fail x) f = .fail x := by simp [bind]
-@[simp] theorem bind_div (f : α → Result β) : bind .div f = .div := by simp [bind]
-
--- Allows using Result in do-blocks
-instance : Bind Result where
-  bind := bind
-
--- Allows using return x in do-blocks
-instance : Pure Result where
-  pure := fun x => ret x
-
-@[simp] theorem bind_tc_ret (x : α) (f : α → Result β) :
-  (do let y ← .ret x; f y) = f x := by simp [Bind.bind, bind]
-
-@[simp] theorem bind_tc_fail (x : Error) (f : α → Result β) :
-  (do let y ← fail x; f y) = fail x := by simp [Bind.bind, bind]
-
-@[simp] theorem bind_tc_div (f : α → Result β) :
-  (do let y ← div; f y) = div := by simp [Bind.bind, bind]
-
-def div? {α: Type u} (r: Result α): Bool :=
-  match r with
-  | div => true
-  | ret _ | fail _ => false
-
-end Primitives
-
 namespace Fix
 
   open Primitives
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index cc580265..41209021 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -174,7 +174,7 @@ private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls :
                  (fun (_ls : List a) => Int)
                  (fun (_scrut1:@Sigma (List a) (fun (_ls : List a) => Int)) => Type)
                  scrut1
-                 (fun (_ls : List a) (_i : Int) => Diverge.Primitives.Result a)
+                 (fun (_ls : List a) (_i : Int) => Primitives.Result a)
 
 private def list_nth_out_ty_outer (scrut0 : @Sigma (Type) (fun (a:Type) =>
                       @Sigma (List a) (fun (_ls : List a) => Int))) :=
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index fd95291e..1c1062c0 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -4,6 +4,7 @@ namespace Diverge
 
 open Lean Elab Term Meta
 
+-- We can't define and use trace classes in the same file
 initialize registerTraceClass `Diverge.elab
 initialize registerTraceClass `Diverge.def
 initialize registerTraceClass `Diverge.def.sigmas
@@ -11,8 +12,8 @@ initialize registerTraceClass `Diverge.def.genBody
 initialize registerTraceClass `Diverge.def.valid
 initialize registerTraceClass `Diverge.def.unfold
 
--- TODO: move
--- TODO: small helper
+-- Useful helper to explore definitions and figure out the variant
+-- of their sub-expressions.
 def explore_term (incr : String) (e : Expr) : MetaM Unit :=
   match e with
   | .bvar _ => do logInfo m!"{incr}bvar: {e}"; return ()
@@ -81,8 +82,7 @@ private def test2 (x : Nat) : Nat := x
 print_decl test1
 print_decl test2
 
--- We adapted this from AbstractNestedProofs.visit
--- A map visitor function for expressions
+-- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
 partial def mapVisit (k : Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
   let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
     let localInstances ← getLocalInstances
diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
index 1185a07d..117f76a2 100644
--- a/backends/lean/Base/Primitives.lean
+++ b/backends/lean/Base/Primitives.lean
@@ -60,12 +60,12 @@ instance Result_Nonempty (α : Type u) : Nonempty (Result α) :=
 
 /- HELPERS -/
 
-def ret? {α: Type} (r: Result α): Bool :=
+def ret? {α: Type u} (r: Result α): Bool :=
   match r with
   | ret _ => true
   | fail _ | div => false
 
-def div? {α: Type} (r: Result α): Bool :=
+def div? {α: Type u} (r: Result α): Bool :=
   match r with
   | div => true
   | ret _ | fail _ => false
@@ -73,14 +73,14 @@ def div? {α: Type} (r: Result α): Bool :=
 def massert (b:Bool) : Result Unit :=
   if b then ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
+def eval_global {α: Type u} (x: Result α) (_: ret? x): α :=
   match x with
   | fail _ | div => by contradiction
   | ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: Result α) (f: α -> Result β) : Result β :=
+def bind {α : Type u} {β : Type v} (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
@@ -111,23 +111,6 @@ def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
   | fail e => fail e
   | div => div
 
-macro "let" e:term " ⟵ " f:term : doElem =>
-  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
-
--- TODO: any way to factorize both definitions?
-macro "let" e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
-
--- We call the hypothesis `h`, in effect making it unavailable to the user
--- (because too much shadowing). But in practice, once can use the French single
--- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
--- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
-#eval do
-  let y <-- .ret (0: Nat)
-  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
-  let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
-  .ret r
-
 @[simp] theorem bind_tc_ret (x : α) (f : α → Result β) :
   (do let y ← .ret x; f y) = f x := by simp [Bind.bind, bind]
 
-- 
cgit v1.3.1


From bd873499f9a8d517cc948c6336a5c6ce856d846d Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Tue, 4 Jul 2023 17:30:35 +0200
Subject: Fix some issues with the extraction to Lean

---
 backends/lean/Base/Diverge/Elab.lean |  63 +++++++++++-----
 compiler/Extract.ml                  | 134 ++++++++++++++++++++++-------------
 2 files changed, 133 insertions(+), 64 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 41209021..4b08fe44 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -255,10 +255,11 @@ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
   preDefs.mapM fun preDef => do
     -- Replace the recursive calls
     let body ← mapVisit visit_e preDef.value
+    trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}"
 
     -- Currify the function by grouping the arguments into a dependent tuple
     -- (over which we match to retrieve the individual arguments).
-    lambdaLetTelescope body fun args body => do
+    lambdaTelescope body fun args body => do
     let body ← mkSigmasMatch args.toList body 0
 
     -- Add the declaration
@@ -376,15 +377,18 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
   | .sort _ => throwError "Unreachable"
   | .lam .. => throwError "Unimplemented"
   | .forallE .. => throwError "Unreachable" -- Shouldn't get there
-  | .letE dName dTy dValue body _nonDep => do
-    -- Introduce a local declaration for the let-binding
-    withLetDecl dName dTy dValue fun decl => do
+  | .letE .. => do
+    -- Telescope all the let-bindings (remark: this also telescopes the lambdas)
+    lambdaLetTelescope e fun xs body => do
+    -- Note that we don't visit the bound values: there shouldn't be
+    -- recursive calls, lambda expressions, etc. inside
+    -- Prove that the body is valid
     let isValid ← proveExprIsValid k_var kk_var body
-    -- Add the let-binding around.
+    -- Add the let-bindings around.
     -- Rem.: the let-binding should be *inside* the `is_valid_p`, not outside,
     -- but because it reduces in the end it doesn't matter. More precisely:
     -- `P (let x := v in y)` and `let x := v in P y` reduce to the same expression.
-    mkLetFVars #[decl] isValid
+    mkLambdaFVars xs isValid (usedLetOnly := false)
   | .mdata _ b => proveExprIsValid k_var kk_var b
   | .proj _ _ _ =>
     -- The projection shouldn't use the continuation
@@ -410,7 +414,7 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
           if isIte then proveExprIsValid k_var kk_var br
           else do
             -- There is a lambda -- TODO: how do we remove exacly *one* lambda?
-            lambdaLetTelescope br fun xs br => do
+            lambdaTelescope br fun xs br => do
             let x := xs.get! 0
             let xs := xs.extract 1 xs.size
             let br ← mkLambdaFVars xs br
@@ -518,7 +522,7 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
         let yValid ← do
           -- This is a lambda expression -- TODO: how do we remove exacly *one* lambda?
-          lambdaLetTelescope y fun xs y => do
+          lambdaTelescope y fun xs y => do
           let x := xs.get! 0
           let xs := xs.extract 1 xs.size
           let y ← mkLambdaFVars xs y
@@ -555,7 +559,7 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     -- binders might come from the match, and some of the binders might come
     -- from the fact that the expression in the match is a lambda expression:
     -- we use the branchesNumParams field for this reason
-    lambdaLetTelescope br fun xs br => do
+    lambdaTelescope br fun xs br => do
     let numParams := me.branchesNumParams.get! idx
     let xs_beg := xs.extract 0 numParams
     let xs_end := xs.extract numParams xs.size
@@ -622,7 +626,7 @@ partial def proveSingleBodyIsValid
   let env ← getEnv
   let body := (env.constants.find! name).value!
   trace[Diverge.def.valid] "body: {body}"
-  lambdaLetTelescope body fun xs body => do
+  lambdaTelescope body fun xs body => do
   assert! xs.size = 2
   let kk_var := xs.get! 0
   let x_var := xs.get! 1
@@ -695,8 +699,10 @@ def proveMutRecIsValid
   let bodiesValid ←
     bodies.mapIdxM fun idx body => do
       let preDef := preDefs.get! idx
+      trace[Diverge.def.valid] "## Proving that the body {body} is valid"
       proveSingleBodyIsValid k_var preDef body
   -- Then prove that the mut rec body is valid
+  trace[Diverge.def.valid] "## Proving that the 'Funs' body is valid"
   let isValid ← proveFunsBodyIsValid inOutTys bodyFuns k_var bodiesValid
   -- Save the theorem
   let thmTy ← mkAppM ``FixI.is_valid #[mutRecBodyConst]
@@ -724,7 +730,7 @@ def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
   TermElabM (Array Name) := do
   let grSize := preDefs.size
   let defs ← preDefs.mapIdxM fun idx preDef => do
-    lambdaLetTelescope preDef.value fun xs _ => do
+    lambdaTelescope preDef.value fun xs _ => do
     -- Create the index
     let idx ← mkFinVal grSize idx.val
     -- Group the inputs into a dependent tuple
@@ -755,7 +761,7 @@ partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinitio
     let preDef := preDefs.get! i
     let defName := decls.get! i
     -- Retrieve the arguments
-    lambdaLetTelescope preDef.value fun xs body => do
+    lambdaTelescope preDef.value fun xs body => do
     trace[Diverge.def.unfold] "proveUnfoldingThms: xs: {xs}"
     trace[Diverge.def.unfold] "proveUnfoldingThms: body: {body}"
     -- The theorem statement
@@ -799,7 +805,7 @@ partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinitio
 
 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
-  trace[Diverge.def] ("divRecursion: defs: " ++ msg)
+  trace[Diverge.def] ("divRecursion: defs:\n" ++ msg)
 
   -- TODO: what is this?
   for preDef in preDefs do
@@ -880,8 +886,11 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
 
   -- Replace the recursive calls in all the function bodies by calls to the
   -- continuation `k` and and generate for those bodies declarations
+  trace[Diverge.def] "# Generating  the unary bodies"
   let bodies ← mkDeclareUnaryBodies grLvlParams kk_var preDefs
+  trace[Diverge.def] "Unary bodies (after decl): {bodies}"
   -- Generate the mutually recursive body
+  trace[Diverge.def] "# Generating  the mut rec body"
   let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var in_ty out_ty inOutTys.toList bodies
   trace[Diverge.def] "mut rec body (after decl): {mutRecBody}"
 
@@ -889,15 +898,18 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
   -- our fixed-point
   let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
   withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do
+  trace[Diverge.def] "# Proving that the mut rec body is valid"
   let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
 
   -- Generate the final definitions
+  trace[Diverge.def] "# Generating the final definitions"
   let decls ← mkDeclareFixDefs mutRecBody preDefs
 
   -- Prove the unfolding theorems
+  trace[Diverge.def] "# Proving the unfolding theorems"
   proveUnfoldingThms isValidThm preDefs decls
 
-  -- Process the definitions - TODO
+  -- Generating code -- TODO
   addAndCompilePartialRec preDefs
 
 -- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
@@ -1064,13 +1076,32 @@ namespace Tests
 
   -- Testing dependent branching and let-bindings
   -- TODO: why the linter warning?
-  divergent def is_non_zero (i : Int) : Result Bool :=
+  divergent def isNonZero (i : Int) : Result Bool :=
     if _h:i = 0 then return false
     else
       let b := true
       return b
 
-  #check is_non_zero.unfold
+  #check isNonZero.unfold
+
+  -- Testing let-bindings
+  divergent def iInBounds {a : Type} (ls : List a) (i : Int) : Result Bool :=
+    let i0 := ls.length
+    if i < i0
+    then Result.ret True
+    else Result.ret False
+
+  #check iInBounds.unfold
+
+  divergent def isCons
+    {a : Type} (ls : List a) : Result Bool :=
+    let ls1 := ls
+    match ls1 with
+    | [] => Result.ret False
+    | x :: tl => Result.ret True
+
+  #check isCons.unfold
+
 end Tests
 
 end Diverge
diff --git a/compiler/Extract.ml b/compiler/Extract.ml
index a54a2299..b18d4743 100644
--- a/compiler/Extract.ml
+++ b/compiler/Extract.ml
@@ -618,9 +618,12 @@ let mk_formatter (ctx : trans_ctx) (crate_name : string)
   let struct_constructor (basename : name) : string =
     let tname = type_name basename in
     let prefix =
-      match !backend with FStar -> "Mk" | Lean | Coq | HOL4 -> "mk"
+      match !backend with FStar -> "Mk" | Coq | HOL4 -> "mk" | Lean -> ""
     in
-    prefix ^ tname
+    let suffix =
+      match !backend with FStar | Coq | HOL4 -> "" | Lean -> ".mk"
+    in
+    prefix ^ tname ^ suffix
   in
   let get_fun_name = get_name in
   let fun_name_to_snake_case (fname : fun_name) : string =
@@ -1326,7 +1329,8 @@ let extract_type_decl_struct_body (ctx : extraction_ctx) (fmt : F.formatter)
       F.pp_print_string fmt (unit_name ()))
     else if !backend = Lean && fields = [] then ()
       (* If the definition is recursive, we may need to extract it as an inductive
-         (instead of a record) *)
+         (instead of a record). We start with the "normal" case: we extract it
+         as a record. *)
     else if (not is_rec) || (!backend <> Coq && !backend <> Lean) then (
       if !backend <> Lean then F.pp_print_space fmt ();
       (* If Coq: print the constructor name *)
@@ -1379,7 +1383,14 @@ let extract_type_decl_struct_body (ctx : extraction_ctx) (fmt : F.formatter)
          a group of mutually recursive types: we extract it as an inductive type *)
       assert (is_rec && (!backend = Coq || !backend = Lean));
       let with_opaque_pre = false in
-      let cons_name = ctx_get_struct with_opaque_pre (AdtId def.def_id) ctx in
+      (* Small trick: in Lean we use namespaces, meaning we don't need to prefix
+         the constructor name with the name of the type at definition site,
+         i.e., instead of generating `inductive Foo := | MkFoo ...` like in Coq
+         we generate `inductive Foo := | mk ... *)
+      let cons_name =
+        if !backend = Lean then "mk"
+        else ctx_get_struct with_opaque_pre (AdtId def.def_id) ctx
+      in
       let def_name = ctx_get_local_type with_opaque_pre def.def_id ctx in
       extract_type_decl_variant ctx fmt type_decl_group def_name type_params
         cons_name fields)
@@ -1950,14 +1961,17 @@ let extract_global_decl_register_names (ctx : extraction_ctx)
     Note that patterns can introduce new variables: we thus return an extraction
     context updated with new bindings.
 
+    [is_single_pat]: are we extracting a single pattern (a pattern for a let-binding
+    or a lambda).
+
     TODO: we don't need something very generic anymore (some definitions used
     to be polymorphic).
  *)
 let extract_adt_g_value
     (extract_value : extraction_ctx -> bool -> 'v -> extraction_ctx)
-    (fmt : F.formatter) (ctx : extraction_ctx) (inside : bool)
-    (variant_id : VariantId.id option) (field_values : 'v list) (ty : ty) :
-    extraction_ctx =
+    (fmt : F.formatter) (ctx : extraction_ctx) (is_single_pat : bool)
+    (inside : bool) (variant_id : VariantId.id option) (field_values : 'v list)
+    (ty : ty) : extraction_ctx =
   match ty with
   | Adt (Tuple, type_args) ->
       (* Tuple *)
@@ -1982,36 +1996,57 @@ let extract_adt_g_value
         ctx)
   | Adt (adt_id, _) ->
       (* "Regular" ADT *)
-      (* We print something of the form: [Cons field0 ... fieldn].
-       * We could update the code to print something of the form:
-       * [{ field0=...; ...; fieldn=...; }] in case of structures.
-       *)
-      let cons =
-        (* The ADT shouldn't be opaque *)
-        let with_opaque_pre = false in
-        match variant_id with
-        | Some vid -> (
-            (* In the case of Lean, we might have to add the type name as a prefix *)
-            match (!backend, adt_id) with
-            | Lean, Assumed _ ->
-                ctx_get_type with_opaque_pre adt_id ctx
-                ^ "."
-                ^ ctx_get_variant adt_id vid ctx
-            | _ -> ctx_get_variant adt_id vid ctx)
-        | None -> ctx_get_struct with_opaque_pre adt_id ctx
-      in
-      let use_parentheses = inside && field_values <> [] in
-      if use_parentheses then F.pp_print_string fmt "(";
-      F.pp_print_string fmt cons;
-      let ctx =
-        Collections.List.fold_left
-          (fun ctx v ->
-            F.pp_print_space fmt ();
-            extract_value ctx true v)
-          ctx field_values
-      in
-      if use_parentheses then F.pp_print_string fmt ")";
-      ctx
+
+      (* If we are generating a pattern for a let-binding and we target Lean,
+         the syntax is: `let ⟨ x0, ..., xn ⟩ := ...`.
+
+         Otherwise, it is: `let Cons x0 ... xn = ...`
+      *)
+      if is_single_pat && !Config.backend = Lean then (
+        F.pp_print_string fmt "⟨";
+        F.pp_print_space fmt ();
+        let ctx =
+          Collections.List.fold_left_link
+            (fun _ ->
+              F.pp_print_string fmt ",";
+              F.pp_print_space fmt ())
+            (fun ctx v -> extract_value ctx true v)
+            ctx field_values
+        in
+        F.pp_print_space fmt ();
+        F.pp_print_string fmt "⟩";
+        ctx)
+      else
+        (* We print something of the form: [Cons field0 ... fieldn].
+         * We could update the code to print something of the form:
+         * [{ field0=...; ...; fieldn=...; }] in case of structures.
+         *)
+        let cons =
+          (* The ADT shouldn't be opaque *)
+          let with_opaque_pre = false in
+          match variant_id with
+          | Some vid -> (
+              (* In the case of Lean, we might have to add the type name as a prefix *)
+              match (!backend, adt_id) with
+              | Lean, Assumed _ ->
+                  ctx_get_type with_opaque_pre adt_id ctx
+                  ^ "."
+                  ^ ctx_get_variant adt_id vid ctx
+              | _ -> ctx_get_variant adt_id vid ctx)
+          | None -> ctx_get_struct with_opaque_pre adt_id ctx
+        in
+        let use_parentheses = inside && field_values <> [] in
+        if use_parentheses then F.pp_print_string fmt "(";
+        F.pp_print_string fmt cons;
+        let ctx =
+          Collections.List.fold_left
+            (fun ctx v ->
+              F.pp_print_space fmt ();
+              extract_value ctx true v)
+            ctx field_values
+        in
+        if use_parentheses then F.pp_print_string fmt ")";
+        ctx
   | _ -> raise (Failure "Inconsistent typed value")
 
 (* Extract globals in the same way as variables *)
@@ -2026,7 +2061,7 @@ let extract_global (ctx : extraction_ctx) (fmt : F.formatter)
     updated with new bindings.
  *)
 let rec extract_typed_pattern (ctx : extraction_ctx) (fmt : F.formatter)
-    (inside : bool) (v : typed_pattern) : extraction_ctx =
+    (is_let : bool) (inside : bool) (v : typed_pattern) : extraction_ctx =
   match v.value with
   | PatConstant cv ->
       ctx.fmt.extract_primitive_value fmt inside cv;
@@ -2042,8 +2077,10 @@ let rec extract_typed_pattern (ctx : extraction_ctx) (fmt : F.formatter)
       F.pp_print_string fmt "_";
       ctx
   | PatAdt av ->
-      let extract_value ctx inside v = extract_typed_pattern ctx fmt inside v in
-      extract_adt_g_value extract_value fmt ctx inside av.variant_id
+      let extract_value ctx inside v =
+        extract_typed_pattern ctx fmt is_let inside v
+      in
+      extract_adt_g_value extract_value fmt ctx is_let inside av.variant_id
         av.field_values v.ty
 
 (** [inside]: controls the introduction of parentheses. See [extract_ty]
@@ -2173,12 +2210,13 @@ and extract_adt_cons (ctx : extraction_ctx) (fmt : F.formatter) (inside : bool)
     (adt_cons : adt_cons_id) (type_args : ty list) (args : texpression list) :
     unit =
   let e_ty = Adt (adt_cons.adt_id, type_args) in
+  let is_single_pat = false in
   let _ =
     extract_adt_g_value
       (fun ctx inside e ->
         extract_texpression ctx fmt inside e;
         ctx)
-      fmt ctx inside adt_cons.variant_id args e_ty
+      fmt ctx is_single_pat inside adt_cons.variant_id args e_ty
   in
   ()
 
@@ -2226,7 +2264,7 @@ and extract_Abs (ctx : extraction_ctx) (fmt : F.formatter) (inside : bool)
     List.fold_left
       (fun ctx x ->
         F.pp_print_space fmt ();
-        extract_typed_pattern ctx fmt true x)
+        extract_typed_pattern ctx fmt true true x)
       ctx xl
   in
   F.pp_print_space fmt ();
@@ -2295,7 +2333,7 @@ and extract_lets (ctx : extraction_ctx) (fmt : F.formatter) (inside : bool)
        * TODO: cleanup
        * *)
       if monadic && (!backend = Coq || !backend = HOL4) then (
-        let ctx = extract_typed_pattern ctx fmt true lv in
+        let ctx = extract_typed_pattern ctx fmt true true lv in
         F.pp_print_space fmt ();
         let arrow =
           match !backend with
@@ -2321,7 +2359,7 @@ and extract_lets (ctx : extraction_ctx) (fmt : F.formatter) (inside : bool)
         else (
           F.pp_print_string fmt "let";
           F.pp_print_space fmt ());
-        let ctx = extract_typed_pattern ctx fmt true lv in
+        let ctx = extract_typed_pattern ctx fmt true true lv in
         F.pp_print_space fmt ();
         let eq =
           match !backend with
@@ -2468,7 +2506,7 @@ and extract_Switch (ctx : extraction_ctx) (fmt : F.formatter) (_inside : bool)
         match !backend with
         | FStar -> "begin match"
         | Coq -> "match"
-        | Lean -> "match h:"
+        | Lean -> if ctx.use_dep_ite then "match h:" else "match"
         | HOL4 ->
             (* We're being extra safe in the case of HOL4 *)
             "(case"
@@ -2495,7 +2533,7 @@ and extract_Switch (ctx : extraction_ctx) (fmt : F.formatter) (_inside : bool)
         (* Print the pattern *)
         F.pp_print_string fmt "|";
         F.pp_print_space fmt ();
-        let ctx = extract_typed_pattern ctx fmt false br.pat in
+        let ctx = extract_typed_pattern ctx fmt false false br.pat in
         F.pp_print_space fmt ();
         let arrow =
           match !backend with FStar -> "->" | Coq | Lean | HOL4 -> "=>"
@@ -2687,7 +2725,7 @@ let extract_fun_parameters (space : bool ref) (ctx : extraction_ctx)
             (* Open a box for the input parameter *)
             F.pp_open_hovbox fmt 0;
             F.pp_print_string fmt "(";
-            let ctx = extract_typed_pattern ctx fmt false lv in
+            let ctx = extract_typed_pattern ctx fmt true false lv in
             F.pp_print_space fmt ();
             F.pp_print_string fmt ":";
             F.pp_print_space fmt ();
@@ -3032,7 +3070,7 @@ let extract_fun_decl_gen (ctx : extraction_ctx) (fmt : F.formatter)
         List.fold_left
           (fun ctx (lv : typed_pattern) ->
             F.pp_print_space fmt ();
-            let ctx = extract_typed_pattern ctx fmt false lv in
+            let ctx = extract_typed_pattern ctx fmt true false lv in
             ctx)
           ctx inputs_lvs
       in
-- 
cgit v1.3.1


From 442caaf62e4a217b9a10116c4e529c49f83c4efd Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Tue, 4 Jul 2023 22:45:02 +0200
Subject: Fix an issue with mkSigmasVal

---
 backends/lean/Base/Diverge/Elab.lean     | 228 +++++++++++++++++++------------
 backends/lean/Base/Diverge/ElabBase.lean |  47 ++++---
 backends/lean/lake-manifest.json         |   2 +-
 tests/lean/lake-manifest.json            |   2 +-
 4 files changed, 171 insertions(+), 108 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 4b08fe44..1af06fea 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -26,38 +26,42 @@ def mkInOutTy (x y : Expr) : MetaM Expr :=
   mkAppM ``FixI.mk_in_out_ty #[x, y]
 
 -- Return the `a` in `Return a`
-def get_result_ty (ty : Expr) : MetaM Expr :=
+def getResultTy (ty : Expr) : MetaM Expr :=
   ty.withApp fun f args => do
   if ¬ f.isConstOf ``Result ∨ args.size ≠ 1 then
-    throwError "Invalid argument to get_result_ty: {ty}"
+    throwError "Invalid argument to getResultTy: {ty}"
   else
     pure (args.get! 0)
 
-/- Group a list of expressions into a dependent tuple.
+/- Deconstruct a sigma type.
 
-   Example:
-   xl = [`a : Type`, `ls : List a`]
-   returns:
-   `⟨ (a:Type), (ls: List a) ⟩`
+   For instance, deconstructs `(a : Type) × List a` into
+   `Type` and `λ a => List a`.
  -/
-def mkSigmasVal (xl : List Expr) : MetaM Expr :=
-  match xl with
-  | [] => do
-    trace[Diverge.def.sigmas] "mkSigmasVal: []"
-    pure (Expr.const ``PUnit.unit [])
-  | [x] => do
-    trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
-    pure x
-  | fst :: xl => do
-    trace[Diverge.def.sigmas] "mkSigmasVal: [{fst}::{xl}]"
-    let alpha ← Lean.Meta.inferType fst
-    let snd ← mkSigmasVal xl
-    let snd_ty ← inferType snd
-    let beta ← mkLambdaFVars #[fst] snd_ty
-    trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
-    mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
-
-/- Generate a Sigma type from a list of expressions.
+def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
+  ty.withApp fun f args => do
+  if ¬ f.isConstOf ``Sigma ∨ args.size ≠ 2 then
+    throwError "Invalid argument to getSigmaTypes: {ty}"
+  else
+    pure (args.get! 0, args.get! 1)
+
+/- Like `lambdaTelescopeN` but only destructs a fixed number of lambdas -/
+def lambdaTelescopeN (e : Expr) (n : Nat) (k : Array Expr → Expr → MetaM α) : MetaM α :=
+  lambdaTelescope e fun xs body => do
+  if xs.size < n then throwError "lambdaTelescopeN: not enough lambdas";
+  let xs := xs.extract 0 n
+  let ys := xs.extract n xs.size
+  let body ← mkLambdaFVars ys body
+  k xs body
+
+/- Like `lambdaTelescope`, but only destructs one lambda
+   TODO: is there an equivalent of this function somewhere in the
+   standard library? -/
+def lambdaOne (e : Expr) (k : Expr → Expr → MetaM α) : MetaM α :=
+  lambdaTelescopeN e 1 λ xs b => k (xs.get! 0) b
+
+/- Generate a Sigma type from a list of *variables* (all the expressions
+   must be variables).
 
    Example:
    - xl = [(a:Type), (ls:List a), (i:Int)]
@@ -84,6 +88,53 @@ def mkSigmasType (xl : List Expr) : MetaM Expr :=
     trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
     mkAppOptM ``Sigma #[some alpha, some beta]
 
+/- Apply a lambda expression to some arguments, simplifying the lambdas -/
+def applyLambdaToArgs (e : Expr) (xs : Array Expr) : MetaM Expr := do
+  lambdaTelescopeN e xs.size fun vars body =>
+  -- Create the substitution
+  let s : HashMap FVarId Expr := HashMap.ofList (List.zip (vars.toList.map Expr.fvarId!) xs.toList)
+  -- Substitute in the body
+  pure (body.replace fun e =>
+    match e with
+    | Expr.fvar fvarId => match s.find? fvarId with
+      | none   => e
+      | some v => v
+    | _ => none)
+
+/- Group a list of expressions into a dependent tuple.
+
+   Example:
+   xl = [`a : Type`, `ls : List a`]
+   returns:
+   `⟨ (a:Type), (ls: List a) ⟩`
+
+   We need the type argument because as the elements in the tuple are
+   "concrete", we can't in all generality figure out the type of the tuple.
+
+   Example:
+   `⟨ True, 3 ⟩ : (x : Bool) × (if x then Int else Unit)`
+ -/
+def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: []"
+    pure (Expr.const ``PUnit.unit [])
+  | [x] => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
+    pure x
+  | fst :: xl => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{fst}::{xl}]"
+    -- Deconstruct the type
+    let (alpha, beta) ← getSigmaTypes ty
+    -- Compute the "second" field
+    -- Specialize beta for fst
+    let nty ← applyLambdaToArgs beta #[fst]
+    -- Recursive call
+    let snd ← mkSigmasVal nty xl
+    -- Put everything together
+    trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
+    mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
+
 def mkAnonymous (s : String) (i : Nat) : Name :=
   .num (.str .anonymous s) i
 
@@ -208,52 +259,57 @@ def mkFinVal (n i : Nat) : MetaM Expr := do
    We return the new declarations.
  -/
 def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
-  (preDefs : Array PreDefinition) :
+  (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
   MetaM (Array Expr) := do
   let grSize := preDefs.size
 
-  -- Compute the map from name to index - the continuation has an indexed type:
-  -- we use the index (a finite number of type `Fin`) to control which function
-  -- we call at the recursive call site.
-  let nameToId : HashMap Name Nat :=
-    let namesIds := preDefs.mapIdx (fun i d => (d.declName, i.val))
-    HashMap.ofList namesIds.toList
+  -- Compute the map from name to (index × input type).
+  -- Remark: the continuation has an indexed type; we use the index (a finite number of
+  -- type `Fin`) to control which function we call at the recursive call site.
+  let nameToInfo : HashMap Name (Nat × Expr) :=
+    let bl := preDefs.mapIdx fun i d => (d.declName, (i.val, (inOutTys.get! i.val).fst))
+    HashMap.ofList bl.toList
 
-  trace[Diverge.def.genBody] "nameToId: {nameToId.toList}"
+  trace[Diverge.def.genBody] "nameToId: {nameToInfo.toList}"
 
   -- Auxiliary function to explore the function bodies and replace the
   -- recursive calls
-  let visit_e (e : Expr) : MetaM Expr := do
-    trace[Diverge.def.genBody] "visiting expression: {e}"
-    match e with
-    | .app .. => do
-      e.withApp fun f args => do
-        trace[Diverge.def.genBody] "this is an app: {f} {args}"
-        -- Check if this is a recursive call
-        if f.isConst then
-          let name := f.constName!
-          match nameToId.find? name with
-          | none => pure e
-          | some id =>
-            -- This is a recursive call: replace it
-            -- Compute the index
-            let i ← mkFinVal grSize id
-            -- Put the arguments in one big dependent tuple
-            let args ← mkSigmasVal args.toList
-            mkAppM' kk_var #[i, args]
-        else
-          -- Not a recursive call: do nothing
-          pure e
-     | .const name _ =>
-       -- Sanity check: we eliminated all the recursive calls
-       if (nameToId.find? name).isSome then
-         throwError "mkUnaryBodies: a recursive call was not eliminated"
-       else pure e
-     | _ => pure e
+  let visit_e (i : Nat) (e : Expr) : MetaM Expr := do
+    trace[Diverge.def.genBody] "visiting expression (dept: {i}): {e}"
+    let ne ← do
+      match e with
+      | .app .. => do
+        e.withApp fun f args => do
+          trace[Diverge.def.genBody] "this is an app: {f} {args}"
+          -- Check if this is a recursive call
+          if f.isConst then
+            let name := f.constName!
+            match nameToInfo.find? name with
+            | none => pure e
+            | some (id, in_ty) =>
+              trace[Diverge.def.genBody] "this is a recursive call"
+              -- This is a recursive call: replace it
+              -- Compute the index
+              let i ← mkFinVal grSize id
+              -- Put the arguments in one big dependent tuple
+              let args ← mkSigmasVal in_ty args.toList
+              mkAppM' kk_var #[i, args]
+          else
+            -- Not a recursive call: do nothing
+            pure e
+       | .const name _ =>
+         -- Sanity check: we eliminated all the recursive calls
+         if (nameToInfo.find? name).isSome then
+           throwError "mkUnaryBodies: a recursive call was not eliminated"
+         else pure e
+       | _ => pure e
+    trace[Diverge.def.genBody] "done with expression (depth: {i}): {e}"
+    pure ne
 
   -- Explore the bodies
   preDefs.mapM fun preDef => do
     -- Replace the recursive calls
+    trace[Diverge.def.genBody] "About to replace recursive calls in {preDef.declName}"
     let body ← mapVisit visit_e preDef.value
     trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}"
 
@@ -413,11 +469,8 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         let proveBranchValid (br : Expr) : MetaM Expr :=
           if isIte then proveExprIsValid k_var kk_var br
           else do
-            -- There is a lambda -- TODO: how do we remove exacly *one* lambda?
-            lambdaTelescope br fun xs br => do
-            let x := xs.get! 0
-            let xs := xs.extract 1 xs.size
-            let br ← mkLambdaFVars xs br
+            -- There is a lambda
+            lambdaOne br fun x br => do
             let brValid ← proveExprIsValid k_var kk_var br
             mkLambdaFVars #[x] brValid
         let br0Valid ← proveBranchValid br0
@@ -521,11 +574,8 @@ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
         let xValid ← proveExprIsValid k_var kk_var x
         trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
         let yValid ← do
-          -- This is a lambda expression -- TODO: how do we remove exacly *one* lambda?
-          lambdaTelescope y fun xs y => do
-          let x := xs.get! 0
-          let xs := xs.extract 1 xs.size
-          let y ← mkLambdaFVars xs y
+          -- This is a lambda expression
+          lambdaOne y fun x y => do
           trace[Diverge.def.valid] "bind: y: {y}"
           let yValid ← proveExprIsValid k_var kk_var y
           trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
@@ -559,15 +609,12 @@ partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Exp
     -- binders might come from the match, and some of the binders might come
     -- from the fact that the expression in the match is a lambda expression:
     -- we use the branchesNumParams field for this reason
-    lambdaTelescope br fun xs br => do
     let numParams := me.branchesNumParams.get! idx
-    let xs_beg := xs.extract 0 numParams
-    let xs_end := xs.extract numParams xs.size
-    let br ← mkLambdaFVars xs_end br
+    lambdaTelescopeN br numParams fun xs br => do
     -- Prove that the branch expression is valid
     let brValid ← proveExprIsValid k_var kk_var br
     -- Reconstruct the lambda expression
-    mkLambdaFVars xs_beg brValid
+    mkLambdaFVars xs brValid
   trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
   -- Compute the motive, which has the following shape:
   -- ```
@@ -726,15 +773,17 @@ def proveMutRecIsValid
 -- def is_even (i : Int) : Result Bool := mut_rec_body 0 i
 -- def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
 -- ```
-def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
+def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
   TermElabM (Array Name) := do
   let grSize := preDefs.size
   let defs ← preDefs.mapIdxM fun idx preDef => do
     lambdaTelescope preDef.value fun xs _ => do
+    -- Retrieve the input type
+    let in_ty := (inOutTys.get! idx.val).fst
     -- Create the index
     let idx ← mkFinVal grSize idx.val
     -- Group the inputs into a dependent tuple
-    let input ← mkSigmasVal xs.toList
+    let input ← mkSigmasVal in_ty xs.toList
     -- Apply the fixed point
     let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
     let fixedBody ← mkLambdaFVars xs fixedBody
@@ -754,8 +803,8 @@ def mkDeclareFixDefs (mutRecBody : Expr) (preDefs : Array PreDefinition) :
   pure defs
 
 -- Prove the equations that we will use as unfolding theorems
-partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinition)
-  (decls : Array Name) : MetaM Unit := do
+partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Expr))
+  (preDefs : Array PreDefinition) (decls : Array Name) : MetaM Unit := do
   let grSize := preDefs.size
   let proveIdx (i : Nat) : MetaM Unit := do
     let preDef := preDefs.get! i
@@ -779,7 +828,7 @@ partial def proveUnfoldingThms (isValidThm : Expr) (preDefs : Array PreDefinitio
     let idx ← mkFinVal grSize i
     let proof ← mkAppM ``congr_fun #[proof, idx]
     -- Add the input argument
-    let arg ← mkSigmasVal xs.toList
+    let arg ← mkSigmasVal (inOutTys.get! i).fst xs.toList
     let proof ← mkAppM ``congr_fun #[proof, arg]
     -- Abstract the arguments away
     let proof ← mkLambdaFVars xs proof
@@ -833,7 +882,7 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
         forallTelescope preDef.type (fun in_tys out_ty => do
           let in_ty ← liftM (mkSigmasType in_tys.toList)
           -- Retrieve the type in the "Result"
-          let out_ty ← get_result_ty out_ty
+          let out_ty ← getResultTy out_ty
           let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
           pure (in_ty, out_ty)
         )
@@ -886,8 +935,8 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
 
   -- Replace the recursive calls in all the function bodies by calls to the
   -- continuation `k` and and generate for those bodies declarations
-  trace[Diverge.def] "# Generating  the unary bodies"
-  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var preDefs
+  trace[Diverge.def] "# Generating the unary bodies"
+  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var inOutTys preDefs
   trace[Diverge.def] "Unary bodies (after decl): {bodies}"
   -- Generate the mutually recursive body
   trace[Diverge.def] "# Generating  the mut rec body"
@@ -903,11 +952,11 @@ def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
 
   -- Generate the final definitions
   trace[Diverge.def] "# Generating the final definitions"
-  let decls ← mkDeclareFixDefs mutRecBody preDefs
+  let decls ← mkDeclareFixDefs mutRecBody inOutTys preDefs
 
   -- Prove the unfolding theorems
   trace[Diverge.def] "# Proving the unfolding theorems"
-  proveUnfoldingThms isValidThm preDefs decls
+  proveUnfoldingThms isValidThm inOutTys preDefs decls
 
   -- Generating code -- TODO
   addAndCompilePartialRec preDefs
@@ -1102,6 +1151,15 @@ namespace Tests
 
   #check isCons.unfold
 
+  -- Testing what happens when we use concrete arguments in dependent tuples
+  divergent def test1
+    (_ : Option Bool) (_ : Unit) :
+    Result Unit
+    :=
+    test1 Option.none ()
+
+  #check test1.unfold
+
 end Tests
 
 end Diverge
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
index 1c1062c0..aaaea6f7 100644
--- a/backends/lean/Base/Diverge/ElabBase.lean
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -83,7 +83,10 @@ print_decl test1
 print_decl test2
 
 -- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
-partial def mapVisit (k : Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+-- The continuation takes as parameters:
+-- - the current depth of the expression (useful for printing/debugging)
+-- - the expression to explore
+partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
   let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
     let localInstances ← getLocalInstances
     let mut lctx ← getLCtx
@@ -98,25 +101,27 @@ partial def mapVisit (k : Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
       lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
     withLCtx lctx localInstances k2
   -- TODO: use a cache? (Lean.checkCache)
-  -- Explore
-  let e ← k e
-  match e with
-  | .bvar _
-  | .fvar _
-  | .mvar _
-  | .sort _
-  | .lit _
-  | .const _ _ => pure e
-  | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (mapVisit k))
-  | .lam .. =>
-    lambdaLetTelescope e fun xs b =>
-      mapVisitBinders xs do mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
-  | .forallE .. => do
-    forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← mapVisit k b)
-  | .letE .. => do
-    lambdaLetTelescope e fun xs b => mapVisitBinders xs do
-      mkLambdaFVars xs (← mapVisit k b) (usedLetOnly := false)
-  | .mdata _ b => return e.updateMData! (← mapVisit k b)
-  | .proj _ _ b => return e.updateProj! (← mapVisit k b)
+  let rec visit (i : Nat) (e : Expr) : MetaM Expr := do
+    -- Explore
+    let e ← k i e
+    match e with
+    | .bvar _
+    | .fvar _
+    | .mvar _
+    | .sort _
+    | .lit _
+    | .const _ _ => pure e
+    | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (visit (i + 1)))
+    | .lam .. =>
+      lambdaLetTelescope e fun xs b =>
+        mapVisitBinders xs do mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .forallE .. => do
+      forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← visit (i + 1) b)
+    | .letE .. => do
+      lambdaLetTelescope e fun xs b => mapVisitBinders xs do
+        mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .mdata _ b => return e.updateMData! (← visit (i + 1) b)
+    | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
+  visit 0 e
 
 end Diverge
diff --git a/backends/lean/lake-manifest.json b/backends/lean/lake-manifest.json
index 40eb1682..f4759ad3 100644
--- a/backends/lean/lake-manifest.json
+++ b/backends/lean/lake-manifest.json
@@ -10,7 +10,7 @@
   {"git":
    {"url": "https://github.com/leanprover-community/mathlib4.git",
     "subDir?": null,
-    "rev": "4f103b3696795c62e76fb89d177efb91c29afdf5",
+    "rev": "cc5d11f24e1b92db65ec3389bb5142f4b2d7670e",
     "name": "mathlib",
     "inputRev?": null}},
   {"git":
diff --git a/tests/lean/lake-manifest.json b/tests/lean/lake-manifest.json
index ccd6d61a..637bda23 100644
--- a/tests/lean/lake-manifest.json
+++ b/tests/lean/lake-manifest.json
@@ -11,7 +11,7 @@
   {"git":
    {"url": "https://github.com/leanprover-community/mathlib4.git",
     "subDir?": null,
-    "rev": "409bee4eabf8072c4569950c3c2f310afd203abf",
+    "rev": "cc5d11f24e1b92db65ec3389bb5142f4b2d7670e",
     "name": "mathlib",
     "inputRev?": null}},
   {"git":
-- 
cgit v1.3.1


From 2496a08691809683e256af7c479588a2fae8e3d7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 6 Jul 2023 14:23:21 +0200
Subject: Register the unfolding theorems in the Lean equation compilers and
 solve a "unused variable" warning

---
 backends/lean/Base/Diverge/Elab.lean | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 1af06fea..e5b39440 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -843,6 +843,8 @@ partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Ex
       all := [name]
     }
     addDecl decl
+    -- Add the unfolding theorem to the equation compiler
+    eqnsAttribute.add preDef.declName #[name]
     trace[Diverge.def.unfold] "proveUnfoldingThms: added thm: {name}:\n{thmTy}"
   let rec prove (i : Nat) : MetaM Unit := do
     if i = preDefs.size then pure ()
@@ -1011,6 +1013,13 @@ def Term.elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM U
     withFunLocalDecls headers fun funFVars => do
       for view in views, funFVar in funFVars do
         addLocalVarInfo view.declId funFVar
+        -- Add fake use site to prevent "unused variable" warning (if the
+        -- function is actually not recursive, Lean would print this warning).
+        -- Remark: we could detect this case and encode the function without
+        -- using the fixed-point. In practice it shouldn't happen however:
+        -- we define non-recursive functions with the `divergent` keyword
+        -- only for testing purposes.
+        addTermInfo' view.declId funFVar
       let values ←
         try
           let values ← elabFunValues headers
@@ -1091,7 +1100,8 @@ namespace Tests
     . intro i hpos h; simp at h; linarith
     . rename_i hd tl ih
       intro i hpos h
-      rw [list_nth.unfold]; simp
+      -- We can directly use `rw [list_nth]`!
+      rw [list_nth]; simp
       split <;> simp [*]
       . tauto
       . -- TODO: we shouldn't have to do that
@@ -1147,7 +1157,7 @@ namespace Tests
     let ls1 := ls
     match ls1 with
     | [] => Result.ret False
-    | x :: tl => Result.ret True
+    | _ :: _ => Result.ret True
 
   #check isCons.unfold
 
-- 
cgit v1.3.1


From 9515bbad5b58ed1c51ac6d9fc9d7a4e5884b6273 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Thu, 6 Jul 2023 15:23:53 +0200
Subject: Reorganize a bit the lean backend files

---
 backends/lean/Base/Diverge/Elab.lean |   2 +
 backends/lean/Base/Utils.lean        | 119 +++++++++++++++++++++++++++++++++++
 2 files changed, 121 insertions(+)
 create mode 100644 backends/lean/Base/Utils.lean

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index e5b39440..96f7abc0 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -2,6 +2,7 @@ import Lean
 import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
+import Base.Utils
 import Base.Diverge.Base
 import Base.Diverge.ElabBase
 
@@ -13,6 +14,7 @@ namespace Diverge
 syntax (name := divergentDef)
   declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command
 
+open Utils
 open Lean Elab Term Meta Primitives Lean.Meta
 
 /- The following was copied from the `wfRecursion` function. -/
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
new file mode 100644
index 00000000..161b9ddb
--- /dev/null
+++ b/backends/lean/Base/Utils.lean
@@ -0,0 +1,119 @@
+import Lean
+
+namespace Utils
+
+open Lean Elab Term Meta
+
+-- Useful helper to explore definitions and figure out the variant
+-- of their sub-expressions.
+def explore_term (incr : String) (e : Expr) : MetaM Unit :=
+  match e with
+  | .bvar _ => do logInfo m!"{incr}bvar: {e}"; return ()
+  | .fvar _ => do logInfo m!"{incr}fvar: {e}"; return ()
+  | .mvar _ => do logInfo m!"{incr}mvar: {e}"; return ()
+  | .sort _ => do logInfo m!"{incr}sort: {e}"; return ()
+  | .const _ _ => do logInfo m!"{incr}const: {e}"; return ()
+  | .app fn arg => do
+    logInfo m!"{incr}app: {e}"
+    explore_term (incr ++ "  ") fn
+    explore_term (incr ++ "  ") arg
+  | .lam _bName bTy body _binfo => do
+    logInfo m!"{incr}lam: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .forallE _bName bTy body _bInfo => do
+    logInfo m!"{incr}forallE: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .letE _dName ty val body _nonDep => do
+    logInfo m!"{incr}letE: {e}"
+    explore_term (incr ++ "  ") ty
+    explore_term (incr ++ "  ") val
+    explore_term (incr ++ "  ") body
+  | .lit _ => do logInfo m!"{incr}lit: {e}"; return ()
+  | .mdata _ e => do
+    logInfo m!"{incr}mdata: {e}"
+    explore_term (incr ++ "  ") e
+  | .proj _ _ struct => do
+    logInfo m!"{incr}proj: {e}"
+    explore_term (incr ++ "  ") struct
+
+def explore_decl (n : Name) : TermElabM Unit := do
+  logInfo m!"Name: {n}"
+  let env ← getEnv
+  let decl := env.constants.find! n
+  match decl with
+  | .defnInfo val =>
+     logInfo m!"About to explore defn: {decl.name}"
+     logInfo m!"# Type:"
+     explore_term "" val.type
+     logInfo m!"# Value:"
+     explore_term "" val.value
+  | .axiomInfo _  => throwError m!"axiom: {n}"
+  | .thmInfo _    => throwError m!"thm: {n}"
+  | .opaqueInfo _ => throwError m!"opaque: {n}"
+  | .quotInfo _   => throwError m!"quot: {n}"
+  | .inductInfo _ => throwError m!"induct: {n}"
+  | .ctorInfo _   => throwError m!"ctor: {n}"
+  | .recInfo _    => throwError m!"rec: {n}"
+
+syntax (name := printDecl) "print_decl " ident : command
+
+open Lean.Elab.Command
+
+@[command_elab printDecl] def elabPrintDecl : CommandElab := fun stx => do
+  liftTermElabM do
+    let id := stx[1]
+    addCompletionInfo <| CompletionInfo.id id id.getId (danglingDot := false) {} none
+    let cs ← resolveGlobalConstWithInfos id
+    explore_decl cs[0]!
+
+private def test1 : Nat := 0
+private def test2 (x : Nat) : Nat := x
+
+print_decl test1
+print_decl test2
+
+-- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- The continuation takes as parameters:
+-- - the current depth of the expression (useful for printing/debugging)
+-- - the expression to explore
+partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+  let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
+    let localInstances ← getLocalInstances
+    let mut lctx ← getLCtx
+    for x in xs do
+      let xFVarId := x.fvarId!
+      let localDecl ← xFVarId.getDecl
+      let type      ← mapVisit k localDecl.type
+      let localDecl := localDecl.setType type
+      let localDecl ← match localDecl.value? with
+         | some value => let value ← mapVisit k value; pure <| localDecl.setValue value
+         | none       => pure localDecl
+      lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
+    withLCtx lctx localInstances k2
+  -- TODO: use a cache? (Lean.checkCache)
+  let rec visit (i : Nat) (e : Expr) : MetaM Expr := do
+    -- Explore
+    let e ← k i e
+    match e with
+    | .bvar _
+    | .fvar _
+    | .mvar _
+    | .sort _
+    | .lit _
+    | .const _ _ => pure e
+    | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (visit (i + 1)))
+    | .lam .. =>
+      lambdaLetTelescope e fun xs b =>
+        mapVisitBinders xs do mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .forallE .. => do
+      forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← visit (i + 1) b)
+    | .letE .. => do
+      lambdaLetTelescope e fun xs b => mapVisitBinders xs do
+        mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .mdata _ b => return e.updateMData! (← visit (i + 1) b)
+    | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
+  visit 0 e
+
+end Utils
-- 
cgit v1.3.1


From 7206b48a73d6204baea99f4f4675be2518a8f8c2 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Mon, 10 Jul 2023 15:06:12 +0200
Subject: Start working on the progress tactic

---
 backends/lean/Base.lean              |   2 +
 backends/lean/Base/Arith.lean        | 464 -----------------------------------
 backends/lean/Base/Arith/Arith.lean  | 409 ++++++++++++++++++++++++++++++
 backends/lean/Base/Arith/Base.lean   |  10 +
 backends/lean/Base/ArithBase.lean    |  10 -
 backends/lean/Base/Diverge/Elab.lean |  17 +-
 backends/lean/Base/Utils.lean        |  95 +++++++
 7 files changed, 517 insertions(+), 490 deletions(-)
 delete mode 100644 backends/lean/Base/Arith.lean
 create mode 100644 backends/lean/Base/Arith/Arith.lean
 create mode 100644 backends/lean/Base/Arith/Base.lean
 delete mode 100644 backends/lean/Base/ArithBase.lean

(limited to 'backends/lean/Base/Diverge/Elab.lean')

diff --git a/backends/lean/Base.lean b/backends/lean/Base.lean
index 1f8cbc8e..51211704 100644
--- a/backends/lean/Base.lean
+++ b/backends/lean/Base.lean
@@ -1,3 +1,5 @@
+import Base.Utils
 import Base.Primitives
 import Base.Diverge
 import Base.Arith
+import Base.Progress
diff --git a/backends/lean/Base/Arith.lean b/backends/lean/Base/Arith.lean
deleted file mode 100644
index 364447ed..00000000
--- a/backends/lean/Base/Arith.lean
+++ /dev/null
@@ -1,464 +0,0 @@
-/- This file contains tactics to solve arithmetic goals -/
-
-import Lean
-import Lean.Meta.Tactic.Simp
-import Init.Data.List.Basic
-import Mathlib.Tactic.RunCmd
-import Mathlib.Tactic.Linarith
--- TODO: there is no Omega tactic for now - it seems it hasn't been ported yet
---import Mathlib.Tactic.Omega
-import Base.Primitives
-import Base.ArithBase
-
-/-
-Mathlib tactics:
-- rcases: https://leanprover-community.github.io/mathlib_docs/tactics.html#rcases
-- split_ifs: https://leanprover-community.github.io/mathlib_docs/tactics.html#split_ifs
-- norm_num: https://leanprover-community.github.io/mathlib_docs/tactics.html#norm_num
-- should we use linarith or omega?
-- hint: https://leanprover-community.github.io/mathlib_docs/tactics.html#hint
-- classical: https://leanprover-community.github.io/mathlib_docs/tactics.html#classical
--/
-
-namespace List
-
-  -- TODO: I could not find this function??
-  @[simp] def flatten {a : Type u} : List (List a) → List a
-  | [] => []
-  | x :: ls => x ++ flatten ls
-
-end List
-
-namespace Lean
-
-namespace LocalContext
-
-  open Lean Lean.Elab Command Term Lean.Meta
-
-  -- Small utility: return the list of declarations in the context, from
-  -- the last to the first.
-  def getAllDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) :=
-    lctx.foldrM (fun d ls => do let d ← instantiateLocalDeclMVars d; pure (d :: ls)) []
-
-  -- Return the list of declarations in the context, but filter the
-  -- declarations which are considered as implementation details
-  def getDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) := do
-    let ls ← lctx.getAllDecls
-    pure (ls.filter (fun d => not d.isImplementationDetail))
-
-end LocalContext
-
-end Lean
-
-namespace Arith
-
-open Primitives
-
--- TODO: move?
-theorem ne_zero_is_lt_or_gt {x : Int} (hne : x ≠ 0) : x < 0 ∨ x > 0 := by
-  cases h: x <;> simp_all
-  . rename_i n;
-    cases n <;> simp_all
-  . apply Int.negSucc_lt_zero
-
--- TODO: move?
-theorem ne_is_lt_or_gt {x y : Int} (hne : x ≠ y) : x < y ∨ x > y := by
-  have hne : x - y ≠ 0 := by
-    simp
-    intro h
-    have: x = y := by linarith
-    simp_all
-  have h := ne_zero_is_lt_or_gt hne
-  match h with
-  | .inl _ => left; linarith
-  | .inr _ => right; linarith
-
--- TODO: move
-instance Vec.cast (a : Type): Coe (Vec a) (List a)  where coe := λ v => v.val
-
--- TODO: move
-/- Remark: we can't write the following instance because of restrictions about
-   the type class parameters (`ty` doesn't appear in the return type, which is
-   forbidden):
-
-   ```
-   instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
-   ```
- -/
-def Scalar.toInt {ty : ScalarTy} (x : Scalar ty) : Int := x.val
-
--- Remark: I tried a version of the shape `HasProp {a : Type} (x : a)`
--- but the lookup didn't work
-class HasProp (a : Sort u) where
-  prop_ty : a → Prop
-  prop : ∀ x:a, prop_ty x
-
-instance (ty : ScalarTy) : HasProp (Scalar ty) where
-  -- prop_ty is inferred
-  prop := λ x => And.intro x.hmin x.hmax
-
-instance (a : Type) : HasProp (Vec a) where
-  prop_ty := λ v => v.val.length ≤ Scalar.max ScalarTy.Usize
-  prop := λ ⟨ _, l ⟩ => l
-
-class PropHasImp (x : Prop) where
-  concl : Prop
-  prop : x → concl
-
--- This also works for `x ≠ y` because this expression reduces to `¬ x = y`
--- and `Ne` is marked as `reducible`
-instance (x y : Int) : PropHasImp (¬ x = y) where
-  concl := x < y ∨ x > y
-  prop := λ (h:x ≠ y) => ne_is_lt_or_gt h
-
-open Lean Lean.Elab Command Term Lean.Meta
-
--- Small utility: print all the declarations in the context
-elab "print_all_decls" : tactic => do
-  let ctx ← Lean.MonadLCtx.getLCtx
-  for decl in ← ctx.getDecls do
-    let ty ← Lean.Meta.inferType decl.toExpr
-    logInfo m!"{decl.toExpr} : {ty}"
-  pure ()
-
--- Explore a term by decomposing the applications (we explore the applied
--- functions and their arguments, but ignore lambdas, forall, etc. -
--- should we go inside?).
-partial def foldTermApps (k : α → Expr → MetaM α) (s : α) (e : Expr) : MetaM α := do
-  -- We do it in a very simpler manner: we deconstruct applications,
-  -- and recursively explore the sub-expressions. Note that we do
-  -- not go inside foralls and abstractions (should we?).
-  e.withApp fun f args => do
-    let s ← k s f
-    args.foldlM (foldTermApps k) s
-
--- Provided a function `k` which lookups type class instances on an expression,
--- collect all the instances lookuped by applying `k` on the sub-expressions of `e`.
-def collectInstances
-  (k : Expr → MetaM (Option Expr)) (s : HashSet Expr) (e : Expr) : MetaM (HashSet Expr) := do
-  let k s e := do
-    match ← k e with
-    | none => pure s
-    | some i => pure (s.insert i)
-  foldTermApps k s e
-
--- Similar to `collectInstances`, but explores all the local declarations in the
--- main context.
-def collectInstancesFromMainCtx (k : Expr → MetaM (Option Expr)) : Tactic.TacticM (HashSet Expr) := do
-  Lean.Elab.Tactic.withMainContext do
-  -- Get the local context
-  let ctx ← Lean.MonadLCtx.getLCtx
-  -- Just a matter of precaution
-  let ctx ← instantiateLCtxMVars ctx
-  -- Initialize the hashset
-  let hs := HashSet.empty
-  -- Explore the declarations
-  let decls ← ctx.getDecls
-  decls.foldlM (fun hs d => collectInstances k hs d.toExpr) hs
-
--- Return an instance of `HasProp` for `e` if it has some
-def lookupHasProp (e : Expr) : MetaM (Option Expr) := do
-  trace[Arith] "lookupHasProp"
-  -- TODO: do we need Lean.observing?
-  -- This actually eliminates the error messages
-  Lean.observing? do
-    trace[Arith] "lookupHasProp: observing"
-    let ty ← Lean.Meta.inferType e
-    let hasProp ← mkAppM ``HasProp #[ty]
-    let hasPropInst ← trySynthInstance hasProp
-    match hasPropInst with
-    | LOption.some i =>
-      trace[Arith] "Found HasProp instance"
-      let i_prop ← mkProjection i (Name.mkSimple "prop")
-      some (← mkAppM' i_prop #[e])
-    | _ => none
-
--- Collect the instances of `HasProp` for the subexpressions in the context
-def collectHasPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
-  collectInstancesFromMainCtx lookupHasProp
-
-elab "display_has_prop_instances" : tactic => do
-  trace[Arith] "Displaying the HasProp instances"
-  let hs ← collectHasPropInstancesFromMainCtx
-  hs.forM fun e => do
-    trace[Arith] "+ HasProp instance: {e}"
-
-example (x : U32) : True := by
-  let i : HasProp U32 := inferInstance
-  have p := @HasProp.prop _ i x
-  simp only [HasProp.prop_ty] at p
-  display_has_prop_instances
-  simp
-
--- Return an instance of `PropHasImp` for `e` if it has some
-def lookupPropHasImp (e : Expr) : MetaM (Option Expr) := do
-  trace[Arith] "lookupPropHasImp"
-  -- TODO: do we need Lean.observing?
-  -- This actually eliminates the error messages
-  Lean.observing? do
-    trace[Arith] "lookupPropHasImp: observing"
-    let ty ← Lean.Meta.inferType e
-    trace[Arith] "lookupPropHasImp: ty: {ty}"
-    let cl ← mkAppM ``PropHasImp #[ty]
-    let inst ← trySynthInstance cl
-    match inst with
-    | LOption.some i =>
-      trace[Arith] "Found PropHasImp instance"
-      let i_prop ←  mkProjection i (Name.mkSimple "prop")
-      some (← mkAppM' i_prop #[e])
-    | _ => none
-
--- Collect the instances of `PropHasImp` for the subexpressions in the context
-def collectPropHasImpInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
-  collectInstancesFromMainCtx lookupPropHasImp
-
-elab "display_prop_has_imp_instances" : tactic => do
-  trace[Arith] "Displaying the PropHasImp instances"
-  let hs ← collectPropHasImpInstancesFromMainCtx
-  hs.forM fun e => do
-    trace[Arith] "+ PropHasImp instance: {e}"
-
-example (x y : Int) (_ : x ≠ y) (_ : ¬ x = y) : True := by
-  display_prop_has_imp_instances
-  simp
-
-def addDecl (name : Name) (val : Expr) (type : Expr) (asLet : Bool) : Tactic.TacticM Expr :=
-  -- I don't think we need that
-  Lean.Elab.Tactic.withMainContext do
-  -- Insert the new declaration
-  let withDecl := if asLet then withLetDecl name type val else withLocalDeclD name type
-  withDecl fun nval => do
-    -- For debugging
-    let lctx ← Lean.MonadLCtx.getLCtx
-    let fid := nval.fvarId!
-    let decl := lctx.get! fid
-    trace[Arith] "  new decl: \"{decl.userName}\" ({nval}) : {decl.type} := {decl.value}"
-    --
-    -- Tranform the main goal `?m0` to `let x = nval in ?m1`
-    let mvarId ← Tactic.getMainGoal
-    let newMVar ← mkFreshExprSyntheticOpaqueMVar (← mvarId.getType)
-    let newVal ← mkLetFVars #[nval] newMVar
-    -- There are two cases:
-    -- - asLet is true: newVal is `let $name := $val in $newMVar`
-    -- - asLet is false: ewVal is `λ $name => $newMVar`
-    --   We need to apply it to `val`
-    let newVal := if asLet then newVal else mkAppN newVal #[val]
-    -- Assign the main goal and update the current goal
-    mvarId.assign newVal
-    let goals ← Tactic.getUnsolvedGoals
-    Lean.Elab.Tactic.setGoals (newMVar.mvarId! :: goals)
-    -- Return the new value - note: we are in the *new* context, created
-    -- after the declaration was added, so it will persist
-    pure nval
-
-def addDeclSyntax (name : Name) (val : Syntax) (asLet : Bool) : Tactic.TacticM Unit :=
-  -- I don't think we need that
-  Lean.Elab.Tactic.withMainContext do
-  --
-  let val ← elabTerm val .none
-  let type ← inferType val
-  -- In some situations, the type will be left as a metavariable (for instance,
-  -- if the term is `3`, Lean has the choice between `Nat` and `Int` and will
-  -- not choose): we force the instantiation of the meta-variable
-  synthesizeSyntheticMVarsUsingDefault
-  --
-  let _ ← addDecl name val type asLet
-
-elab "custom_let " n:ident " := " v:term : tactic => do
-  addDeclSyntax n.getId v (asLet := true)
-
-elab "custom_have " n:ident " := " v:term : tactic =>
-  addDeclSyntax n.getId v (asLet := false)
-
-example : Nat := by
-  custom_let x := 4
-  custom_have y := 4
-  apply y
-
-example (x : Bool) : Nat := by
-  cases x <;> custom_let x := 3 <;> apply x
-
--- Lookup instances in a context and introduce them with additional declarations.
-def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr)) : Tactic.TacticM (Array Expr) := do
-  let hs ← collectInstancesFromMainCtx lookup
-  hs.toArray.mapM fun e => do
-    let type ← inferType e
-    let name ← mkFreshUserName `h
-    -- Add a declaration
-    let nval ← addDecl name e type (asLet := false)
-    -- Simplify to unfold the declaration to unfold (i.e., the projector)
-    let simpTheorems ← Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) ({} : SimpTheorems)
-    -- Add the equational theorem for the decl to unfold
-    let simpTheorems ← simpTheorems.addDeclToUnfold declToUnfold
-    let congrTheorems ← getSimpCongrTheorems
-    let ctx : Simp.Context := { simpTheorems := #[simpTheorems], congrTheorems }
-    -- Where to apply the simplifier
-    let loc := Tactic.Location.targets #[mkIdent name] false
-    -- Apply the simplifier
-    let _ ← Tactic.simpLocation ctx (discharge? := .none) loc
-    -- Return the new value
-    pure nval
-
--- Lookup the instances of `HasProp for all the sub-expressions in the context,
--- and introduce the corresponding assumptions
-elab "intro_has_prop_instances" : tactic => do
-  trace[Arith] "Introducing the HasProp instances"
-  let _ ← introInstances ``HasProp.prop_ty lookupHasProp
-
-example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
-  intro_has_prop_instances
-  simp [*]
-
-example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
-  intro_has_prop_instances
-  simp_all [Scalar.max, Scalar.min]
-
--- Tactic to split on a disjunction.
--- The expression `h` should be an fvar.
-def splitDisj (h : Expr) (kleft kright : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
-  trace[Arith] "assumption on which to split: {h}"
-  -- Retrieve the main goal
-  Lean.Elab.Tactic.withMainContext do
-  let goalType ← Lean.Elab.Tactic.getMainTarget
-  let hDecl := (← getLCtx).get! h.fvarId!
-  let hName := hDecl.userName
-  -- Case disjunction
-  let hTy ← inferType h
-  hTy.withApp fun f xs => do
-  trace[Arith] "as app: {f} {xs}"
-  -- Sanity check
-  if ¬ (f.isConstOf ``Or ∧ xs.size = 2) then throwError "Invalid argument to splitDisj"
-  let a := xs.get! 0
-  let b := xs.get! 1
-  -- Introduce the new goals
-  -- Returns:
-  -- - the match branch
-  -- - a fresh new mvar id
-  let mkGoal (hTy : Expr) (nGoalName : String) : MetaM (Expr × MVarId) := do
-    -- Introduce a variable for the assumption (`a` or `b`). Note that we reuse
-    -- the name of the assumption we split.
-    withLocalDeclD hName hTy fun var => do
-    -- The new goal
-    let mgoal ← mkFreshExprSyntheticOpaqueMVar goalType (tag := Name.mkSimple nGoalName)
-    -- Clear the assumption that we split
-    let mgoal ← mgoal.mvarId!.tryClearMany #[h.fvarId!]
-    -- The branch expression
-    let branch ← mkLambdaFVars #[var] (mkMVar mgoal)
-    pure (branch, mgoal)
-  let (inl, mleft) ← mkGoal a "left"
-  let (inr, mright) ← mkGoal b "right"
-  trace[Arith] "left: {inl}: {mleft}"
-  trace[Arith] "right: {inr}: {mright}"
-  -- Create the match expression
-  withLocalDeclD (← mkFreshUserName `h) hTy fun hVar => do
-  let motive ← mkLambdaFVars #[hVar] goalType
-  let casesExpr ← mkAppOptM ``Or.casesOn #[a, b, motive, h, inl, inr]
-  let mgoal ← Tactic.getMainGoal
-  trace[Arith] "goals: {← Tactic.getUnsolvedGoals}"
-  trace[Arith] "main goal: {mgoal}"
-  mgoal.assign casesExpr
-  let goals ← Tactic.getUnsolvedGoals
-  -- Focus on the left
-  Tactic.setGoals [mleft]
-  kleft
-  let leftGoals ← Tactic.getUnsolvedGoals
-  -- Focus on the right
-  Tactic.setGoals [mright]
-  kright
-  let rightGoals ← Tactic.getUnsolvedGoals
-  -- Put all the goals back
-  Tactic.setGoals (leftGoals ++ rightGoals ++ goals)
-  trace[Arith] "new goals: {← Tactic.getUnsolvedGoals}"
-
-elab "split_disj " n:ident : tactic => do
-  Lean.Elab.Tactic.withMainContext do
-  let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
-  let fvar := mkFVar decl.fvarId
-  splitDisj fvar (fun _ => pure ()) (fun _ => pure ())
-
-example (x y : Int) (h0 : x ≤ y ∨ x ≥ y) : x ≤ y ∨ x ≥ y := by
-  split_disj h0
-  . left; assumption
-  . right; assumption
-
--- Lookup the instances of `PropHasImp for all the sub-expressions in the context,
--- and introduce the corresponding assumptions
-elab "intro_prop_has_imp_instances" : tactic => do
-  trace[Arith] "Introducing the PropHasImp instances"
-  let _ ← introInstances ``PropHasImp.concl lookupPropHasImp
-
-example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
-  intro_prop_has_imp_instances
-  rename_i h
-  split_disj h
-  . linarith
-  . linarith
-
-/- Boosting a bit the linarith tac.
-
-   We do the following:
-   - for all the assumptions of the shape `(x : Int) ≠ y` or `¬ (x = y), we
-     introduce two goals with the assumptions `x < y` and `x > y`
-     TODO: we could create a PR for mathlib.
- -/
-def intTacPreprocess : Tactic.TacticM Unit := do
-  Lean.Elab.Tactic.withMainContext do
-  -- Lookup the instances of PropHasImp (this is how we detect assumptions
-  -- of the proper shape), introduce assumptions in the context and split
-  -- on those
-  -- TODO: get rid of the assumptions that we split
-  let rec splitOnAsms (asms : List Expr) : Tactic.TacticM Unit :=
-    match asms with
-    | [] => pure ()
-    | asm :: asms =>
-      let k := splitOnAsms asms
-      splitDisj asm k k
-  -- Introduce
-  let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
-  -- Split
-  splitOnAsms asms.toList
-
-elab "int_tac_preprocess" : tactic =>
-  intTacPreprocess
-
-example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
-  int_tac_preprocess
-  linarith
-  linarith
-
-syntax "int_tac" : tactic
-macro_rules
-  | `(tactic| int_tac) =>
-    `(tactic|
-      (repeat (apply And.intro)) <;> -- TODO: improve this
-      int_tac_preprocess <;>
-      linarith)
-
-example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
-  int_tac
-
--- Checking that things append correctly when there are several disjunctions
-example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := by
-  int_tac
-
--- A tactic to solve linear arithmetic goals in the presence of scalars
-syntax "scalar_tac" : tactic
-macro_rules
-  | `(tactic| scalar_tac) =>
-    `(tactic|
-      intro_has_prop_instances;
-      have := Scalar.cMin_bound ScalarTy.Usize;
-      have := Scalar.cMin_bound ScalarTy.Isize;
-      have := Scalar.cMax_bound ScalarTy.Usize;
-      have := Scalar.cMax_bound ScalarTy.Isize;
-      -- TODO: not too sure about that
-      simp only [*, Scalar.max, Scalar.min, Scalar.cMin, Scalar.cMax] at *;
-      int_tac)
-
-example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
-  scalar_tac
-
-example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
-  scalar_tac
-
-end Arith
diff --git a/backends/lean/Base/Arith/Arith.lean b/backends/lean/Base/Arith/Arith.lean
new file mode 100644
index 00000000..0ba73d18
--- /dev/null
+++ b/backends/lean/Base/Arith/Arith.lean
@@ -0,0 +1,409 @@
+/- This file contains tactics to solve arithmetic goals -/
+
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+-- TODO: there is no Omega tactic for now - it seems it hasn't been ported yet
+--import Mathlib.Tactic.Omega
+import Base.Primitives
+import Base.Utils
+import Base.Arith.Base
+
+/-
+Mathlib tactics:
+- rcases: https://leanprover-community.github.io/mathlib_docs/tactics.html#rcases
+- split_ifs: https://leanprover-community.github.io/mathlib_docs/tactics.html#split_ifs
+- norm_num: https://leanprover-community.github.io/mathlib_docs/tactics.html#norm_num
+- should we use linarith or omega?
+- hint: https://leanprover-community.github.io/mathlib_docs/tactics.html#hint
+- classical: https://leanprover-community.github.io/mathlib_docs/tactics.html#classical
+-/
+
+namespace List
+
+  -- TODO: I could not find this function??
+  @[simp] def flatten {a : Type u} : List (List a) → List a
+  | [] => []
+  | x :: ls => x ++ flatten ls
+
+end List
+
+namespace Lean
+
+namespace LocalContext
+
+  open Lean Lean.Elab Command Term Lean.Meta
+
+  -- Small utility: return the list of declarations in the context, from
+  -- the last to the first.
+  def getAllDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) :=
+    lctx.foldrM (fun d ls => do let d ← instantiateLocalDeclMVars d; pure (d :: ls)) []
+
+  -- Return the list of declarations in the context, but filter the
+  -- declarations which are considered as implementation details
+  def getDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) := do
+    let ls ← lctx.getAllDecls
+    pure (ls.filter (fun d => not d.isImplementationDetail))
+
+end LocalContext
+
+end Lean
+
+namespace Arith
+
+open Primitives
+
+-- TODO: move?
+theorem ne_zero_is_lt_or_gt {x : Int} (hne : x ≠ 0) : x < 0 ∨ x > 0 := by
+  cases h: x <;> simp_all
+  . rename_i n;
+    cases n <;> simp_all
+  . apply Int.negSucc_lt_zero
+
+-- TODO: move?
+theorem ne_is_lt_or_gt {x y : Int} (hne : x ≠ y) : x < y ∨ x > y := by
+  have hne : x - y ≠ 0 := by
+    simp
+    intro h
+    have: x = y := by linarith
+    simp_all
+  have h := ne_zero_is_lt_or_gt hne
+  match h with
+  | .inl _ => left; linarith
+  | .inr _ => right; linarith
+
+-- TODO: move
+instance Vec.cast (a : Type): Coe (Vec a) (List a)  where coe := λ v => v.val
+
+-- TODO: move
+/- Remark: we can't write the following instance because of restrictions about
+   the type class parameters (`ty` doesn't appear in the return type, which is
+   forbidden):
+
+   ```
+   instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
+   ```
+ -/
+def Scalar.toInt {ty : ScalarTy} (x : Scalar ty) : Int := x.val
+
+-- Remark: I tried a version of the shape `HasProp {a : Type} (x : a)`
+-- but the lookup didn't work
+class HasProp (a : Sort u) where
+  prop_ty : a → Prop
+  prop : ∀ x:a, prop_ty x
+
+instance (ty : ScalarTy) : HasProp (Scalar ty) where
+  -- prop_ty is inferred
+  prop := λ x => And.intro x.hmin x.hmax
+
+instance (a : Type) : HasProp (Vec a) where
+  prop_ty := λ v => v.val.length ≤ Scalar.max ScalarTy.Usize
+  prop := λ ⟨ _, l ⟩ => l
+
+class PropHasImp (x : Prop) where
+  concl : Prop
+  prop : x → concl
+
+-- This also works for `x ≠ y` because this expression reduces to `¬ x = y`
+-- and `Ne` is marked as `reducible`
+instance (x y : Int) : PropHasImp (¬ x = y) where
+  concl := x < y ∨ x > y
+  prop := λ (h:x ≠ y) => ne_is_lt_or_gt h
+
+open Lean Lean.Elab Command Term Lean.Meta
+
+-- Small utility: print all the declarations in the context
+elab "print_all_decls" : tactic => do
+  let ctx ← Lean.MonadLCtx.getLCtx
+  for decl in ← ctx.getDecls do
+    let ty ← Lean.Meta.inferType decl.toExpr
+    logInfo m!"{decl.toExpr} : {ty}"
+  pure ()
+
+-- Explore a term by decomposing the applications (we explore the applied
+-- functions and their arguments, but ignore lambdas, forall, etc. -
+-- should we go inside?).
+partial def foldTermApps (k : α → Expr → MetaM α) (s : α) (e : Expr) : MetaM α := do
+  -- We do it in a very simpler manner: we deconstruct applications,
+  -- and recursively explore the sub-expressions. Note that we do
+  -- not go inside foralls and abstractions (should we?).
+  e.withApp fun f args => do
+    let s ← k s f
+    args.foldlM (foldTermApps k) s
+
+-- Provided a function `k` which lookups type class instances on an expression,
+-- collect all the instances lookuped by applying `k` on the sub-expressions of `e`.
+def collectInstances
+  (k : Expr → MetaM (Option Expr)) (s : HashSet Expr) (e : Expr) : MetaM (HashSet Expr) := do
+  let k s e := do
+    match ← k e with
+    | none => pure s
+    | some i => pure (s.insert i)
+  foldTermApps k s e
+
+-- Similar to `collectInstances`, but explores all the local declarations in the
+-- main context.
+def collectInstancesFromMainCtx (k : Expr → MetaM (Option Expr)) : Tactic.TacticM (HashSet Expr) := do
+  Lean.Elab.Tactic.withMainContext do
+  -- Get the local context
+  let ctx ← Lean.MonadLCtx.getLCtx
+  -- Just a matter of precaution
+  let ctx ← instantiateLCtxMVars ctx
+  -- Initialize the hashset
+  let hs := HashSet.empty
+  -- Explore the declarations
+  let decls ← ctx.getDecls
+  decls.foldlM (fun hs d => collectInstances k hs d.toExpr) hs
+
+-- Return an instance of `HasProp` for `e` if it has some
+def lookupHasProp (e : Expr) : MetaM (Option Expr) := do
+  trace[Arith] "lookupHasProp"
+  -- TODO: do we need Lean.observing?
+  -- This actually eliminates the error messages
+  Lean.observing? do
+    trace[Arith] "lookupHasProp: observing"
+    let ty ← Lean.Meta.inferType e
+    let hasProp ← mkAppM ``HasProp #[ty]
+    let hasPropInst ← trySynthInstance hasProp
+    match hasPropInst with
+    | LOption.some i =>
+      trace[Arith] "Found HasProp instance"
+      let i_prop ← mkProjection i (Name.mkSimple "prop")
+      some (← mkAppM' i_prop #[e])
+    | _ => none
+
+-- Collect the instances of `HasProp` for the subexpressions in the context
+def collectHasPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
+  collectInstancesFromMainCtx lookupHasProp
+
+elab "display_has_prop_instances" : tactic => do
+  trace[Arith] "Displaying the HasProp instances"
+  let hs ← collectHasPropInstancesFromMainCtx
+  hs.forM fun e => do
+    trace[Arith] "+ HasProp instance: {e}"
+
+example (x : U32) : True := by
+  let i : HasProp U32 := inferInstance
+  have p := @HasProp.prop _ i x
+  simp only [HasProp.prop_ty] at p
+  display_has_prop_instances
+  simp
+
+-- Return an instance of `PropHasImp` for `e` if it has some
+def lookupPropHasImp (e : Expr) : MetaM (Option Expr) := do
+  trace[Arith] "lookupPropHasImp"
+  -- TODO: do we need Lean.observing?
+  -- This actually eliminates the error messages
+  Lean.observing? do
+    trace[Arith] "lookupPropHasImp: observing"
+    let ty ← Lean.Meta.inferType e
+    trace[Arith] "lookupPropHasImp: ty: {ty}"
+    let cl ← mkAppM ``PropHasImp #[ty]
+    let inst ← trySynthInstance cl
+    match inst with
+    | LOption.some i =>
+      trace[Arith] "Found PropHasImp instance"
+      let i_prop ←  mkProjection i (Name.mkSimple "prop")
+      some (← mkAppM' i_prop #[e])
+    | _ => none
+
+-- Collect the instances of `PropHasImp` for the subexpressions in the context
+def collectPropHasImpInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
+  collectInstancesFromMainCtx lookupPropHasImp
+
+elab "display_prop_has_imp_instances" : tactic => do
+  trace[Arith] "Displaying the PropHasImp instances"
+  let hs ← collectPropHasImpInstancesFromMainCtx
+  hs.forM fun e => do
+    trace[Arith] "+ PropHasImp instance: {e}"
+
+example (x y : Int) (_ : x ≠ y) (_ : ¬ x = y) : True := by
+  display_prop_has_imp_instances
+  simp
+
+-- Lookup instances in a context and introduce them with additional declarations.
+def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr)) : Tactic.TacticM (Array Expr) := do
+  let hs ← collectInstancesFromMainCtx lookup
+  hs.toArray.mapM fun e => do
+    let type ← inferType e
+    let name ← mkFreshUserName `h
+    -- Add a declaration
+    let nval ← Utils.addDecl name e type (asLet := false)
+    -- Simplify to unfold the declaration to unfold (i.e., the projector)
+    let simpTheorems ← Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) ({} : SimpTheorems)
+    -- Add the equational theorem for the decl to unfold
+    let simpTheorems ← simpTheorems.addDeclToUnfold declToUnfold
+    let congrTheorems ← getSimpCongrTheorems
+    let ctx : Simp.Context := { simpTheorems := #[simpTheorems], congrTheorems }
+    -- Where to apply the simplifier
+    let loc := Tactic.Location.targets #[mkIdent name] false
+    -- Apply the simplifier
+    let _ ← Tactic.simpLocation ctx (discharge? := .none) loc
+    -- Return the new value
+    pure nval
+
+-- Lookup the instances of `HasProp for all the sub-expressions in the context,
+-- and introduce the corresponding assumptions
+elab "intro_has_prop_instances" : tactic => do
+  trace[Arith] "Introducing the HasProp instances"
+  let _ ← introInstances ``HasProp.prop_ty lookupHasProp
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+  intro_has_prop_instances
+  simp [*]
+
+example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
+  intro_has_prop_instances
+  simp_all [Scalar.max, Scalar.min]
+
+-- Tactic to split on a disjunction.
+-- The expression `h` should be an fvar.
+def splitDisj (h : Expr) (kleft kright : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
+  trace[Arith] "assumption on which to split: {h}"
+  -- Retrieve the main goal
+  Lean.Elab.Tactic.withMainContext do
+  let goalType ← Lean.Elab.Tactic.getMainTarget
+  let hDecl := (← getLCtx).get! h.fvarId!
+  let hName := hDecl.userName
+  -- Case disjunction
+  let hTy ← inferType h
+  hTy.withApp fun f xs => do
+  trace[Arith] "as app: {f} {xs}"
+  -- Sanity check
+  if ¬ (f.isConstOf ``Or ∧ xs.size = 2) then throwError "Invalid argument to splitDisj"
+  let a := xs.get! 0
+  let b := xs.get! 1
+  -- Introduce the new goals
+  -- Returns:
+  -- - the match branch
+  -- - a fresh new mvar id
+  let mkGoal (hTy : Expr) (nGoalName : String) : MetaM (Expr × MVarId) := do
+    -- Introduce a variable for the assumption (`a` or `b`). Note that we reuse
+    -- the name of the assumption we split.
+    withLocalDeclD hName hTy fun var => do
+    -- The new goal
+    let mgoal ← mkFreshExprSyntheticOpaqueMVar goalType (tag := Name.mkSimple nGoalName)
+    -- Clear the assumption that we split
+    let mgoal ← mgoal.mvarId!.tryClearMany #[h.fvarId!]
+    -- The branch expression
+    let branch ← mkLambdaFVars #[var] (mkMVar mgoal)
+    pure (branch, mgoal)
+  let (inl, mleft) ← mkGoal a "left"
+  let (inr, mright) ← mkGoal b "right"
+  trace[Arith] "left: {inl}: {mleft}"
+  trace[Arith] "right: {inr}: {mright}"
+  -- Create the match expression
+  withLocalDeclD (← mkFreshUserName `h) hTy fun hVar => do
+  let motive ← mkLambdaFVars #[hVar] goalType
+  let casesExpr ← mkAppOptM ``Or.casesOn #[a, b, motive, h, inl, inr]
+  let mgoal ← Tactic.getMainGoal
+  trace[Arith] "goals: {← Tactic.getUnsolvedGoals}"
+  trace[Arith] "main goal: {mgoal}"
+  mgoal.assign casesExpr
+  let goals ← Tactic.getUnsolvedGoals
+  -- Focus on the left
+  Tactic.setGoals [mleft]
+  kleft
+  let leftGoals ← Tactic.getUnsolvedGoals
+  -- Focus on the right
+  Tactic.setGoals [mright]
+  kright
+  let rightGoals ← Tactic.getUnsolvedGoals
+  -- Put all the goals back
+  Tactic.setGoals (leftGoals ++ rightGoals ++ goals)
+  trace[Arith] "new goals: {← Tactic.getUnsolvedGoals}"
+
+elab "split_disj " n:ident : tactic => do
+  Lean.Elab.Tactic.withMainContext do
+  let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
+  let fvar := mkFVar decl.fvarId
+  splitDisj fvar (fun _ => pure ()) (fun _ => pure ())
+
+example (x y : Int) (h0 : x ≤ y ∨ x ≥ y) : x ≤ y ∨ x ≥ y := by
+  split_disj h0
+  . left; assumption
+  . right; assumption
+
+-- Lookup the instances of `PropHasImp for all the sub-expressions in the context,
+-- and introduce the corresponding assumptions
+elab "intro_prop_has_imp_instances" : tactic => do
+  trace[Arith] "Introducing the PropHasImp instances"
+  let _ ← introInstances ``PropHasImp.concl lookupPropHasImp
+
+example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
+  intro_prop_has_imp_instances
+  rename_i h
+  split_disj h
+  . linarith
+  . linarith
+
+/- Boosting a bit the linarith tac.
+
+   We do the following:
+   - for all the assumptions of the shape `(x : Int) ≠ y` or `¬ (x = y), we
+     introduce two goals with the assumptions `x < y` and `x > y`
+     TODO: we could create a PR for mathlib.
+ -/
+def intTacPreprocess : Tactic.TacticM Unit := do
+  Lean.Elab.Tactic.withMainContext do
+  -- Lookup the instances of PropHasImp (this is how we detect assumptions
+  -- of the proper shape), introduce assumptions in the context and split
+  -- on those
+  -- TODO: get rid of the assumptions that we split
+  let rec splitOnAsms (asms : List Expr) : Tactic.TacticM Unit :=
+    match asms with
+    | [] => pure ()
+    | asm :: asms =>
+      let k := splitOnAsms asms
+      splitDisj asm k k
+  -- Introduce
+  let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
+  -- Split
+  splitOnAsms asms.toList
+
+elab "int_tac_preprocess" : tactic =>
+  intTacPreprocess
+
+example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
+  int_tac_preprocess
+  linarith
+  linarith
+
+syntax "int_tac" : tactic
+macro_rules
+  | `(tactic| int_tac) =>
+    `(tactic|
+      (repeat (apply And.intro)) <;> -- TODO: improve this
+      int_tac_preprocess <;>
+      linarith)
+
+example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
+  int_tac
+
+-- Checking that things append correctly when there are several disjunctions
+example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := by
+  int_tac
+
+-- A tactic to solve linear arithmetic goals in the presence of scalars
+syntax "scalar_tac" : tactic
+macro_rules
+  | `(tactic| scalar_tac) =>
+    `(tactic|
+      intro_has_prop_instances;
+      have := Scalar.cMin_bound ScalarTy.Usize;
+      have := Scalar.cMin_bound ScalarTy.Isize;
+      have := Scalar.cMax_bound ScalarTy.Usize;
+      have := Scalar.cMax_bound ScalarTy.Isize;
+      -- TODO: not too sure about that
+      simp only [*, Scalar.max, Scalar.min, Scalar.cMin, Scalar.cMax] at *;
+      int_tac)
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+  scalar_tac
+
+example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
+  scalar_tac
+
+end Arith
diff --git a/backends/lean/Base/Arith/Base.lean b/backends/lean/Base/Arith/Base.lean
new file mode 100644
index 00000000..ddd2dc24
--- /dev/null
+++ b/backends/lean/Base/Arith/Base.lean
@@ -0,0 +1,10 @@
+import Lean
+
+namespace Arith
+
+open Lean Elab Term Meta
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Arith
+
+end Arith
diff --git a/backends/lean/Base/ArithBase.lean b/backends/lean/Base/ArithBase.lean
deleted file mode 100644
index ddd2dc24..00000000
--- a/backends/lean/Base/ArithBase.lean
+++ /dev/null
@@ -1,10 +0,0 @@
-import Lean
-
-namespace Arith
-
-open Lean Elab Term Meta
-
--- We can't define and use trace classes in the same file
-initialize registerTraceClass `Arith
-
-end Arith
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
index 96f7abc0..f109e847 100644
--- a/backends/lean/Base/Diverge/Elab.lean
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -14,8 +14,8 @@ namespace Diverge
 syntax (name := divergentDef)
   declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command
 
-open Utils
 open Lean Elab Term Meta Primitives Lean.Meta
+open Utils
 
 /- The following was copied from the `wfRecursion` function. -/
 
@@ -47,21 +47,6 @@ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
   else
     pure (args.get! 0, args.get! 1)
 
-/- Like `lambdaTelescopeN` but only destructs a fixed number of lambdas -/
-def lambdaTelescopeN (e : Expr) (n : Nat) (k : Array Expr → Expr → MetaM α) : MetaM α :=
-  lambdaTelescope e fun xs body => do
-  if xs.size < n then throwError "lambdaTelescopeN: not enough lambdas";
-  let xs := xs.extract 0 n
-  let ys := xs.extract n xs.size
-  let body ← mkLambdaFVars ys body
-  k xs body
-
-/- Like `lambdaTelescope`, but only destructs one lambda
-   TODO: is there an equivalent of this function somewhere in the
-   standard library? -/
-def lambdaOne (e : Expr) (k : Expr → Expr → MetaM α) : MetaM α :=
-  lambdaTelescopeN e 1 λ xs b => k (xs.get! 0) b
-
 /- Generate a Sigma type from a list of *variables* (all the expressions
    must be variables).
 
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
index 161b9ddb..2ce63620 100644
--- a/backends/lean/Base/Utils.lean
+++ b/backends/lean/Base/Utils.lean
@@ -116,4 +116,99 @@ partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr :
     | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
   visit 0 e
 
+section Methods
+  variable [MonadLiftT MetaM m] [MonadControlT MetaM m] [Monad m] [MonadError m]
+  variable {a : Type}
+
+  /- Like `lambdaTelescopeN` but only destructs a fixed number of lambdas -/
+  def lambdaTelescopeN (e : Expr) (n : Nat) (k : Array Expr → Expr → m a) : m a :=
+    lambdaTelescope e fun xs body => do
+    if xs.size < n then throwError "lambdaTelescopeN: not enough lambdas"
+    let xs := xs.extract 0 n
+    let ys := xs.extract n xs.size
+    let body ← liftMetaM (mkLambdaFVars ys body)
+    k xs body
+
+  /- Like `lambdaTelescope`, but only destructs one lambda
+     TODO: is there an equivalent of this function somewhere in the
+     standard library? -/
+  def lambdaOne (e : Expr) (k : Expr → Expr → m a) : m a :=
+    lambdaTelescopeN e 1 λ xs b => k (xs.get! 0) b
+
+  def isExists (e : Expr) : Bool := e.getAppFn.isConstOf ``Exists ∧ e.getAppNumArgs = 2
+
+  -- Remark: Lean doesn't find the inhabited and nonempty instances if we don'
+  -- put them explicitely in the signature
+  partial def existsTelescopeProcess [Inhabited (m a)] [Nonempty (m a)]
+    (fvars : Array Expr) (e : Expr) (k : Array Expr → Expr → m a) : m a := do
+    -- Attempt to deconstruct an existential
+    if isExists e then do
+      let p := e.appArg!
+      lambdaOne p fun x ne =>
+      existsTelescopeProcess (fvars.push x) ne k
+    else
+      -- No existential: call the continuation
+      k fvars e
+
+  def existsTelescope [Inhabited (m a)] [Nonempty (m a)] (e : Expr) (k : Array Expr → Expr → m a) : m a := do
+    existsTelescopeProcess #[] e k
+
+end Methods
+
+def addDecl (name : Name) (val : Expr) (type : Expr) (asLet : Bool) : Tactic.TacticM Expr :=
+  -- I don't think we need that
+  Lean.Elab.Tactic.withMainContext do
+  -- Insert the new declaration
+  let withDecl := if asLet then withLetDecl name type val else withLocalDeclD name type
+  withDecl fun nval => do
+    -- For debugging
+    let lctx ← Lean.MonadLCtx.getLCtx
+    let fid := nval.fvarId!
+    let decl := lctx.get! fid
+    trace[Arith] "  new decl: \"{decl.userName}\" ({nval}) : {decl.type} := {decl.value}"
+    --
+    -- Tranform the main goal `?m0` to `let x = nval in ?m1`
+    let mvarId ← Tactic.getMainGoal
+    let newMVar ← mkFreshExprSyntheticOpaqueMVar (← mvarId.getType)
+    let newVal ← mkLetFVars #[nval] newMVar
+    -- There are two cases:
+    -- - asLet is true: newVal is `let $name := $val in $newMVar`
+    -- - asLet is false: ewVal is `λ $name => $newMVar`
+    --   We need to apply it to `val`
+    let newVal := if asLet then newVal else mkAppN newVal #[val]
+    -- Assign the main goal and update the current goal
+    mvarId.assign newVal
+    let goals ← Tactic.getUnsolvedGoals
+    Lean.Elab.Tactic.setGoals (newMVar.mvarId! :: goals)
+    -- Return the new value - note: we are in the *new* context, created
+    -- after the declaration was added, so it will persist
+    pure nval
+
+def addDeclSyntax (name : Name) (val : Syntax) (asLet : Bool) : Tactic.TacticM Unit :=
+  -- I don't think we need that
+  Lean.Elab.Tactic.withMainContext do
+  --
+  let val ← elabTerm val .none
+  let type ← inferType val
+  -- In some situations, the type will be left as a metavariable (for instance,
+  -- if the term is `3`, Lean has the choice between `Nat` and `Int` and will
+  -- not choose): we force the instantiation of the meta-variable
+  synthesizeSyntheticMVarsUsingDefault
+  --
+  let _ ← addDecl name val type asLet
+
+elab "custom_let " n:ident " := " v:term : tactic => do
+  addDeclSyntax n.getId v (asLet := true)
+
+elab "custom_have " n:ident " := " v:term : tactic =>
+  addDeclSyntax n.getId v (asLet := false)
+
+example : Nat := by
+  custom_let x := 4
+  custom_have y := 4
+  apply y
+
+example (x : Bool) : Nat := by
+  cases x <;> custom_let x := 3 <;> apply x
+
 end Utils
-- 
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