6 files changed, 346 insertions, 13 deletions
diff --git a/backends/lean/Base/Arith.lean b/backends/lean/Base/Arith.lean
new file mode 100644
index 00000000..fd5698c5
--- /dev/null
+++ b/backends/lean/Base/Arith.lean
@@ -0,0 +1 @@
+import Base.Arith.Arith
diff --git a/backends/lean/Base/Arith/Arith.lean b/backends/lean/Base/Arith/Arith.lean
index 0ba73d18..ff628cf3 100644
--- a/backends/lean/Base/Arith/Arith.lean
+++ b/backends/lean/Base/Arith/Arith.lean
@@ -146,7 +146,7 @@ def collectInstances
 -- 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
+  Tactic.withMainContext do
   -- Get the local context
   let ctx ← Lean.MonadLCtx.getLCtx
   -- Just a matter of precaution
@@ -263,8 +263,8 @@ example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
 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
+  Tactic.withMainContext do
+  let goalType ← Tactic.getMainTarget
   let hDecl := (← getLCtx).get! h.fvarId!
   let hName := hDecl.userName
   -- Case disjunction
@@ -316,7 +316,7 @@ def splitDisj (h : Expr) (kleft kright : Tactic.TacticM Unit) : Tactic.TacticM U
   trace[Arith] "new goals: {← Tactic.getUnsolvedGoals}"
 
 elab "split_disj " n:ident : tactic => do
-  Lean.Elab.Tactic.withMainContext do
+  Tactic.withMainContext do
   let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
   let fvar := mkFVar decl.fvarId
   splitDisj fvar (fun _ => pure ()) (fun _ => pure ())
@@ -347,7 +347,7 @@ example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
      TODO: we could create a PR for mathlib.
  -/
 def intTacPreprocess : Tactic.TacticM Unit := do
-  Lean.Elab.Tactic.withMainContext do
+  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
@@ -366,19 +366,31 @@ def intTacPreprocess : Tactic.TacticM Unit := do
 elab "int_tac_preprocess" : tactic =>
   intTacPreprocess
 
+def intTac : Tactic.TacticM Unit := do
+  Tactic.withMainContext do
+  Tactic.focus do
+  -- Preprocess - wondering if we should do this before or after splitting
+  -- the goal. I think before leads to a smaller proof term?
+  Tactic.allGoals intTacPreprocess
+  -- Split the conjunctions in the goal
+  Utils.repeatTac Utils.splitConjTarget
+  -- Call linarith
+  let linarith :=
+    let cfg : Linarith.LinarithConfig := {
+      -- We do this with our custom preprocessing
+      splitNe := false
+    }
+    Tactic.liftMetaFinishingTactic <| Linarith.linarith false [] cfg
+  Tactic.allGoals linarith
+
+elab "int_tac" : tactic =>
+  intTac
+
 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
 
@@ -386,6 +398,10 @@ example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
 example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := 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 ∧ x + y ≥ 2 := by
+  int_tac
+
 -- A tactic to solve linear arithmetic goals in the presence of scalars
 syntax "scalar_tac" : tactic
 macro_rules
diff --git a/backends/lean/Base/Progress.lean b/backends/lean/Base/Progress.lean
new file mode 100644
index 00000000..d812b896
--- /dev/null
+++ b/backends/lean/Base/Progress.lean
@@ -0,0 +1 @@
+import Base.Progress.Progress
diff --git a/backends/lean/Base/Progress/Base.lean b/backends/lean/Base/Progress/Base.lean
new file mode 100644
index 00000000..3f44f46c
--- /dev/null
+++ b/backends/lean/Base/Progress/Base.lean
@@ -0,0 +1,175 @@
+import Lean
+import Base.Utils
+import Base.Primitives
+
+namespace Progress
+
+open Lean Elab Term Meta
+open Utils
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Progress
+
+-- Return the first conjunct if the expression is a conjunction, or the
+-- expression itself otherwise. Also return the second conjunct if it is a
+-- conjunction.
+def getFirstConj (e : Expr) : MetaM (Expr × Option Expr) := do
+  e.withApp fun f args =>
+  if f.isConstOf ``And ∧ args.size = 2 then pure (args.get! 0, some (args.get! 1))
+  else pure (e, none)
+
+-- Destruct an equaliy and return the two sides
+def destEq (e : Expr) : MetaM (Expr × Expr) := do
+  e.withApp fun f args =>
+  if f.isConstOf ``Eq ∧ args.size = 3 then pure (args.get! 1, args.get! 2)
+  else throwError "Not an equality: {e}"
+
+-- Return the set of FVarIds in the expression
+partial def getFVarIds (e : Expr) (hs : HashSet FVarId := HashSet.empty) : MetaM (HashSet FVarId) := do
+  e.withApp fun body args => do
+  let hs := if body.isFVar then hs.insert body.fvarId! else hs
+  args.foldlM (fun hs arg => getFVarIds arg hs) hs
+
+/- # Progress tactic -/
+
+structure PSpecDesc where
+  -- The universally quantified variables
+  fvars : Array Expr
+  -- The existentially quantified variables
+  evars : Array Expr
+  -- The function
+  fName : Name
+  -- The function arguments
+  fLevels : List Level
+  args : Array Expr
+  -- The universally quantified variables which appear in the function arguments
+  argsFVars : Array FVarId
+  -- The returned value
+  ret : Expr
+  -- The postcondition (if there is)
+  post : Option Expr
+
+section Methods
+  variable [MonadLiftT MetaM m] [MonadControlT MetaM m] [Monad m] [MonadOptions m]
+  variable [MonadTrace m] [MonadLiftT IO m] [MonadRef m] [AddMessageContext m]
+  variable [MonadError m]
+  variable {a : Type}
+
+  /- Analyze a pspec theorem to decompose its arguments.
+
+     PSpec theorems should be of the following shape:
+     ```
+     ∀ x1 ... xn, H1 → ... Hn → ∃ y1 ... ym. f x1 ... xn = .ret ... ∧ Post1 ∧ ... ∧ Postk
+     ```
+
+     The continuation `k` receives the following inputs:
+     - universally quantified variables
+     - assumptions
+     - existentially quantified variables
+     - function name
+     - function arguments
+     - return
+     - postconditions
+
+     TODO: generalize for when we do inductive proofs
+  -/
+  partial
+  def withPSpec [Inhabited (m a)] [Nonempty (m a)] (th : Expr) (k : PSpecDesc → m a)
+    (sanityChecks : Bool := false) :
+    m a := do
+    trace[Progress] "Theorem: {th}"
+    -- Dive into the quantified variables and the assumptions
+    forallTelescope th fun fvars th => do
+    trace[Progress] "All argumens: {fvars}"
+  /-  -- Filter the argumens which are not propositions
+    let rec getFirstPropIdx (i : Nat) : MetaM Nat := do
+      if i ≥ fargs.size then pure i
+      else do
+        let x := fargs.get! i
+        if ← Meta.isProp (← inferType x) then pure i
+        else getFirstPropIdx (i + 1)
+    let i ← getFirstPropIdx 0
+    let fvars := fargs.extract 0 i
+    let hyps := fargs.extract i fargs.size
+    trace[Progress] "Quantified variables: {fvars}"
+    trace[Progress] "Assumptions: {hyps}"
+    -- Sanity check: all hypotheses are propositions (in particular, all the
+    -- quantified variables are at the beginning)
+    let hypsAreProp ← hyps.allM fun x => do Meta.isProp (← inferType x)
+    if ¬ hypsAreProp then
+      throwError "The theorem doesn't have the proper shape: all the quantified arguments should be at the beginning"
+      -/
+    -- Dive into the existentials
+    existsTelescope th fun evars th => do
+    trace[Progress] "Existentials: {evars}"
+    -- Take the first conjunct
+    let (th, post) ← getFirstConj th
+    -- Destruct the equality
+    let (th, ret) ← destEq th
+    -- Destruct the application to get the name
+    th.withApp fun f args => do
+    if ¬ f.isConst then throwError "Not a constant: {f}"
+    -- Compute the set of universally quantified variables which appear in the function arguments
+    let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
+    -- Sanity check
+    if sanityChecks then
+      let fvarsSet : HashSet FVarId := HashSet.ofArray (fvars.map (fun x => x.fvarId!))
+      let filtArgsFVars := allArgsFVars.toArray.filter (fun fvar => ¬ fvarsSet.contains fvar)
+      if ¬ filtArgsFVars.isEmpty then
+        let filtArgsFVars := filtArgsFVars.map (fun fvarId => Expr.fvar fvarId)
+        throwError "Some of the function inputs are not universally quantified: {filtArgsFVars}"
+    let argsFVars := fvars.map (fun x => x.fvarId!)
+    let argsFVars := argsFVars.filter (fun fvar => allArgsFVars.contains fvar)
+    -- Return
+    trace[Progress] "Function: {f.constName!}";
+    let thDesc := {
+      fvars := fvars
+      evars := evars
+      fName := f.constName!
+      fLevels := f.constLevels!
+      args := args
+      argsFVars
+      ret := ret
+      post := post
+    }
+    k thDesc
+end Methods
+
+
+def getPSpecFunName (th : Expr) : MetaM Name :=
+  withPSpec th (fun d => do pure d.fName) true
+
+structure PSpecAttr where
+  attr : AttributeImpl
+  ext  : MapDeclarationExtension Name
+  deriving Inhabited
+
+/- The persistent map from function to pspec theorems. -/
+initialize pspecAttr : PSpecAttr ← do
+  let ext ← mkMapDeclarationExtension `pspecMap
+  let attrImpl := {
+    name := `pspec
+    descr := "Marks theorems to use with the `progress` tactic"
+    add := fun thName stx attrKind => do
+      Attribute.Builtin.ensureNoArgs stx
+      -- TODO: use the attribute kind
+      unless attrKind == AttributeKind.global do
+        throwError "invalid attribute 'pspec', must be global"
+      -- Lookup the theorem
+      let env ← getEnv
+      let thDecl := env.constants.find! thName
+      let fName ← MetaM.run' (getPSpecFunName thDecl.type)
+      trace[Progress] "Registering spec theorem for {fName}"
+      let env := ext.addEntry env (fName, thName)
+      setEnv env
+      pure ()
+  }
+  registerBuiltinAttribute attrImpl
+  pure { attr := attrImpl, ext := ext }
+
+def PSpecAttr.find? (s : PSpecAttr) (name : Name) : MetaM (Option Name) := do
+  return (s.ext.getState (← getEnv)).find? name
+  --return s.ext.find? (← getEnv) name
+
+
+end Progress
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
new file mode 100644
index 00000000..1b9ee55c
--- /dev/null
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -0,0 +1,112 @@
+import Lean
+import Base.Arith
+import Base.Progress.Base
+
+namespace Progress
+
+open Lean Elab Term Meta Tactic
+open Utils
+
+namespace Test
+  open Primitives
+
+  set_option trace.Progress true
+
+  @[pspec]
+  theorem vec_index_test (α : Type u) (v: Vec α) (i: Usize) (h: i.val < v.val.length) :
+    ∃ x, v.index α i = .ret x := by
+      apply
+      sorry
+
+  #eval pspecAttr.find? ``Primitives.Vec.index
+end Test
+
+#check isDefEq
+#check allGoals
+
+def progressLookupTheorem (asmTac : TacticM Unit) : TacticM Unit := do
+  withMainContext do
+  -- Retrieve the goal
+  let mgoal ← Tactic.getMainGoal
+  let goalTy ← mgoal.getType
+  -- Dive into the goal to lookup the theorem
+  let (fName, fLevels, args) ← do
+    withPSpec goalTy fun desc =>
+    -- TODO: check that no universally quantified variables in the arguments
+    pure (desc.fName, desc.fLevels, desc.args)
+  -- TODO: also try the assumptions
+  trace[Progress] "Function: {fName}"
+  -- TODO: use a list of theorems, and try them one by one?
+  let thName ← do
+    match ← pspecAttr.find? fName with
+    | none => throwError "Could not find a pspec theorem for {fName}"
+    | some thName => pure thName
+  trace[Progress] "Lookuped up: {thName}"
+  /- Apply the theorem
+     We try to match the theorem with the goal
+     In order to do so, we introduce meta-variables for all the parameters
+     (i.e., quantified variables and assumpions), and unify those with the goal.
+     Remark: we do not introduce meta-variables for the quantified variables
+     which don't appear in the function arguments (we want to let them
+     quantified).
+     We also make sure that all the meta variables which appear in the
+     function arguments have been instantiated
+   -/  
+  let env ← getEnv
+  let thDecl := env.constants.find! thName
+  let thTy := thDecl.type
+  -- TODO: the tactic fails if we uncomment withNewMCtxDepth
+  -- withNewMCtxDepth do
+  let (mvars, binders, thExBody) ← forallMetaTelescope thTy
+  -- Introduce the existentially quantified variables and the post-condition
+  -- in the context
+  let thBody ←
+    existsTelescope thExBody fun _evars thBody => do
+    let (thBody, _) ← destEq thBody
+    -- There shouldn't be any existential variables in thBody
+    pure thBody
+  -- Match the body with the target
+  let target := mkAppN (.const fName fLevels) args
+  trace[Progress] "mvars:\n{mvars.map Expr.mvarId!}"
+  trace[Progress] "thBody: {thBody}"
+  trace[Progress] "target: {target}"
+  let ok ← isDefEq thBody target
+  if ¬ ok then throwError "Could not unify the theorem with the target:\n- theorem: {thBody}\n- target: {target}"
+  postprocessAppMVars `progress mgoal mvars binders true true
+  Term.synthesizeSyntheticMVarsNoPostponing
+  let thBody ← instantiateMVars thBody
+  trace[Progress] "thBody (after instantiation): {thBody}"
+  -- Add the instantiated theorem to the assumptions (we apply it on the metavariables).
+  let th ← mkAppOptM thName (mvars.map some)
+  let asmName ← mkFreshUserName `h
+  let thTy ← inferType th
+  let thAsm ← Utils.addDecl asmName th thTy (asLet := false)
+  -- Update the set of goals
+  let curGoals ← getUnsolvedGoals
+  let newGoals := mvars.map Expr.mvarId!
+  let newGoals ← newGoals.filterM fun mvar => not <$> mvar.isAssigned
+  trace[Progress] "new goals: {newGoals}"
+  setGoals newGoals.toList
+  allGoals asmTac
+  let newGoals ← getUnsolvedGoals
+  setGoals (newGoals ++ curGoals)
+  -- 
+  pure ()
+
+elab "progress" : tactic => do
+  progressLookupTheorem (firstTac [assumptionTac, Arith.intTac])
+
+namespace Test
+  open Primitives
+
+  set_option trace.Progress true
+
+  @[pspec]
+  theorem vec_index_test2 (α : Type u) (v: Vec α) (i: Usize) (h: i.val < v.val.length) :
+    ∃ x, v.index α i = .ret x := by
+      progress
+      tauto
+      
+end Test
+
+end Progress
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
index 2ce63620..1351f3d4 100644
--- a/backends/lean/Base/Utils.lean
+++ b/backends/lean/Base/Utils.lean
@@ -1,4 +1,5 @@
 import Lean
+import Mathlib.Tactic.Core
 
 namespace Utils
 
@@ -211,4 +212,31 @@ example : Nat := by
 example (x : Bool) : Nat := by
   cases x <;> custom_let x := 3 <;> apply x
 
+-- Repeatedly apply a tactic
+partial def repeatTac (tac : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
+  try
+    tac
+    Tactic.allGoals (Tactic.focus (repeatTac tac))
+  -- TODO: does this restore the state?
+  catch _ => pure ()
+
+def firstTac (tacl : List (Tactic.TacticM Unit)) : Tactic.TacticM Unit := do
+  match tacl with
+  | [] => pure ()
+  | tac :: tacl =>
+    try tac
+    catch _ => firstTac tacl
+
+-- Split the goal if it is a conjunction
+def splitConjTarget : Tactic.TacticM Unit := do
+  Tactic.withMainContext do
+  let and_intro := Expr.const ``And.intro []
+  let mvarIds' ← _root_.Lean.MVarId.apply (← Tactic.getMainGoal) and_intro
+  Term.synthesizeSyntheticMVarsNoPostponing
+  Tactic.replaceMainGoal mvarIds'
+
+-- Taken from Lean.Elab.Tactic.evalAssumption
+def assumptionTac : Tactic.TacticM Unit :=
+  Tactic.liftMetaTactic fun mvarId => do mvarId.assumption; pure []
+
 end Utils