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authorSon HO2023-07-31 16:15:58 +0200
committerGitHub2023-07-31 16:15:58 +0200
commit887d0ef1efc8912c6273b5ebcf979384e9d7fa97 (patch)
tree92d6021eb549f7cc25501856edd58859786b7e90 /backends/lean
parent53adf30fe440eb8b6f58ba89f4a4c0acc7877498 (diff)
parent9b3a58e423333fc9a4a5a264c3beb0a3d951e86b (diff)
Merge pull request #31 from AeneasVerif/son_lean_backend
Improve the Lean backend
Diffstat (limited to 'backends/lean')
-rw-r--r--backends/lean/Base.lean6
-rw-r--r--backends/lean/Base/Arith.lean2
-rw-r--r--backends/lean/Base/Arith/Arith.lean0
-rw-r--r--backends/lean/Base/Arith/Base.lean60
-rw-r--r--backends/lean/Base/Arith/Int.lean280
-rw-r--r--backends/lean/Base/Arith/Scalar.lean49
-rw-r--r--backends/lean/Base/Diverge.lean7
-rw-r--r--backends/lean/Base/Diverge/Base.lean1138
-rw-r--r--backends/lean/Base/Diverge/Elab.lean1162
-rw-r--r--backends/lean/Base/Diverge/ElabBase.lean15
-rw-r--r--backends/lean/Base/IList.lean1
-rw-r--r--backends/lean/Base/IList/IList.lean284
-rw-r--r--backends/lean/Base/Primitives.lean3
-rw-r--r--backends/lean/Base/Primitives/Base.lean130
-rw-r--r--backends/lean/Base/Primitives/Scalar.lean831
-rw-r--r--backends/lean/Base/Primitives/Vec.lean145
-rw-r--r--backends/lean/Base/Progress.lean1
-rw-r--r--backends/lean/Base/Progress/Base.lean316
-rw-r--r--backends/lean/Base/Progress/Progress.lean377
-rw-r--r--backends/lean/Base/Utils.lean640
-rw-r--r--backends/lean/Primitives.lean583
-rw-r--r--backends/lean/lake-manifest.json20
-rw-r--r--backends/lean/lakefile.lean3
-rw-r--r--backends/lean/lean-toolchain1
24 files changed, 5466 insertions, 588 deletions
diff --git a/backends/lean/Base.lean b/backends/lean/Base.lean
new file mode 100644
index 00000000..2077d410
--- /dev/null
+++ b/backends/lean/Base.lean
@@ -0,0 +1,6 @@
+import Base.Utils
+import Base.Primitives
+import Base.Diverge
+import Base.Arith
+import Base.Progress
+import Base.IList
diff --git a/backends/lean/Base/Arith.lean b/backends/lean/Base/Arith.lean
new file mode 100644
index 00000000..c0d09fd2
--- /dev/null
+++ b/backends/lean/Base/Arith.lean
@@ -0,0 +1,2 @@
+import Base.Arith.Int
+import Base.Arith.Scalar
diff --git a/backends/lean/Base/Arith/Arith.lean b/backends/lean/Base/Arith/Arith.lean
new file mode 100644
index 00000000..e69de29b
--- /dev/null
+++ b/backends/lean/Base/Arith/Arith.lean
diff --git a/backends/lean/Base/Arith/Base.lean b/backends/lean/Base/Arith/Base.lean
new file mode 100644
index 00000000..9c11ed45
--- /dev/null
+++ b/backends/lean/Base/Arith/Base.lean
@@ -0,0 +1,60 @@
+import Lean
+import Std.Data.Int.Lemmas
+import Mathlib.Tactic.Linarith
+
+namespace Arith
+
+open Lean Elab Term Meta
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Arith
+
+-- 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?
+theorem add_one_le_iff_le_ne (n m : Nat) (h1 : m ≤ n) (h2 : m ≠ n) : m + 1 ≤ n := by
+ -- Damn, those proofs on natural numbers are hard - I wish Omega was in mathlib4...
+ simp [Nat.add_one_le_iff]
+ simp [Nat.lt_iff_le_and_ne]
+ simp_all
+
+/- Induction over positive integers -/
+-- TODO: move
+theorem int_pos_ind (p : Int → Prop) :
+ (zero:p 0) → (pos:∀ i, 0 ≤ i → p i → p (i + 1)) → ∀ i, 0 ≤ i → p i := by
+ intro h0 hr i hpos
+-- have heq : Int.toNat i = i := by
+-- cases i <;> simp_all
+ have ⟨ n, heq ⟩ : {n:Nat // n = i } := ⟨ Int.toNat i, by cases i <;> simp_all ⟩
+ revert i
+ induction n
+ . intro i hpos heq
+ cases i <;> simp_all
+ . rename_i n hi
+ intro i hpos heq
+ cases i <;> simp_all
+ rename_i m
+ cases m <;> simp_all
+
+-- We sometimes need this to make sure no natural numbers appear in the goals
+-- TODO: there is probably something more general to do
+theorem nat_zero_eq_int_zero : (0 : Nat) = (0 : Int) := by simp
+
+end Arith
diff --git a/backends/lean/Base/Arith/Int.lean b/backends/lean/Base/Arith/Int.lean
new file mode 100644
index 00000000..7a5bbe98
--- /dev/null
+++ b/backends/lean/Base/Arith/Int.lean
@@ -0,0 +1,280 @@
+/- 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.Utils
+import Base.Arith.Base
+
+namespace Arith
+
+open Utils
+
+-- Remark: I tried a version of the shape `HasScalarProp {a : Type} (x : a)`
+-- but the lookup didn't work
+class HasIntProp (a : Sort u) where
+ prop_ty : a → Prop
+ prop : ∀ x:a, prop_ty x
+
+class PropHasImp (x : Prop) where
+ concl : Prop
+ prop : x → concl
+
+instance (p : Int → Prop) : HasIntProp (Subtype p) where
+ prop_ty := λ x => p x
+ prop := λ x => x.property
+
+-- 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
+
+-- Check if a proposition is a linear integer proposition.
+-- We notably use this to check the goals.
+class IsLinearIntProp (x : Prop) where
+
+instance (x y : Int) : IsLinearIntProp (x < y) where
+instance (x y : Int) : IsLinearIntProp (x > y) where
+instance (x y : Int) : IsLinearIntProp (x ≤ y) where
+instance (x y : Int) : IsLinearIntProp (x ≥ y) where
+instance (x y : Int) : IsLinearIntProp (x ≥ y) where
+instance (x y : Int) : IsLinearIntProp (x = y) where
+/- It seems we don't need to do any special preprocessing when the *goal*
+ has the following shape - I guess `linarith` automatically calls `intro` -/
+instance (x y : Int) : IsLinearIntProp (¬ x = y) where
+
+open Lean Lean.Elab Lean.Meta
+
+-- 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
+ 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
+
+-- Helper
+def lookupProp (fName : String) (className : Name) (e : Expr) : MetaM (Option Expr) := do
+ trace[Arith] fName
+ -- TODO: do we need Lean.observing?
+ -- This actually eliminates the error messages
+ Lean.observing? do
+ trace[Arith] m!"{fName}: observing"
+ let ty ← Lean.Meta.inferType e
+ let hasProp ← mkAppM className #[ty]
+ let hasPropInst ← trySynthInstance hasProp
+ match hasPropInst with
+ | LOption.some i =>
+ trace[Arith] "Found {fName} instance"
+ let i_prop ← mkProjection i (Name.mkSimple "prop")
+ some (← mkAppM' i_prop #[e])
+ | _ => none
+
+-- Return an instance of `HasIntProp` for `e` if it has some
+def lookupHasIntProp (e : Expr) : MetaM (Option Expr) :=
+ lookupProp "lookupHasIntProp" ``HasIntProp e
+
+-- Collect the instances of `HasIntProp` for the subexpressions in the context
+def collectHasIntPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
+ collectInstancesFromMainCtx lookupHasIntProp
+
+-- 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 ← mkFreshAnonPropUserName
+ -- Add a declaration
+ let nval ← Utils.addDeclTac name e type (asLet := false)
+ -- Simplify to unfold the declaration to unfold (i.e., the projector)
+ Utils.simpAt [declToUnfold] [] [] (Tactic.Location.targets #[mkIdent name] false)
+ -- Return the new value
+ pure nval
+
+def introHasIntPropInstances : Tactic.TacticM (Array Expr) := do
+ trace[Arith] "Introducing the HasIntProp instances"
+ introInstances ``HasIntProp.prop_ty lookupHasIntProp
+
+-- Lookup the instances of `HasIntProp for all the sub-expressions in the context,
+-- and introduce the corresponding assumptions
+elab "intro_has_int_prop_instances" : tactic => do
+ let _ ← introHasIntPropInstances
+
+-- 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 (extraPreprocess : Tactic.TacticM Unit) : Tactic.TacticM Unit := 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
+ -- 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
+ Utils.splitDisjTac asm k k
+ -- Introduce the scalar bounds
+ let _ ← introHasIntPropInstances
+ -- Extra preprocessing, before we split on the disjunctions
+ extraPreprocess
+ -- Split
+ let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
+ splitOnAsms asms.toList
+
+elab "int_tac_preprocess" : tactic =>
+ intTacPreprocess (do pure ())
+
+-- Check if the goal is a linear arithmetic goal
+def goalIsLinearInt : Tactic.TacticM Bool := do
+ Tactic.withMainContext do
+ let gty ← Tactic.getMainTarget
+ match ← trySynthInstance (← mkAppM ``IsLinearIntProp #[gty]) with
+ | .some _ => pure true
+ | _ => pure false
+
+def intTac (splitGoalConjs : Bool) (extraPreprocess : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
+ Tactic.withMainContext do
+ Tactic.focus do
+ let g ← Tactic.getMainGoal
+ trace[Arith] "Original goal: {g}"
+ -- Introduce all the universally quantified variables (includes the assumptions)
+ let (_, g) ← g.intros
+ Tactic.setGoals [g]
+ -- 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 extraPreprocess)
+ -- More preprocessing
+ Tactic.allGoals (Utils.simpAt [] [``nat_zero_eq_int_zero] [] .wildcard)
+ -- Split the conjunctions in the goal
+ if splitGoalConjs then Tactic.allGoals (Utils.repeatTac Utils.splitConjTarget)
+ -- Call linarith
+ let linarith := do
+ let cfg : Linarith.LinarithConfig := {
+ -- We do this with our custom preprocessing
+ splitNe := false
+ }
+ Tactic.liftMetaFinishingTactic <| Linarith.linarith false [] cfg
+ Tactic.allGoals do
+ -- We check if the goal is a linear arithmetic goal: if yes, we directly
+ -- call linarith, otherwise we first apply exfalso (we do this because
+ -- linarith is too general and sometimes fails to do this correctly).
+ if ← goalIsLinearInt then do
+ trace[Arith] "linarith goal: {← Tactic.getMainGoal}"
+ linarith
+ else do
+ let g ← Tactic.getMainGoal
+ let gs ← g.apply (Expr.const ``False.elim [.zero])
+ let goals ← Tactic.getGoals
+ Tactic.setGoals (gs ++ goals)
+ Tactic.allGoals do
+ trace[Arith] "linarith goal: {← Tactic.getMainGoal}"
+ linarith
+
+elab "int_tac" args:(" split_goal"?): tactic =>
+ let split := args.raw.getArgs.size > 0
+ intTac split (do pure ())
+
+example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
+ int_tac_preprocess
+ linarith
+ 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 split_goal
+
+-- 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 split_goal
+
+-- Checking that we can prove exfalso
+example (a : Prop) (x : Int) (h0: 0 < x) (h1: x < 0) : a := by
+ int_tac
+
+end Arith
diff --git a/backends/lean/Base/Arith/Scalar.lean b/backends/lean/Base/Arith/Scalar.lean
new file mode 100644
index 00000000..b792ff21
--- /dev/null
+++ b/backends/lean/Base/Arith/Scalar.lean
@@ -0,0 +1,49 @@
+import Base.Arith.Int
+import Base.Primitives.Scalar
+
+/- Automation for scalars - TODO: not sure it is worth having two files (Int.lean and Scalar.lean) -/
+namespace Arith
+
+open Lean Lean.Elab Lean.Meta
+open Primitives
+
+def scalarTacExtraPreprocess : Tactic.TacticM Unit := do
+ Tactic.withMainContext do
+ -- Inroduce the bounds for the isize/usize types
+ let add (e : Expr) : Tactic.TacticM Unit := do
+ let ty ← inferType e
+ let _ ← Utils.addDeclTac (← Utils.mkFreshAnonPropUserName) e ty (asLet := false)
+ add (← mkAppM ``Scalar.cMin_bound #[.const ``ScalarTy.Isize []])
+ add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Usize []])
+ add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Isize []])
+ -- Reveal the concrete bounds
+ Utils.simpAt [``Scalar.min, ``Scalar.max, ``Scalar.cMin, ``Scalar.cMax,
+ ``I8.min, ``I16.min, ``I32.min, ``I64.min, ``I128.min,
+ ``I8.max, ``I16.max, ``I32.max, ``I64.max, ``I128.max,
+ ``U8.min, ``U16.min, ``U32.min, ``U64.min, ``U128.min,
+ ``U8.max, ``U16.max, ``U32.max, ``U64.max, ``U128.max,
+ ``Usize.min
+ ] [] [] .wildcard
+
+elab "scalar_tac_preprocess" : tactic =>
+ intTacPreprocess scalarTacExtraPreprocess
+
+-- A tactic to solve linear arithmetic goals in the presence of scalars
+def scalarTac (splitGoalConjs : Bool) : Tactic.TacticM Unit := do
+ intTac splitGoalConjs scalarTacExtraPreprocess
+
+elab "scalar_tac" : tactic =>
+ scalarTac false
+
+instance (ty : ScalarTy) : HasIntProp (Scalar ty) where
+ -- prop_ty is inferred
+ prop := λ x => And.intro x.hmin x.hmax
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+ intro_has_int_prop_instances
+ simp [*]
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+ scalar_tac
+
+end Arith
diff --git a/backends/lean/Base/Diverge.lean b/backends/lean/Base/Diverge.lean
new file mode 100644
index 00000000..c9a2eec2
--- /dev/null
+++ b/backends/lean/Base/Diverge.lean
@@ -0,0 +1,7 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.Diverge.Base
+import Base.Diverge.Elab
diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
new file mode 100644
index 00000000..1d548389
--- /dev/null
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -0,0 +1,1138 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+import Base.Arith.Base
+
+/- TODO: this is very useful, but is there more? -/
+set_option profiler true
+set_option profiler.threshold 100
+
+namespace Diverge
+
+namespace Fix
+
+ open Primitives
+ open Result
+
+ variable {a : Type u} {b : a → Type v}
+ variable {c d : Type w} -- TODO: why do we have to make them both : Type w?
+
+ /-! # The least fixed point definition and its properties -/
+
+ def least_p (p : Nat → Prop) (n : Nat) : Prop := p n ∧ (∀ m, m < n → ¬ p m)
+ noncomputable def least (p : Nat → Prop) : Nat :=
+ Classical.epsilon (least_p p)
+
+ -- Auxiliary theorem for [least_spec]: if there exists an `n` satisfying `p`,
+ -- there there exists a least `m` satisfying `p`.
+ theorem least_spec_aux (p : Nat → Prop) : ∀ (n : Nat), (hn : p n) → ∃ m, least_p p m := by
+ apply Nat.strongRec'
+ intros n hi hn
+ -- Case disjunction on: is n the smallest n satisfying p?
+ match Classical.em (∀ m, m < n → ¬ p m) with
+ | .inl hlt =>
+ -- Yes: trivial
+ exists n
+ | .inr hlt =>
+ simp at *
+ let ⟨ m, ⟨ hmlt, hm ⟩ ⟩ := hlt
+ have hi := hi m hmlt hm
+ apply hi
+
+ -- The specification of [least]: either `p` is never satisfied, or it is satisfied
+ -- by `least p` and no `n < least p` satisfies `p`.
+ theorem least_spec (p : Nat → Prop) : (∀ n, ¬ p n) ∨ (p (least p) ∧ ∀ n, n < least p → ¬ p n) := by
+ -- Case disjunction on the existence of an `n` which satisfies `p`
+ match Classical.em (∀ n, ¬ p n) with
+ | .inl h =>
+ -- There doesn't exist: trivial
+ apply (Or.inl h)
+ | .inr h =>
+ -- There exists: we simply use `least_spec_aux` in combination with the property
+ -- of the epsilon operator
+ simp at *
+ let ⟨ n, hn ⟩ := h
+ apply Or.inr
+ have hl := least_spec_aux p n hn
+ have he := Classical.epsilon_spec hl
+ apply he
+
+ /-! # The fixed point definitions -/
+
+ def fix_fuel (n : Nat) (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+ Result (b x) :=
+ match n with
+ | 0 => .div
+ | n + 1 =>
+ f (fix_fuel n f) x
+
+ @[simp] def fix_fuel_pred (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (x : a) (n : Nat) :=
+ not (div? (fix_fuel n f x))
+
+ def fix_fuel_P (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (x : a) (n : Nat) : Prop :=
+ fix_fuel_pred f x n
+
+ partial
+ def fixImpl (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
+ f (fixImpl f) x
+
+ -- The fact that `fix` is implemented by `fixImpl` allows us to not mark the
+ -- functions defined with the fixed-point as noncomputable. One big advantage
+ -- is that it allows us to evaluate those functions, for instance with #eval.
+ @[implemented_by fixImpl]
+ def fix (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
+ fix_fuel (least (fix_fuel_P f x)) f x
+
+ /-! # The validity property -/
+
+ -- Monotonicity relation over results
+ -- TODO: generalize (we should parameterize the definition by a relation over `a`)
+ def result_rel {a : Type u} (x1 x2 : Result a) : Prop :=
+ match x1 with
+ | div => True
+ | fail _ => x2 = x1
+ | ret _ => x2 = x1 -- TODO: generalize
+
+ -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+ def karrow_rel (k1 k2 : (x:a) → Result (b x)) : Prop :=
+ ∀ x, result_rel (k1 x) (k2 x)
+
+ -- Monotonicity property for function bodies
+ def is_mono (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+ ∀ {{k1 k2}}, karrow_rel k1 k2 → karrow_rel (f k1) (f k2)
+
+ -- "Continuity" property.
+ -- We need this, and this looks a lot like continuity. Also see this paper:
+ -- https://inria.hal.science/file/index/docid/216187/filename/tarski.pdf
+ -- We define our "continuity" criteria so that it gives us what we need to
+ -- prove the fixed-point equation, and we can also easily manipulate it.
+ def is_cont (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+ ∀ x, (Hdiv : ∀ n, fix_fuel (.succ n) f x = div) → f (fix f) x = div
+
+ /-! # The proof of the fixed-point equation -/
+ theorem fix_fuel_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}
+ (Hmono : is_mono f) :
+ ∀ {{n m}}, n ≤ m → karrow_rel (fix_fuel n f) (fix_fuel m f) := by
+ intros n
+ induction n
+ case zero => simp [karrow_rel, fix_fuel, result_rel]
+ case succ n1 Hi =>
+ intros m Hle x
+ simp [result_rel]
+ match m with
+ | 0 =>
+ exfalso
+ zify at *
+ linarith
+ | Nat.succ m1 =>
+ simp_arith at Hle
+ simp [fix_fuel]
+ have Hi := Hi Hle
+ have Hmono := Hmono Hi x
+ simp [result_rel] at Hmono
+ apply Hmono
+
+ @[simp] theorem neg_fix_fuel_P
+ {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} {x : a} {n : Nat} :
+ ¬ fix_fuel_P f x n ↔ (fix_fuel n f x = div) := by
+ simp [fix_fuel_P, div?]
+ cases fix_fuel n f x <;> simp
+
+ theorem fix_fuel_fix_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
+ ∀ n, karrow_rel (fix_fuel n f) (fix f) := by
+ intros n x
+ simp [result_rel]
+ have Hl := least_spec (fix_fuel_P f x)
+ simp at Hl
+ match Hl with
+ | .inl Hl => simp [*]
+ | .inr ⟨ Hl, Hn ⟩ =>
+ match Classical.em (fix_fuel n f x = div) with
+ | .inl Hd =>
+ simp [*]
+ | .inr Hd =>
+ have Hineq : least (fix_fuel_P f x) ≤ n := by
+ -- Proof by contradiction
+ cases Classical.em (least (fix_fuel_P f x) ≤ n) <;> simp [*]
+ simp at *
+ rename_i Hineq
+ have Hn := Hn n Hineq
+ contradiction
+ have Hfix : ¬ (fix f x = div) := by
+ simp [fix]
+ -- By property of the least upper bound
+ revert Hd Hl
+ -- TODO: there is no conversion to select the head of a function!
+ conv => lhs; apply congr_fun; apply congr_fun; apply congr_fun; simp [fix_fuel_P, div?]
+ cases fix_fuel (least (fix_fuel_P f x)) f x <;> simp
+ have Hmono := fix_fuel_mono Hmono Hineq x
+ simp [result_rel] at Hmono
+ simp [fix] at *
+ cases Heq: fix_fuel (least (fix_fuel_P f x)) f x <;>
+ cases Heq':fix_fuel n f x <;>
+ simp_all
+
+ theorem fix_fuel_P_least {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
+ ∀ {{x n}}, fix_fuel_P f x n → fix_fuel_P f x (least (fix_fuel_P f x)) := by
+ intros x n Hf
+ have Hfmono := fix_fuel_fix_mono Hmono n x
+ -- TODO: there is no conversion to select the head of a function!
+ conv => apply congr_fun; simp [fix_fuel_P]
+ simp [fix_fuel_P] at Hf
+ revert Hf Hfmono
+ simp [div?, result_rel, fix]
+ cases fix_fuel n f x <;> simp_all
+
+ -- Prove the fixed point equation in the case there exists some fuel for which
+ -- the execution terminates
+ theorem fix_fixed_eq_terminates (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (Hmono : is_mono f)
+ (x : a) (n : Nat) (He : fix_fuel_P f x n) :
+ fix f x = f (fix f) x := by
+ have Hl := fix_fuel_P_least Hmono He
+ -- TODO: better control of simplification
+ conv at Hl =>
+ apply congr_fun
+ simp [fix_fuel_P]
+ -- The least upper bound is > 0
+ have ⟨ n, Hsucc ⟩ : ∃ n, least (fix_fuel_P f x) = Nat.succ n := by
+ revert Hl
+ simp [div?]
+ cases least (fix_fuel_P f x) <;> simp [fix_fuel]
+ simp [Hsucc] at Hl
+ revert Hl
+ simp [*, div?, fix, fix_fuel]
+ -- Use the monotonicity
+ have Hfixmono := fix_fuel_fix_mono Hmono n
+ have Hvm := Hmono Hfixmono x
+ -- Use functional extensionality
+ simp [result_rel, fix] at Hvm
+ revert Hvm
+ split <;> simp [*] <;> intros <;> simp [*]
+
+ theorem fix_fixed_eq_forall {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+ (Hmono : is_mono f) (Hcont : is_cont f) :
+ ∀ x, fix f x = f (fix f) x := by
+ intros x
+ -- Case disjunction: is there a fuel such that the execution successfully execute?
+ match Classical.em (∃ n, fix_fuel_P f x n) with
+ | .inr He =>
+ -- No fuel: the fixed point evaluates to `div`
+ --simp [fix] at *
+ simp at *
+ conv => lhs; simp [fix]
+ have Hel := He (Nat.succ (least (fix_fuel_P f x))); simp [*, fix_fuel] at *; clear Hel
+ -- Use the "continuity" of `f`
+ have He : ∀ n, fix_fuel (.succ n) f x = div := by intros; simp [*]
+ have Hcont := Hcont x He
+ simp [Hcont]
+ | .inl ⟨ n, He ⟩ => apply fix_fixed_eq_terminates f Hmono x n He
+
+ -- The final fixed point equation
+ theorem fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+ (Hmono : is_mono f) (Hcont : is_cont f) :
+ fix f = f (fix f) := by
+ have Heq := fix_fixed_eq_forall Hmono Hcont
+ have Heq1 : fix f = (λ x => fix f x) := by simp
+ rw [Heq1]
+ conv => lhs; ext; simp [Heq]
+
+ /-! # Making the proofs of validity manageable (and automatable) -/
+
+ -- Monotonicity property for expressions
+ def is_mono_p (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+ ∀ {{k1 k2}}, karrow_rel k1 k2 → result_rel (e k1) (e k2)
+
+ theorem is_mono_p_same (x : Result c) :
+ @is_mono_p a b c (λ _ => x) := by
+ simp [is_mono_p, karrow_rel, result_rel]
+ split <;> simp
+
+ theorem is_mono_p_rec (x : a) :
+ @is_mono_p a b (b x) (λ f => f x) := by
+ simp_all [is_mono_p, karrow_rel, result_rel]
+
+ -- The important lemma about `is_mono_p`
+ theorem is_mono_p_bind
+ (g : ((x:a) → Result (b x)) → Result c)
+ (h : c → ((x:a) → Result (b x)) → Result d) :
+ is_mono_p g →
+ (∀ y, is_mono_p (h y)) →
+ @is_mono_p a b d (λ k => @Bind.bind Result _ c d (g k) (fun y => h y k)) := by
+-- @is_mono_p a b d (λ k => do let (y : c) ← g k; h y k) := by
+ intro hg hh
+ simp [is_mono_p]
+ intro fg fh Hrgh
+ simp [karrow_rel, result_rel]
+ have hg := hg Hrgh; simp [result_rel] at hg
+ cases heq0: g fg <;> simp_all
+ rename_i y _
+ have hh := hh y Hrgh; simp [result_rel] at hh
+ simp_all
+
+ -- Continuity property for expressions - note that we take the continuation
+ -- as parameter
+ def is_cont_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+ (Hc : ∀ n, e (fix_fuel n k) = .div) →
+ e (fix k) = .div
+
+ theorem is_cont_p_same (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (x : Result c) :
+ is_cont_p k (λ _ => x) := by
+ simp [is_cont_p]
+
+ theorem is_cont_p_rec (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+ is_cont_p f (λ f => f x) := by
+ simp_all [is_cont_p, fix]
+
+ -- The important lemma about `is_cont_p`
+ theorem is_cont_p_bind
+ (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (Hkmono : is_mono k)
+ (g : ((x:a) → Result (b x)) → Result c)
+ (h : c → ((x:a) → Result (b x)) → Result d) :
+ is_mono_p g →
+ is_cont_p k g →
+ (∀ y, is_mono_p (h y)) →
+ (∀ y, is_cont_p k (h y)) →
+ is_cont_p k (λ k => do let y ← g k; h y k) := by
+ intro Hgmono Hgcont Hhmono Hhcont
+ simp [is_cont_p]
+ intro Hdiv
+ -- Case on `g (fix... k)`: is there an n s.t. it terminates?
+ cases Classical.em (∀ n, g (fix_fuel n k) = .div) <;> rename_i Hn
+ . -- Case 1: g diverges
+ have Hgcont := Hgcont Hn
+ simp_all
+ . -- Case 2: g doesn't diverge
+ simp at Hn
+ let ⟨ n, Hn ⟩ := Hn
+ have Hdivn := Hdiv n
+ have Hffmono := fix_fuel_fix_mono Hkmono n
+ have Hgeq := Hgmono Hffmono
+ simp [result_rel] at Hgeq
+ cases Heq: g (fix_fuel n k) <;> rename_i y <;> simp_all
+ -- Remains the .ret case
+ -- Use Hdiv to prove that: ∀ n, h y (fix_fuel n f) = div
+ -- We do this in two steps: first we prove it for m ≥ n
+ have Hhdiv: ∀ m, h y (fix_fuel m k) = .div := by
+ have Hhdiv : ∀ m, n ≤ m → h y (fix_fuel m k) = .div := by
+ -- We use the fact that `g (fix_fuel n f) = .div`, combined with Hdiv
+ intro m Hle
+ have Hdivm := Hdiv m
+ -- Monotonicity of g
+ have Hffmono := fix_fuel_mono Hkmono Hle
+ have Hgmono := Hgmono Hffmono
+ -- We need to clear Hdiv because otherwise simp_all rewrites Hdivm with Hdiv
+ clear Hdiv
+ simp_all [result_rel]
+ intro m
+ -- TODO: we shouldn't need the excluded middle here because it is decidable
+ cases Classical.em (n ≤ m) <;> rename_i Hl
+ . apply Hhdiv; assumption
+ . simp at Hl
+ -- Make a case disjunction on `h y (fix_fuel m k)`: if it is not equal
+ -- to div, use the monotonicity of `h y`
+ have Hle : m ≤ n := by linarith
+ have Hffmono := fix_fuel_mono Hkmono Hle
+ have Hmono := Hhmono y Hffmono
+ simp [result_rel] at Hmono
+ cases Heq: h y (fix_fuel m k) <;> simp_all
+ -- We can now use the continuity hypothesis for h
+ apply Hhcont; assumption
+
+ -- The validity property for an expression
+ def is_valid_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+ (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+ is_mono_p e ∧
+ (is_mono k → is_cont_p k e)
+
+ @[simp] theorem is_valid_p_same
+ (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : Result c) :
+ is_valid_p k (λ _ => x) := by
+ simp [is_valid_p, is_mono_p_same, is_cont_p_same]
+
+ @[simp] theorem is_valid_p_rec
+ (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+ 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)
+ theorem is_valid_p_bind
+ {{k : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+ {{g : ((x:a) → Result (b x)) → Result c}}
+ {{h : c → ((x:a) → Result (b x)) → Result d}}
+ (Hgvalid : is_valid_p k g)
+ (Hhvalid : ∀ y, is_valid_p k (h y)) :
+ is_valid_p k (λ k => do let y ← g k; h y k) := by
+ let ⟨ Hgmono, Hgcont ⟩ := Hgvalid
+ simp [is_valid_p, forall_and] at Hhvalid
+ have ⟨ Hhmono, Hhcont ⟩ := Hhvalid
+ simp [← imp_forall_iff] at Hhcont
+ simp [is_valid_p]; constructor
+ . -- Monotonicity
+ apply is_mono_p_bind <;> assumption
+ . -- Continuity
+ intro Hkmono
+ have Hgcont := Hgcont Hkmono
+ have Hhcont := Hhcont Hkmono
+ apply is_cont_p_bind <;> assumption
+
+ def is_valid (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+ ∀ k x, is_valid_p k (λ k => f k x)
+
+ theorem is_valid_p_imp_is_valid {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+ (Hvalid : is_valid f) :
+ is_mono f ∧ is_cont f := by
+ have Hmono : is_mono f := by
+ intro f h Hr x
+ have Hmono := Hvalid (λ _ _ => .div) x
+ have Hmono := Hmono.left
+ apply Hmono; assumption
+ have Hcont : is_cont f := by
+ intro x Hdiv
+ have Hcont := (Hvalid f x).right Hmono
+ simp [is_cont_p] at Hcont
+ apply Hcont
+ intro n
+ have Hdiv := Hdiv n
+ simp [fix_fuel] at Hdiv
+ simp [*]
+ simp [*]
+
+ theorem is_valid_fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+ (Hvalid : is_valid f) :
+ fix f = f (fix f) := by
+ have ⟨ Hmono, Hcont ⟩ := is_valid_p_imp_is_valid Hvalid
+ exact fix_fixed_eq Hmono Hcont
+
+end Fix
+
+namespace FixI
+ /- Indexed fixed-point: definitions with indexed types, convenient to use for mutually
+ recursive definitions. We simply port the definitions and proofs from Fix to a more
+ specific case.
+ -/
+ open Primitives Fix
+
+ -- The index type
+ variable {id : Type u}
+
+ -- The input/output types
+ variable {a : id → Type v} {b : (i:id) → a i → Type w}
+
+ -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+ def karrow_rel (k1 k2 : (i:id) → (x:a i) → Result (b i x)) : Prop :=
+ ∀ i x, result_rel (k1 i x) (k2 i x)
+
+ def kk_to_gen (k : (i:id) → (x:a i) → Result (b i x)) :
+ (x: (i:id) × a i) → Result (b x.fst x.snd) :=
+ λ ⟨ i, x ⟩ => k i x
+
+ def kk_of_gen (k : (x: (i:id) × a i) → Result (b x.fst x.snd)) :
+ (i:id) → (x:a i) → Result (b i x) :=
+ λ i x => k ⟨ i, x ⟩
+
+ def k_to_gen (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
+ ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd) :=
+ λ kk => kk_to_gen (k (kk_of_gen kk))
+
+ def k_of_gen (k : ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd)) :
+ ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x) :=
+ λ kk => kk_of_gen (k (kk_to_gen kk))
+
+ def e_to_gen (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) :
+ ((x: (i:id) × a i) → Result (b x.fst x.snd)) → Result c :=
+ λ k => e (kk_of_gen k)
+
+ def is_valid_p (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+ (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) : Prop :=
+ Fix.is_valid_p (k_to_gen k) (e_to_gen e)
+
+ def is_valid (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) : Prop :=
+ ∀ k i x, is_valid_p k (λ k => f k i x)
+
+ def fix
+ (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
+ (i:id) → (x:a i) → Result (b i x) :=
+ kk_of_gen (Fix.fix (k_to_gen f))
+
+ theorem is_valid_fix_fixed_eq
+ {{f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
+ (Hvalid : is_valid f) :
+ fix f = f (fix f) := by
+ have Hvalid' : Fix.is_valid (k_to_gen f) := by
+ intro k x
+ simp only [is_valid, is_valid_p] at Hvalid
+ let ⟨ i, x ⟩ := x
+ have Hvalid := Hvalid (k_of_gen k) i x
+ simp only [k_to_gen, k_of_gen, kk_to_gen, kk_of_gen] at Hvalid
+ refine Hvalid
+ have Heq := Fix.is_valid_fix_fixed_eq Hvalid'
+ simp [fix]
+ conv => lhs; rw [Heq]
+
+ /- Some utilities to define the mutually recursive functions -/
+
+ -- TODO: use more
+ 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)
+ 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
+
+ abbrev in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
+ -- TODO: remove?
+ abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
+ in_out_ty :=
+ Sigma.mk in_ty out_ty
+
+ -- Initially, we had left out the parameters id, a and b.
+ -- However, by parameterizing Funs with those parameters, we can state
+ -- and prove lemmas like Funs.is_valid_p_is_valid_p
+ inductive Funs (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :
+ List in_out_ty.{v, w} → Type (max (u + 1) (max (v + 1) (w + 1))) :=
+ | Nil : Funs id a b []
+ | Cons {ity : Type v} {oty : ity → Type w} {tys : List in_out_ty}
+ (f : kk_ty id a b → (x:ity) → Result (oty x)) (tl : Funs id a b tys) :
+ Funs id a b (⟨ ity, oty ⟩ :: tys)
+
+ def get_fun {tys : List in_out_ty} (fl : Funs id a b tys) :
+ (i : Fin tys.length) → kk_ty id a b → (x : (tys.get i).fst) →
+ Result ((tys.get i).snd x) :=
+ match fl with
+ | .Nil => λ i => by have h:= i.isLt; simp at h
+ | @Funs.Cons id a b ity oty tys1 f tl =>
+ λ ⟨ i, iLt ⟩ =>
+ match i with
+ | 0 =>
+ Eq.mp (by simp [List.get]) f
+ | .succ j =>
+ have jLt: j < tys1.length := by
+ simp at iLt
+ revert iLt
+ simp_arith
+ let j: Fin tys1.length := ⟨ j, jLt ⟩
+ Eq.mp (by simp) (get_fun tl j)
+
+ def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
+ if heq: m = n then True
+ else
+ f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
+ for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
+ termination_by for_all_fin_aux n _ m h => n - m
+ decreasing_by
+ simp_wf
+ apply Nat.sub_add_lt_sub <;> simp
+ simp_all [Arith.add_one_le_iff_le_ne]
+
+ def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
+
+ theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
+ (h : m ≤ n) →
+ for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
+ := by
+ generalize h: (n - m) = k
+ revert m
+ induction k -- TODO: induction h rather?
+ case zero =>
+ simp_all
+ intro m h1 h2
+ have h: n = m := by
+ linarith
+ unfold for_all_fin_aux; simp_all
+ simp_all
+ -- There is no i s.t. m ≤ i
+ intro i h3; cases i; simp_all
+ linarith
+ case succ k hi =>
+ simp_all
+ intro m hk hmn
+ intro hf i hmi
+ have hne: m ≠ n := by
+ have hineq := Nat.lt_of_sub_eq_succ hk
+ linarith
+ -- m = i?
+ if heq: m = i then
+ -- Yes: simply use the `for_all_fin_aux` hyp
+ unfold for_all_fin_aux at hf
+ simp_all
+ tauto
+ else
+ -- No: use the induction hypothesis
+ have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
+ have hineq: m + 1 ≤ n := by
+ have hineq := Nat.lt_of_sub_eq_succ hk
+ simp [*, Nat.add_one_le_iff]
+ have heq1: n - (m + 1) = k := by
+ -- TODO: very annoying arithmetic proof
+ simp [Nat.sub_eq_iff_eq_add hineq]
+ have hineq1: m ≤ n := by linarith
+ simp [Nat.sub_eq_iff_eq_add hineq1] at hk
+ simp_arith [hk]
+ have hi := hi (m + 1) heq1 hineq
+ apply hi <;> simp_all
+ . unfold for_all_fin_aux at hf
+ simp_all
+ . simp_all [Arith.add_one_le_iff_le_ne]
+
+ -- TODO: this is not necessary anymore
+ theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
+ for_all_fin f → ∀ i, f i
+ := by
+ intro Hf i
+ apply for_all_fin_aux_imp_forall <;> try assumption
+ simp
+
+ /- Automating the proofs -/
+ @[simp] theorem is_valid_p_same
+ (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (x : Result c) :
+ is_valid_p k (λ _ => x) := by
+ simp [is_valid_p, k_to_gen, e_to_gen]
+
+ @[simp] theorem is_valid_p_rec
+ (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (i : id) (x : a i) :
+ 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}}
+ {{h : c → ((i:id) → (x:a i) → Result (b i x)) → Result d}}
+ (Hgvalid : is_valid_p k g)
+ (Hhvalid : ∀ y, is_valid_p k (h y)) :
+ is_valid_p k (λ k => do let y ← g k; h y k) := by
+ apply Fix.is_valid_p_bind
+ . apply Hgvalid
+ . apply Hhvalid
+
+ def Funs.is_valid_p
+ (k : k_ty id a b)
+ (fl : Funs id a b tys) :
+ Prop :=
+ match fl with
+ | .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}
+ (fl : Funs id a b tys) (Hvalid : is_valid_p k fl) :
+ ∀ (i : Fin tys.length) (x : (tys.get i).fst), FixI.is_valid_p k (fun k => get_fun fl i k x) := by
+ -- Prepare the induction
+ have ⟨ n, Hn ⟩ : { n : Nat // tys.length = n } := ⟨ tys.length, by rfl ⟩
+ revert tys fl Hvalid
+ induction n
+ --
+ case zero =>
+ intro tys fl Hvalid Hlen;
+ have Heq: tys = [] := by cases tys <;> simp_all
+ intro i x
+ simp_all
+ have Hi := i.isLt
+ simp_all
+ case succ n Hn =>
+ intro tys fl Hvalid Hlen i x;
+ cases fl <;> simp at Hlen i x Hvalid
+ rename_i ity oty tys f fl
+ have ⟨ Hvf, Hvalid ⟩ := Hvalid
+ have Hvf1: is_valid_p k fl := by
+ simp [Hvalid, Funs.is_valid_p]
+ have Hn := @Hn tys fl Hvf1 (by simp [*])
+ -- Case disjunction on i
+ match i with
+ | ⟨ 0, _ ⟩ =>
+ simp at x
+ simp [get_fun]
+ apply (Hvf x)
+ | ⟨ .succ j, HiLt ⟩ =>
+ simp_arith at HiLt
+ simp at x
+ let j : Fin (List.length tys) := ⟨ j, by simp_arith [HiLt] ⟩
+ have Hn := Hn j x
+ apply Hn
+
+ def Funs.is_valid_p_is_valid_p
+ (tys : List in_out_ty)
+ (k : k_ty (Fin (List.length tys)) (λ i => (tys.get i).fst) (fun i x => (List.get tys i).snd x))
+ (fl : Funs (Fin tys.length) (λ i => (tys.get i).fst) (λ i x => (tys.get i).snd x) tys) :
+ fl.is_valid_p k →
+ ∀ (i : Fin tys.length) (x : (tys.get i).fst),
+ FixI.is_valid_p k (fun k => get_fun fl i k x)
+ := by
+ intro Hvalid
+ apply is_valid_p_is_valid_p_aux; simp [*]
+
+end FixI
+
+namespace Ex1
+ /- An example of use of the fixed-point -/
+ open Primitives Fix
+
+ variable {a : Type} (k : (List a × Int) → Result a)
+
+ def list_nth_body (x : (List a × Int)) : Result a :=
+ let (ls, i) := x
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else k (tl, i - 1)
+
+ theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
+ intro k x
+ simp [list_nth_body]
+ split <;> simp
+ split <;> simp
+
+ def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (ls, i)
+
+ -- The unfolding equation - diverges if `i < 0`
+ theorem list_nth_eq (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)
+ := by
+ have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
+ simp [list_nth]
+ conv => lhs; rw [Heq]
+
+end Ex1
+
+namespace Ex2
+ /- Same as Ex1, but we make the body of nth non tail-rec (this is mostly
+ to see what happens when there are let-bindings) -/
+ open Primitives Fix
+
+ variable {a : Type} (k : (List a × Int) → Result a)
+
+ def list_nth_body (x : (List a × Int)) : Result a :=
+ let (ls, i) := x
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else
+ do
+ let y ← k (tl, i - 1)
+ .ret y
+
+ theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
+ intro k x
+ simp [list_nth_body]
+ split <;> simp
+ split <;> simp
+ apply is_valid_p_bind <;> intros <;> simp_all
+
+ def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (ls, i)
+
+ -- The unfolding equation - diverges if `i < 0`
+ theorem list_nth_eq (ls : List a) (i : Int) :
+ (list_nth ls i =
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else
+ do
+ let y ← list_nth tl (i - 1)
+ .ret y)
+ := by
+ have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
+ simp [list_nth]
+ conv => lhs; rw [Heq]
+
+end Ex2
+
+namespace Ex3
+ /- Mutually recursive functions - first encoding (see Ex4 for a better encoding) -/
+ open Primitives Fix
+
+ /- Because we have mutually recursive functions, we use a sum for the inputs
+ and the output types:
+ - inputs: the sum allows to select the function to call in the recursive
+ calls (and the functions may not have the same input types)
+ - outputs: this case is degenerate because `even` and `odd` have the same
+ return type `Bool`, but generally speaking we need a sum type because
+ the functions in the mutually recursive group may have different
+ return types.
+ -/
+ variable (k : (Int ⊕ Int) → Result (Bool ⊕ Bool))
+
+ def is_even_is_odd_body (x : (Int ⊕ Int)) : Result (Bool ⊕ Bool) :=
+ match x with
+ | .inl i =>
+ -- Body of `is_even`
+ if i = 0
+ then .ret (.inl true) -- We use .inl because this is `is_even`
+ else
+ do
+ let b ←
+ do
+ -- Call `odd`: we need to wrap the input value in `.inr`, then
+ -- extract the output value
+ let r ← k (.inr (i- 1))
+ match r with
+ | .inl _ => .fail .panic -- Invalid output
+ | .inr b => .ret b
+ -- Wrap the return value
+ .ret (.inl b)
+ | .inr i =>
+ -- Body of `is_odd`
+ if i = 0
+ then .ret (.inr false) -- We use .inr because this is `is_odd`
+ else
+ do
+ let b ←
+ do
+ -- Call `is_even`: we need to wrap the input value in .inr, then
+ -- extract the output value
+ let r ← k (.inl (i- 1))
+ match r with
+ | .inl b => .ret b
+ | .inr _ => .fail .panic -- Invalid output
+ -- Wrap the return value
+ .ret (.inr b)
+
+ theorem is_even_is_odd_body_is_valid:
+ ∀ k x, is_valid_p k (λ k => is_even_is_odd_body k x) := by
+ intro k x
+ simp [is_even_is_odd_body]
+ split <;> simp <;> split <;> simp
+ apply is_valid_p_bind; simp
+ intros; split <;> simp
+ apply is_valid_p_bind; simp
+ intros; split <;> simp
+
+ def is_even (i : Int): Result Bool :=
+ do
+ let r ← fix is_even_is_odd_body (.inl i)
+ match r with
+ | .inl b => .ret b
+ | .inr _ => .fail .panic
+
+ def is_odd (i : Int): Result Bool :=
+ do
+ let r ← fix is_even_is_odd_body (.inr i)
+ match r with
+ | .inl _ => .fail .panic
+ | .inr b => .ret b
+
+ -- The unfolding equation for `is_even` - diverges if `i < 0`
+ theorem is_even_eq (i : Int) :
+ is_even i = (if i = 0 then .ret true else is_odd (i - 1))
+ := by
+ have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
+ simp [is_even, is_odd]
+ conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
+ -- Very annoying: we need to swap the matches
+ -- Doing this with rewriting lemmas is hard generally speaking
+ -- (especially as we may have to generate lemmas for user-defined
+ -- inductives on the fly).
+ -- The simplest is to repeatedly split then simplify (we identify
+ -- the outer match or monadic let-binding, and split on its scrutinee)
+ split <;> simp
+ cases H0 : fix is_even_is_odd_body (Sum.inr (i - 1)) <;> simp
+ rename_i v
+ split <;> simp
+
+ -- The unfolding equation for `is_odd` - diverges if `i < 0`
+ theorem is_odd_eq (i : Int) :
+ is_odd i = (if i = 0 then .ret false else is_even (i - 1))
+ := by
+ have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
+ simp [is_even, is_odd]
+ conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
+ -- Same remark as for `even`
+ split <;> simp
+ cases H0 : fix is_even_is_odd_body (Sum.inl (i - 1)) <;> simp
+ rename_i v
+ split <;> simp
+
+end Ex3
+
+namespace Ex4
+ /- Mutually recursive functions - 2nd encoding -/
+ open Primitives FixI
+
+ /- We make the input type and output types dependent on a parameter -/
+ @[simp] def tys : List in_out_ty := [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)]
+ @[simp] def input_ty (i : Fin 2) : Type := (tys.get i).fst
+ @[simp] def output_ty (i : Fin 2) (x : input_ty i) : Type :=
+ (tys.get i).snd x
+
+ /- The bodies are more natural -/
+ def is_even_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
+ if i = 0
+ then .ret true
+ else do
+ let b ← k 1 (i - 1)
+ .ret b
+
+ def is_odd_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
+ if i = 0
+ then .ret false
+ else do
+ let b ← k 0 (i - 1)
+ .ret b
+
+ @[simp] def bodies :
+ Funs (Fin 2) input_ty output_ty
+ [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)] :=
+ Funs.Cons (is_even_body) (Funs.Cons (is_odd_body) Funs.Nil)
+
+ def body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 2) :
+ (x : input_ty i) → Result (output_ty i x) := get_fun bodies i k
+
+ 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]
+ intro k
+ apply (Funs.is_valid_p_is_valid_p tys)
+ simp [Funs.is_valid_p]
+ (repeat (apply And.intro)) <;> intro x <;> simp at x <;>
+ simp only [is_even_body, is_odd_body]
+ -- Prove the validity of the individual bodies
+ . split <;> simp
+ apply is_valid_p_bind <;> simp
+ . split <;> simp
+ apply is_valid_p_bind <;> simp
+
+ theorem body_fix_eq : fix body = body (fix body) :=
+ is_valid_fix_fixed_eq body_is_valid
+
+ def is_even (i : Int) : Result Bool := fix body 0 i
+ def is_odd (i : Int) : Result Bool := fix body 1 i
+
+ theorem is_even_eq (i : Int) : is_even i =
+ (if i = 0
+ then .ret true
+ else do
+ let b ← is_odd (i - 1)
+ .ret b) := by
+ simp [is_even, is_odd];
+ conv => lhs; rw [body_fix_eq]
+
+ theorem is_odd_eq (i : Int) : is_odd i =
+ (if i = 0
+ then .ret false
+ else do
+ let b ← is_even (i - 1)
+ .ret b) := by
+ simp [is_even, is_odd];
+ conv => lhs; rw [body_fix_eq]
+end Ex4
+
+namespace Ex5
+ /- Higher-order example -/
+ open Primitives Fix
+
+ variable {a b : Type}
+
+ /- An auxiliary function, which doesn't require the fixed-point -/
+ def map (f : a → Result b) (ls : List a) : Result (List b) :=
+ match ls with
+ | [] => .ret []
+ | hd :: tl =>
+ do
+ let hd ← f hd
+ let tl ← map f tl
+ .ret (hd :: tl)
+
+ /- The validity theorem for `map`, generic in `f` -/
+ theorem map_is_valid
+ {{f : (a → Result b) → a → Result c}}
+ (Hfvalid : ∀ k x, is_valid_p k (λ k => f k x))
+ (k : (a → Result b) → a → Result b)
+ (ls : List a) :
+ is_valid_p k (λ k => map (f k) ls) := by
+ induction ls <;> simp [map]
+ apply is_valid_p_bind <;> simp_all
+ intros
+ apply is_valid_p_bind <;> simp_all
+
+ /- An example which uses map -/
+ inductive Tree (a : Type) :=
+ | leaf (x : a)
+ | node (tl : List (Tree a))
+
+ def id_body (k : Tree a → Result (Tree a)) (t : Tree a) : Result (Tree a) :=
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map k tl
+ .ret (.node tl)
+
+ theorem id_body_is_valid :
+ ∀ k x, is_valid_p k (λ k => @id_body a k x) := by
+ intro k x
+ simp only [id_body]
+ split <;> simp
+ apply is_valid_p_bind <;> simp [*]
+ -- We have to show that `map k tl` is valid
+ apply map_is_valid;
+ -- Remark: if we don't do the intro, then the last step is expensive:
+ -- "typeclass inference of Nonempty took 119ms"
+ intro k x
+ simp only [is_valid_p_same, is_valid_p_rec]
+
+ def id (t : Tree a) := fix id_body t
+
+ -- The unfolding equation
+ theorem id_eq (t : Tree a) :
+ (id t =
+ match t with
+ | .leaf x => .ret (.leaf x)
+ | .node tl =>
+ do
+ let tl ← map id tl
+ .ret (.node tl))
+ := by
+ have Heq := is_valid_fix_fixed_eq (@id_body_is_valid a)
+ simp [id]
+ conv => lhs; rw [Heq]; simp; rw [id_body]
+
+end Ex5
+
+namespace Ex6
+ /- `list_nth` again, but this time we use FixI -/
+ open Primitives FixI
+
+ @[simp] def tys.{u} : List in_out_ty :=
+ [mk_in_out_ty ((a:Type u) × (List a × Int)) (λ ⟨ a, _ ⟩ => a)]
+
+ @[simp] def input_ty (i : Fin 1) := (tys.get i).fst
+ @[simp] def output_ty (i : Fin 1) (x : input_ty i) :=
+ (tys.get i).snd x
+
+ def list_nth_body.{u} (k : (i:Fin 1) → (x:input_ty i) → Result (output_ty i x))
+ (x : (a : Type u) × List a × Int) : Result x.fst :=
+ let ⟨ a, ls, i ⟩ := x
+ match ls with
+ | [] => .fail .panic
+ | hd :: tl =>
+ if i = 0 then .ret hd
+ else k 0 ⟨ a, tl, i - 1 ⟩
+
+ @[simp] def bodies :
+ Funs (Fin 1) input_ty output_ty tys :=
+ Funs.Cons list_nth_body Funs.Nil
+
+ 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 body_is_valid: is_valid body := by
+ -- Split the proof into proofs of validity of the individual bodies
+ rw [is_valid]
+ simp only [body]
+ intro k
+ apply (Funs.is_valid_p_is_valid_p tys)
+ simp [Funs.is_valid_p]
+ (repeat (apply And.intro)); intro x; simp at x
+ simp only [list_nth_body]
+ -- Prove the validity of the individual bodies
+ intro k x
+ simp [list_nth_body]
+ 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))
+
+ def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
+ fix body 0 ⟨ a, ls , i ⟩
+
+ -- The unfolding equation - diverges if `i < 0`
+ theorem list_nth_eq (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)
+ := by
+ 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
new file mode 100644
index 00000000..f109e847
--- /dev/null
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -0,0 +1,1162 @@
+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
+
+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 Lean.Meta
+open Utils
+
+/- The following was copied from the `wfRecursion` function. -/
+
+open WF in
+
+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]
+
+-- Return the `a` in `Return a`
+def getResultTy (ty : Expr) : MetaM Expr :=
+ ty.withApp fun f args => do
+ if ¬ f.isConstOf ``Result ∨ args.size ≠ 1 then
+ throwError "Invalid argument to getResultTy: {ty}"
+ else
+ pure (args.get! 0)
+
+/- Deconstruct a sigma type.
+
+ For instance, deconstructs `(a : Type) × List a` into
+ `Type` and `λ a => List a`.
+ -/
+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)
+
+/- Generate a Sigma type from a list of *variables* (all the expressions
+ must be variables).
+
+ Example:
+ - xl = [(a:Type), (ls:List a), (i:Int)]
+
+ Generates:
+ `(a:Type) × (ls:List a) × (i:Int)`
+
+ -/
+def mkSigmasType (xl : List Expr) : MetaM Expr :=
+ match xl with
+ | [] => do
+ trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
+ pure (Expr.const ``PUnit.unit [])
+ | [x] => do
+ trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
+ let ty ← Lean.Meta.inferType x
+ pure ty
+ | x :: xl => do
+ trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
+ let alpha ← Lean.Meta.inferType x
+ 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]
+
+/- 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
+
+/- 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.
+
+ We use this function to currify functions (the function bodies given to the
+ fixed-point operator must be unary functions).
+
+ Example:
+ ========
+ - xl = `[a:Type, ls:List a, i:Int]`
+ - out = `a`
+ - index = 0
+
+ generates (getting rid of most of the syntactic sugar):
+ ```
+ λ scrut0 => match scrut0 with
+ | Sigma.mk x scrut1 =>
+ match scrut1 with
+ | Sigma.mk ls i =>
+ a
+ ```
+-/
+partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr :=
+ match xl with
+ | [] => do
+ -- This would be unexpected
+ throwError "mkSigmasMatch: empyt list of input parameters"
+ | [x] => do
+ -- 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 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` (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 ← mkSigmasType xl
+ let beta ← mkLambdaFVars #[fst] snd_ty
+ let snd ← mkSigmasMatch xl out (index + 1)
+ 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
+ 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
+ -- 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_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) => Primitives.Result a)
+
+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_ty_inner a scrut1)
+/- -/
+
+-- 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]
+
+/- 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) (kk_var : Expr)
+ (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+ MetaM (Array Expr) := do
+ let grSize := preDefs.size
+
+ -- 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: {nameToInfo.toList}"
+
+ -- Auxiliary function to explore the function bodies and replace the
+ -- recursive calls
+ 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}"
+
+ -- Currify the function by grouping the arguments into a dependent tuple
+ -- (over which we match to retrieve the individual arguments).
+ lambdaTelescope 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 "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 #[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.
+-- 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))
+ (bodies : Array Expr) : MetaM (Expr × 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 ← 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"
+ let bodyFuns ← mkFuns inOutTys bodies.toList
+ -- Wrap in `get_fun`
+ 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 #[kk_var, i_var] body
+ trace[Diverge.def] "mkDeclareMutRecBody: body: {body}"
+ -- Add the declaration
+ let name := grName.append "mut_rec_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 bodies and the constant
+ 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}"
+
+-- 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]
+ trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}"
+ pure eIsValid
+
+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
+ | .const _ _ => throwError "Unimplemented" -- Shouldn't get there?
+ | .bvar _
+ | .fvar _
+ | .lit _
+ | .mvar _
+ | .sort _ => throwError "Unreachable"
+ | .lam .. => throwError "Unimplemented"
+ | .forallE .. => throwError "Unreachable" -- Shouldn't get there
+ | .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-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.
+ mkLambdaFVars xs isValid (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
+ -- 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
+ 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
+ lambdaOne br fun x br => do
+ 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
+ -- Check if 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 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 := {
+ 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
+ -- 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)
+ -- - 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
+ 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 (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
+ 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
+ -- 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
+ 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
+ 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}"
+ 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]
+ -- 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}"
+ 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
+
+-- 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
+ 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
+ let numParams := me.branchesNumParams.get! idx
+ 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 brValid
+ trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
+ -- 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 := mkAnonymous "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
+ -- 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
+ -- 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
+ 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.
+--
+-- 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 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!
+ trace[Diverge.def.valid] "body: {body}"
+ lambdaTelescope 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 ++ "body_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))
+
+-- 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
+ 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 (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)
+ (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
+ 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]
+ 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.
+--
+-- 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) (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 in_ty xs.toList
+ -- Apply the fixed point
+ let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, 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 name
+ pure defs
+
+-- Prove the equations that we will use as unfolding theorems
+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
+ let defName := decls.get! i
+ -- Retrieve the arguments
+ lambdaTelescope 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 ← mkSigmasVal (inOutTys.get! i).fst 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
+ -- 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 ()
+ 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)
+ trace[Diverge.def] ("divRecursion: defs:\n" ++ msg)
+
+ -- TODO: what is this?
+ 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 input/output types of the continuation `k`. -/
+ let grLvlParams := def0.levelParams
+ trace[Diverge.def] "def0 universe levels: {def0.levelParams}"
+
+ -- 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.
+ 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 (mkSigmasType in_tys.toList)
+ -- Retrieve the type in the "Result"
+ let out_ty ← getResultTy out_ty
+ let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
+ pure (in_ty, out_ty)
+ )
+ )
+ 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 ← 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 (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`
+ 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 (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}"
+
+ -- Introduce the continuation `k`
+ let in_ty ← mkLambdaFVars #[i_var] in_ty
+ 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 (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
+ -- continuation `k` and and generate for those bodies declarations
+ 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"
+ 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
+ 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 inOutTys preDefs
+
+ -- Prove the unfolding theorems
+ trace[Diverge.def] "# Proving the unfolding theorems"
+ proveUnfoldingThms isValidThm inOutTys preDefs decls
+
+ -- Generating code -- TODO
+ 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[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[Diverge.elab] "{preDefs.map (·.declName)}"
+ try
+ withRef (preDefs[0]!.ref) do
+ divRecursion preDefs
+ catch ex =>
+ -- If it failed, we add the functions as partial functions
+ 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
+ -- 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
+ 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[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[Diverge.elab] "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))
+
+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
+ -- We can directly use `rw [list_nth]`!
+ rw [list_nth]; 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 isNonZero (i : Int) : Result Bool :=
+ if _h:i = 0 then return false
+ else
+ let b := true
+ return b
+
+ #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
+ | _ :: _ => Result.ret True
+
+ #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
new file mode 100644
index 00000000..fedb1c74
--- /dev/null
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -0,0 +1,15 @@
+import Lean
+
+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
+initialize registerTraceClass `Diverge.def.genBody
+initialize registerTraceClass `Diverge.def.valid
+initialize registerTraceClass `Diverge.def.unfold
+
+end Diverge
diff --git a/backends/lean/Base/IList.lean b/backends/lean/Base/IList.lean
new file mode 100644
index 00000000..31b66ffa
--- /dev/null
+++ b/backends/lean/Base/IList.lean
@@ -0,0 +1 @@
+import Base.IList.IList
diff --git a/backends/lean/Base/IList/IList.lean b/backends/lean/Base/IList/IList.lean
new file mode 100644
index 00000000..93047a1b
--- /dev/null
+++ b/backends/lean/Base/IList/IList.lean
@@ -0,0 +1,284 @@
+/- Complementary list functions and lemmas which operate on integers rather
+ than natural numbers. -/
+
+import Std.Data.Int.Lemmas
+import Base.Arith
+
+namespace List
+
+def len (ls : List α) : Int :=
+ match ls with
+ | [] => 0
+ | _ :: tl => 1 + len tl
+
+@[simp] theorem len_nil : len ([] : List α) = 0 := by simp [len]
+@[simp] theorem len_cons : len ((x :: tl) : List α) = 1 + len tl := by simp [len]
+
+theorem len_pos : 0 ≤ (ls : List α).len := by
+ induction ls <;> simp [*]
+ linarith
+
+instance (a : Type u) : Arith.HasIntProp (List a) where
+ prop_ty := λ ls => 0 ≤ ls.len
+ prop := λ ls => ls.len_pos
+
+-- Remark: if i < 0, then the result is none
+def indexOpt (ls : List α) (i : Int) : Option α :=
+ match ls with
+ | [] => none
+ | hd :: tl => if i = 0 then some hd else indexOpt tl (i - 1)
+
+@[simp] theorem indexOpt_nil : indexOpt ([] : List α) i = none := by simp [indexOpt]
+@[simp] theorem indexOpt_zero_cons : indexOpt ((x :: tl) : List α) 0 = some x := by simp [indexOpt]
+@[simp] theorem indexOpt_nzero_cons (hne : i ≠ 0) : indexOpt ((x :: tl) : List α) i = indexOpt tl (i - 1) := by simp [*, indexOpt]
+
+-- Remark: if i < 0, then the result is the defaul element
+def index [Inhabited α] (ls : List α) (i : Int) : α :=
+ match ls with
+ | [] => Inhabited.default
+ | x :: tl =>
+ if i = 0 then x else index tl (i - 1)
+
+@[simp] theorem index_zero_cons [Inhabited α] : index ((x :: tl) : List α) 0 = x := by simp [index]
+@[simp] theorem index_nzero_cons [Inhabited α] (hne : i ≠ 0) : index ((x :: tl) : List α) i = index tl (i - 1) := by simp [*, index]
+
+theorem indexOpt_bounds (ls : List α) (i : Int) :
+ ls.indexOpt i = none ↔ i < 0 ∨ ls.len ≤ i :=
+ match ls with
+ | [] =>
+ have : ¬ (i < 0) → 0 ≤ i := by int_tac
+ by simp; tauto
+ | _ :: tl =>
+ have := indexOpt_bounds tl (i - 1)
+ if h: i = 0 then
+ by
+ simp [*];
+ int_tac
+ else by
+ simp [*]
+ constructor <;> intros <;>
+ casesm* _ ∨ _ <;> -- splits all the disjunctions
+ first | left; int_tac | right; int_tac
+
+theorem indexOpt_eq_index [Inhabited α] (ls : List α) (i : Int) :
+ 0 ≤ i →
+ i < ls.len →
+ ls.indexOpt i = some (ls.index i) :=
+ match ls with
+ | [] => by simp; intros; linarith
+ | hd :: tl =>
+ if h: i = 0 then
+ by simp [*]
+ else
+ have hi := indexOpt_eq_index tl (i - 1)
+ by simp [*]; intros; apply hi <;> int_tac
+
+-- Remark: the list is unchanged if the index is not in bounds (in particular
+-- if it is < 0)
+def update (ls : List α) (i : Int) (y : α) : List α :=
+ match ls with
+ | [] => []
+ | x :: tl => if i = 0 then y :: tl else x :: update tl (i - 1) y
+
+-- Remark: the whole list is dropped if the index is not in bounds (in particular
+-- if it is < 0)
+def idrop (i : Int) (ls : List α) : List α :=
+ match ls with
+ | [] => []
+ | x :: tl => if i = 0 then x :: tl else idrop (i - 1) tl
+
+section Lemmas
+
+variable {α : Type u}
+
+@[simp] theorem update_nil : update ([] : List α) i y = [] := by simp [update]
+@[simp] theorem update_zero_cons : update ((x :: tl) : List α) 0 y = y :: tl := by simp [update]
+@[simp] theorem update_nzero_cons (hne : i ≠ 0) : update ((x :: tl) : List α) i y = x :: update tl (i - 1) y := by simp [*, update]
+
+@[simp] theorem idrop_nil : idrop i ([] : List α) = [] := by simp [idrop]
+@[simp] theorem idrop_zero : idrop 0 (ls : List α) = ls := by cases ls <;> simp [idrop]
+@[simp] theorem idrop_nzero_cons (hne : i ≠ 0) : idrop i ((x :: tl) : List α) = idrop (i - 1) tl := by simp [*, idrop]
+
+theorem len_eq_length (ls : List α) : ls.len = ls.length := by
+ induction ls
+ . rfl
+ . simp [*, Int.ofNat_succ, Int.add_comm]
+
+@[simp] theorem len_append (l1 l2 : List α) : (l1 ++ l2).len = l1.len + l2.len := by
+ -- Remark: simp loops here because of the following rewritings:
+ -- @Nat.cast_add: ↑(List.length l1 + List.length l2) ==> ↑(List.length l1) + ↑(List.length l2)
+ -- Int.ofNat_add_ofNat: ↑(List.length l1) + ↑(List.length l2) ==> ↑(List.length l1 + List.length l2)
+ -- TODO: post an issue?
+ simp only [len_eq_length]
+ simp only [length_append]
+ simp only [Int.ofNat_add]
+
+@[simp]
+theorem length_update (ls : List α) (i : Int) (x : α) : (ls.update i x).length = ls.length := by
+ revert i
+ induction ls <;> simp_all [length, update]
+ intro; split <;> simp [*]
+
+@[simp]
+theorem len_update (ls : List α) (i : Int) (x : α) : (ls.update i x).len = ls.len := by
+ simp [len_eq_length]
+
+@[simp]
+theorem len_map (ls : List α) (f : α → β) : (ls.map f).len = ls.len := by
+ simp [len_eq_length]
+
+theorem left_length_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l1.length = l1'.length) :
+ l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+ revert l1'
+ induction l1
+ . intro l1'; cases l1' <;> simp [*]
+ . intro l1'; cases l1' <;> simp_all; tauto
+
+theorem right_length_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l2.length = l2'.length) :
+ l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+ have := left_length_eq_append_eq l1 l2 l1' l2'
+ constructor <;> intro heq2 <;>
+ have : l1.length + l2.length = l1'.length + l2'.length := by
+ have : (l1 ++ l2).length = (l1' ++ l2').length := by simp [*]
+ simp only [length_append] at this
+ apply this
+ . simp [heq] at this
+ tauto
+ . tauto
+
+theorem left_len_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l1.len = l1'.len) :
+ l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+ simp [len_eq_length] at heq
+ apply left_length_eq_append_eq
+ assumption
+
+theorem right_len_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l2.len = l2'.len) :
+ l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+ simp [len_eq_length] at heq
+ apply right_length_eq_append_eq
+ assumption
+
+open Arith in
+theorem idrop_eq_nil_of_le (hineq : ls.len ≤ i) : idrop i ls = [] := by
+ revert i
+ induction ls <;> simp [*]
+ rename_i hd tl hi
+ intro i hineq
+ if heq: i = 0 then
+ simp [*] at *
+ have := tl.len_pos
+ linarith
+ else
+ simp at hineq
+ have : 0 < i := by int_tac
+ simp [*]
+ apply hi
+ linarith
+
+@[simp]
+theorem index_ne
+ {α : Type u} [Inhabited α] (l: List α) (i: ℤ) (j: ℤ) (x: α) :
+ 0 ≤ i → i < l.len → 0 ≤ j → j < l.len → j ≠ i →
+ (l.update i x).index j = l.index j
+ :=
+ λ _ _ _ _ _ => match l with
+ | [] => by simp at *
+ | hd :: tl =>
+ if h: i = 0 then
+ have : j ≠ 0 := by scalar_tac
+ by simp [*]
+ else if h : j = 0 then
+ have : i ≠ 0 := by scalar_tac
+ by simp [*]
+ else
+ by
+ simp [*]
+ simp at *
+ apply index_ne <;> scalar_tac
+
+@[simp]
+theorem index_eq
+ {α : Type u} [Inhabited α] (l: List α) (i: ℤ) (x: α) :
+ 0 ≤ i → i < l.len →
+ (l.update i x).index i = x
+ :=
+ fun _ _ => match l with
+ | [] => by simp at *; scalar_tac
+ | hd :: tl =>
+ if h: i = 0 then
+ by
+ simp [*]
+ else
+ by
+ simp [*]
+ simp at *
+ apply index_eq <;> scalar_tac
+
+theorem update_map_eq {α : Type u} {β : Type v} (ls : List α) (i : Int) (x : α) (f : α → β) :
+ (ls.update i x).map f = (ls.map f).update i (f x) :=
+ match ls with
+ | [] => by simp
+ | hd :: tl =>
+ if h : i = 0 then by simp [*]
+ else
+ have hi := update_map_eq tl (i - 1) x f
+ by simp [*]
+
+theorem len_flatten_update_eq {α : Type u} (ls : List (List α)) (i : Int) (x : List α)
+ (h0 : 0 ≤ i) (h1 : i < ls.len) :
+ (ls.update i x).flatten.len = ls.flatten.len + x.len - (ls.index i).len :=
+ match ls with
+ | [] => by simp at h1; int_tac
+ | hd :: tl => by
+ simp at h1
+ if h : i = 0 then simp [*]; int_tac
+ else
+ have hi := len_flatten_update_eq tl (i - 1) x (by int_tac) (by int_tac)
+ simp [*]
+ int_tac
+
+@[simp]
+theorem index_map_eq {α : Type u} {β : Type v} [Inhabited α] [Inhabited β] (ls : List α) (i : Int) (f : α → β)
+ (h0 : 0 ≤ i) (h1 : i < ls.len) :
+ (ls.map f).index i = f (ls.index i) :=
+ match ls with
+ | [] => by simp at h1; int_tac
+ | hd :: tl =>
+ if h : i = 0 then by
+ simp [*]
+ else
+ have hi := index_map_eq tl (i - 1) f (by int_tac) (by simp at h1; int_tac)
+ by
+ simp [*]
+
+def allP {α : Type u} (l : List α) (p: α → Prop) : Prop :=
+ foldr (fun a r => p a ∧ r) True l
+
+@[simp]
+theorem allP_nil {α : Type u} (p: α → Prop) : allP [] p :=
+ by simp [allP, foldr]
+
+@[simp]
+theorem allP_cons {α : Type u} (hd: α) (tl : List α) (p: α → Prop) :
+ allP (hd :: tl) p ↔ p hd ∧ allP tl p
+ := by simp [allP, foldr]
+
+def pairwise_rel
+ {α : Type u} (rel : α → α → Prop) (l: List α) : Prop
+ := match l with
+ | [] => True
+ | hd :: tl => allP tl (rel hd) ∧ pairwise_rel rel tl
+
+@[simp]
+theorem pairwise_rel_nil {α : Type u} (rel : α → α → Prop) :
+ pairwise_rel rel []
+ := by simp [pairwise_rel]
+
+@[simp]
+theorem pairwise_rel_cons {α : Type u} (rel : α → α → Prop) (hd: α) (tl: List α) :
+ pairwise_rel rel (hd :: tl) ↔ allP tl (rel hd) ∧ pairwise_rel rel tl
+ := by simp [pairwise_rel]
+
+end Lemmas
+
+end List
diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
new file mode 100644
index 00000000..91823cb6
--- /dev/null
+++ b/backends/lean/Base/Primitives.lean
@@ -0,0 +1,3 @@
+import Base.Primitives.Base
+import Base.Primitives.Scalar
+import Base.Primitives.Vec
diff --git a/backends/lean/Base/Primitives/Base.lean b/backends/lean/Base/Primitives/Base.lean
new file mode 100644
index 00000000..7c0fa3bb
--- /dev/null
+++ b/backends/lean/Base/Primitives/Base.lean
@@ -0,0 +1,130 @@
+import Lean
+
+namespace Primitives
+
+--------------------
+-- ASSERT COMMAND --Std.
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+ runTermElabM (fun _ => do
+ let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+ if not r then
+ logInfo ("Assertion failed for:\n" ++ _stx[1])
+ throwError ("Expression reduced to false:\n" ++ _stx[1])
+ pure ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+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
+
+/- HELPERS -/
+
+def ret? {α: Type u} (r: Result α): Bool :=
+ match r with
+ | ret _ => true
+ | fail _ | div => false
+
+def div? {α: Type u} (r: Result α): Bool :=
+ match r with
+ | div => true
+ | ret _ | fail _ => false
+
+def massert (b:Bool) : Result Unit :=
+ if b then ret () else fail assertionFailure
+
+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 {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
+
+-- 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_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]
+
+/- CUSTOM-DSL SUPPORT -/
+
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
+-- because we may need a hypothesis for equational reasoning in the scope. We
+-- rely on subtype, and a custom let-binding operator, in effect recreating our
+-- own variant of the do-dsl
+
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+ match o with
+ | ret x => ret ⟨x, rfl⟩
+ | fail e => fail e
+ | div => div
+
+@[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]
+
+----------
+-- MISC --
+----------
+
+@[simp] def mem.replace (a : Type) (x : a) (_ : a) : a := x
+@[simp] def mem.replace_back (a : Type) (_ : a) (y : a) : a := y
+
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
new file mode 100644
index 00000000..2e5be8bf
--- /dev/null
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -0,0 +1,831 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+import Base.Diverge.Base
+import Base.Progress.Base
+import Base.Arith.Int
+
+namespace Primitives
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- We redefine our machine integers types.
+
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits`
+-- using the simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at
+-- least officially, 16-bit microcontrollers, so this seems like a fine design decision
+-- for now.)
+
+-- Note from Chris Bailey: "If there's more than one salient property of your
+-- definition then the subtyping strategy might get messy, and the property part
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
+
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
+
+open Result Error
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+
+-- The "structured" bounds
+def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1))
+def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+def I8.smin : Int := - (HPow.hPow 2 7)
+def I8.smax : Int := HPow.hPow 2 7 - 1
+def I16.smin : Int := - (HPow.hPow 2 15)
+def I16.smax : Int := HPow.hPow 2 15 - 1
+def I32.smin : Int := -(HPow.hPow 2 31)
+def I32.smax : Int := HPow.hPow 2 31 - 1
+def I64.smin : Int := -(HPow.hPow 2 63)
+def I64.smax : Int := HPow.hPow 2 63 - 1
+def I128.smin : Int := -(HPow.hPow 2 127)
+def I128.smax : Int := HPow.hPow 2 127 - 1
+def Usize.smin : Int := 0
+def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1
+def U8.smin : Int := 0
+def U8.smax : Int := HPow.hPow 2 8 - 1
+def U16.smin : Int := 0
+def U16.smax : Int := HPow.hPow 2 16 - 1
+def U32.smin : Int := 0
+def U32.smax : Int := HPow.hPow 2 32 - 1
+def U64.smin : Int := 0
+def U64.smax : Int := HPow.hPow 2 64 - 1
+def U128.smin : Int := 0
+def U128.smax : Int := HPow.hPow 2 128 - 1
+
+-- The "normalized" bounds, that we use in practice
+def I8.min : Int := -128
+def I8.max : Int := 127
+def I16.min : Int := -32768
+def I16.max : Int := 32767
+def I32.min : Int := -2147483648
+def I32.max : Int := 2147483647
+def I64.min : Int := -9223372036854775808
+def I64.max : Int := 9223372036854775807
+def I128.min : Int := -170141183460469231731687303715884105728
+def I128.max : Int := 170141183460469231731687303715884105727
+@[simp]
+def U8.min : Int := 0
+def U8.max : Int := 255
+@[simp]
+def U16.min : Int := 0
+def U16.max : Int := 65535
+@[simp]
+def U32.min : Int := 0
+def U32.max : Int := 4294967295
+@[simp]
+def U64.min : Int := 0
+def U64.max : Int := 18446744073709551615
+@[simp]
+def U128.min : Int := 0
+def U128.max : Int := 340282366920938463463374607431768211455
+@[simp]
+def Usize.min : Int := 0
+
+def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
+ ⟨ Isize.smin, by
+ simp [Isize.smin]
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } :=
+ ⟨ Isize.smax, by
+ simp [Isize.smax]
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } :=
+ ⟨ Usize.smax, by
+ simp [Usize.smax]
+ cases System.Platform.numBits_eq <;>
+ unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.min := Isize.refined_min.val
+def Isize.max := Isize.refined_max.val
+def Usize.max := Usize.refined_max.val
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def ScalarTy.isSigned (ty : ScalarTy) : Bool :=
+ match ty with
+ | Isize
+ | I8
+ | I16
+ | I32
+ | I64
+ | I128 => true
+ | Usize
+ | U8
+ | U16
+ | U32
+ | U64
+ | U128 => false
+
+
+def Scalar.smin (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.smin
+ | .I8 => I8.smin
+ | .I16 => I16.smin
+ | .I32 => I32.smin
+ | .I64 => I64.smin
+ | .I128 => I128.smin
+ | .Usize => Usize.smin
+ | .U8 => U8.smin
+ | .U16 => U16.smin
+ | .U32 => U32.smin
+ | .U64 => U64.smin
+ | .U128 => U128.smin
+
+def Scalar.smax (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.smax
+ | .I8 => I8.smax
+ | .I16 => I16.smax
+ | .I32 => I32.smax
+ | .I64 => I64.smax
+ | .I128 => I128.smax
+ | .Usize => Usize.smax
+ | .U8 => U8.smax
+ | .U16 => U16.smax
+ | .U32 => U32.smax
+ | .U64 => U64.smax
+ | .U128 => U128.smax
+
+def Scalar.min (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.min
+ | .I8 => I8.min
+ | .I16 => I16.min
+ | .I32 => I32.min
+ | .I64 => I64.min
+ | .I128 => I128.min
+ | .Usize => Usize.min
+ | .U8 => U8.min
+ | .U16 => U16.min
+ | .U32 => U32.min
+ | .U64 => U64.min
+ | .U128 => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Isize.max
+ | .I8 => I8.max
+ | .I16 => I16.max
+ | .I32 => I32.max
+ | .I64 => I64.max
+ | .I128 => I128.max
+ | .Usize => Usize.max
+ | .U8 => U8.max
+ | .U16 => U16.max
+ | .U32 => U32.max
+ | .U64 => U64.max
+ | .U128 => U128.max
+
+def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by
+ cases ty <;> rfl
+
+def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by
+ cases ty <;> rfl
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Scalar.min .I32
+ | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+ match ty with
+ | .Isize => Scalar.max .I32
+ | .Usize => Scalar.max .U32
+ | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+ have h := Isize.refined_min.property
+ cases h <;> simp [*, Isize.min]
+
+theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
+ cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+ . have h := Isize.refined_max.property
+ cases h <;> simp [*, Isize.max]
+ . have h := Usize.refined_max.property
+ cases h <;> simp [*, Usize.max]
+
+theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
+ have := Scalar.cMin_bound ty
+ linarith
+
+theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
+ have := Scalar.cMax_bound ty
+ linarith
+
+structure Scalar (ty : ScalarTy) where
+ val : Int
+ hmin : Scalar.min ty ≤ val
+ hmax : val ≤ Scalar.max ty
+deriving Repr
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+ Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
+ Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
+ :=
+ λ h => by
+ apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+ (hmin : Scalar.min ty ≤ x) (hmax : x ≤ Scalar.max ty) : Scalar ty :=
+ { val := x, hmin := hmin, hmax := hmax }
+
+-- Tactic to prove that integers are in bounds
+-- TODO: use this: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam
+syntax "intlit" : tactic
+macro_rules
+ | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices; decide)
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+ (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by intlit) : Scalar ty :=
+ -- Remark: we initially wrote:
+ -- let ⟨ hmin, hmax ⟩ := h
+ -- Scalar.ofIntCore x hmin hmax
+ -- We updated to the line below because a similar pattern in `Scalar.tryMk`
+ -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
+ -- TODO: investigate
+ Scalar.ofIntCore x h.left h.right
+
+@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
+ (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
+
+theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
+ Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
+ simp at *
+ have ⟨ hmin, hmax ⟩ := h
+ have hbmin := Scalar.cMin_bound ty
+ have hbmax := Scalar.cMax_bound ty
+ cases hmin <;> cases hmax <;> apply And.intro <;> linarith
+
+-- Further thoughts: look at what has been done here:
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
+-- and
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
+-- which both contain a fair amount of reasoning already!
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+ if h:Scalar.check_bounds ty x then
+ -- If we do:
+ -- ```
+ -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
+ -- Scalar.ofIntCore x hmin hmax
+ -- ```
+ -- then normalization blocks (for instance, some proofs which use reflexivity fail).
+ -- However, the version below doesn't block reduction (TODO: investigate):
+ return Scalar.ofInt x (Scalar.check_bounds_prop h)
+ else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+-- Our custom remainder operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_rem (x y : Int) : Int :=
+ if 0 ≤ x then x % y
+ else - (|x| % |y|)
+
+@[simp]
+def scalar_rem_nonneg {x y : Int} (hx : 0 ≤ x) : scalar_rem x y = x % y := by
+ intros
+ simp [*, scalar_rem]
+
+-- Our custom division operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_div (x y : Int) : Int :=
+ if 0 ≤ x && 0 ≤ y then x / y
+ else if 0 ≤ x && y < 0 then - (|x| / |y|)
+ else if x < 0 && 0 ≤ y then - (|x| / |y|)
+ else |x| / |y|
+
+@[simp]
+def scalar_div_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : scalar_div x y = x / y := by
+ intros
+ simp [*, scalar_div]
+
+-- Checking that the remainder operation is correct
+#assert scalar_rem 1 2 = 1
+#assert scalar_rem (-1) 2 = -1
+#assert scalar_rem 1 (-2) = 1
+#assert scalar_rem (-1) (-2) = -1
+#assert scalar_rem 7 3 = (1:Int)
+#assert scalar_rem (-7) 3 = -1
+#assert scalar_rem 7 (-3) = 1
+#assert scalar_rem (-7) (-3) = -1
+
+-- Checking that the division operation is correct
+#assert scalar_div 3 2 = 1
+#assert scalar_div (-3) 2 = -1
+#assert scalar_div 3 (-2) = -1
+#assert scalar_div (-3) (-2) = 1
+#assert scalar_div 7 3 = 2
+#assert scalar_div (-7) 3 = -2
+#assert scalar_div 7 (-3) = -2
+#assert scalar_div (-7) (-3) = 2
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ if y.val != 0 then Scalar.tryMk ty (scalar_div x.val y.val) else fail divisionByZero
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ if y.val != 0 then Scalar.tryMk ty (scalar_rem x.val y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+ Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+ Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8 := Scalar .I8
+@[reducible] def I16 := Scalar .I16
+@[reducible] def I32 := Scalar .I32
+@[reducible] def I64 := Scalar .I64
+@[reducible] def I128 := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8 := Scalar .U8
+@[reducible] def U16 := Scalar .U16
+@[reducible] def U32 := Scalar .U32
+@[reducible] def U64 := Scalar .U64
+@[reducible] def U128 := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+ /-- `- a` computes the negation of `a`.
+ The meaning of this notation is type-dependent. -/
+ hNeg : α → β
+
+prefix:75 "-" => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+ hMod x y := Scalar.rem x y
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.add_spec {ty} {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val + y.val)
+ (hmax : x.val + y.val ≤ Scalar.max ty) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ simp [HAdd.hAdd, add, Add.add]
+ simp [tryMk]
+ split
+ . simp [pure]
+ rfl
+ . tauto
+
+theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+ (hmax : x.val + y.val ≤ Scalar.max ty) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ have hmin : Scalar.min ty ≤ x.val + y.val := by
+ have hx := x.hmin
+ have hy := y.hmin
+ cases ty <;> simp [min] at * <;> linarith
+ apply add_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.sub_spec {ty} {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val - y.val)
+ (hmax : x.val - y.val ≤ Scalar.max ty) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ simp [HSub.hSub, sub, Sub.sub]
+ simp [tryMk]
+ split
+ . simp [pure]
+ rfl
+ . tauto
+
+theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ have : x.val - y.val ≤ Scalar.max ty := by
+ have hx := x.hmin
+ have hxm := x.hmax
+ have hy := y.hmin
+ cases ty <;> simp [min, max] at * <;> linarith
+ intros
+ apply sub_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
+ ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+ apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+-- Generic theorem - shouldn't be used much
+theorem Scalar.mul_spec {ty} {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val * y.val)
+ (hmax : x.val * y.val ≤ Scalar.max ty) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ simp [HMul.hMul, mul, Mul.mul]
+ simp [tryMk]
+ split
+ . simp [pure]
+ rfl
+ . tauto
+
+theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+ (hmax : x.val * y.val ≤ Scalar.max ty) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ have : Scalar.min ty ≤ x.val * y.val := by
+ have hx := x.hmin
+ have hy := y.hmin
+ cases ty <;> simp at * <;> apply mul_nonneg hx hy
+ apply mul_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
+ ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+ apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.div_spec {ty} {x y : Scalar ty}
+ (hnz : y.val ≠ 0)
+ (hmin : Scalar.min ty ≤ scalar_div x.val y.val)
+ (hmax : scalar_div x.val y.val ≤ Scalar.max ty) :
+ ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := by
+ simp [HDiv.hDiv, div, Div.div]
+ simp [tryMk, *]
+ simp [pure]
+ rfl
+
+theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+ (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+ have hx := x.hmin
+ have hy := y.hmin
+ simp [h] at hx hy
+ have hmin : 0 ≤ x.val / y.val := Int.ediv_nonneg hx hy
+ have hmax : x.val / y.val ≤ Scalar.max ty := by
+ have := Int.ediv_le_self y.val hx
+ have := x.hmax
+ linarith
+ have hs := @div_spec ty x y hnz
+ simp [*] at hs
+ apply hs
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+ ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+ apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.rem_spec {ty} {x y : Scalar ty}
+ (hnz : y.val ≠ 0)
+ (hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
+ (hmax : scalar_rem x.val y.val ≤ Scalar.max ty) :
+ ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
+ simp [HMod.hMod, rem]
+ simp [tryMk, *]
+ simp [pure]
+ rfl
+
+theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+ (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+ have hx := x.hmin
+ have hy := y.hmin
+ simp [h] at hx hy
+ have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz
+ have hmax : x.val % y.val ≤ Scalar.max ty := by
+ have h : 0 < y.val := by int_tac
+ have h := Int.emod_lt_of_pos x.val h
+ have := y.hmax
+ linarith
+ have hs := @rem_spec ty x y hnz
+ simp [*] at hs
+ simp [*]
+
+@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+ ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+ apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore := @Scalar.ofIntCore .I8
+def I16.ofIntCore := @Scalar.ofIntCore .I16
+def I32.ofIntCore := @Scalar.ofIntCore .I32
+def I64.ofIntCore := @Scalar.ofIntCore .I64
+def I128.ofIntCore := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore := @Scalar.ofIntCore .U8
+def U16.ofIntCore := @Scalar.ofIntCore .U16
+def U32.ofIntCore := @Scalar.ofIntCore .U32
+def U64.ofIntCore := @Scalar.ofIntCore .U64
+def U128.ofIntCore := @Scalar.ofIntCore .U128
+
+-- ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt := @Scalar.ofInt .I8
+def I16.ofInt := @Scalar.ofInt .I16
+def I32.ofInt := @Scalar.ofInt .I32
+def I64.ofInt := @Scalar.ofInt .I64
+def I128.ofInt := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt := @Scalar.ofInt .U8
+def U16.ofInt := @Scalar.ofInt .U16
+def U32.ofInt := @Scalar.ofInt .U32
+def U64.ofInt := @Scalar.ofInt .U64
+def U128.ofInt := @Scalar.ofInt .U128
+
+-- TODO: factor those lemmas out
+@[simp] theorem Scalar.ofInt_val_eq {ty} (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : (Scalar.ofInt x h).val = x := by
+ simp [Scalar.ofInt, Scalar.ofIntCore]
+
+@[simp] theorem Isize.ofInt_val_eq (h : Scalar.min ScalarTy.Isize ≤ x ∧ x ≤ Scalar.max ScalarTy.Isize) : (Isize.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I8.ofInt_val_eq (h : Scalar.min ScalarTy.I8 ≤ x ∧ x ≤ Scalar.max ScalarTy.I8) : (I8.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I16.ofInt_val_eq (h : Scalar.min ScalarTy.I16 ≤ x ∧ x ≤ Scalar.max ScalarTy.I16) : (I16.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I32.ofInt_val_eq (h : Scalar.min ScalarTy.I32 ≤ x ∧ x ≤ Scalar.max ScalarTy.I32) : (I32.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I64.ofInt_val_eq (h : Scalar.min ScalarTy.I64 ≤ x ∧ x ≤ Scalar.max ScalarTy.I64) : (I64.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I128.ofInt_val_eq (h : Scalar.min ScalarTy.I128 ≤ x ∧ x ≤ Scalar.max ScalarTy.I128) : (I128.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem Usize.ofInt_val_eq (h : Scalar.min ScalarTy.Usize ≤ x ∧ x ≤ Scalar.max ScalarTy.Usize) : (Usize.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U8.ofInt_val_eq (h : Scalar.min ScalarTy.U8 ≤ x ∧ x ≤ Scalar.max ScalarTy.U8) : (U8.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U16.ofInt_val_eq (h : Scalar.min ScalarTy.U16 ≤ x ∧ x ≤ Scalar.max ScalarTy.U16) : (U16.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U32.ofInt_val_eq (h : Scalar.min ScalarTy.U32 ≤ x ∧ x ≤ Scalar.max ScalarTy.U32) : (U32.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U64.ofInt_val_eq (h : Scalar.min ScalarTy.U64 ≤ x ∧ x ≤ Scalar.max ScalarTy.U64) : (U64.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U128.ofInt_val_eq (h : Scalar.min ScalarTy.U128 ≤ x ∧ x ≤ Scalar.max ScalarTy.U128) : (U128.ofInt x h).val = x := by
+ apply Scalar.ofInt_val_eq h
+
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+ lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+ | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+ h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+ fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+ fun i j =>
+ match decEq i.val j.val with
+ | isTrue h => isTrue (Scalar.eq_of_val_eq h)
+ | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+/- 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} (n : Scalar ty) : Int := n.val
+
+-- -- We now define a type class that subsumes the various machine integer types, so
+-- -- as to write a concise definition for scalar_cast, rather than exhaustively
+-- -- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- -- and fails if a cast operation would involve a truncation or modulo.
+
+-- class MachineInteger (t: Type) where
+-- size: Nat
+-- val: t -> Fin size
+-- ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+-- for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+-- Lean.Elab.Command.elabCommand (← `(
+-- namespace $typeName
+-- instance: MachineInteger $typeName where
+-- size := size
+-- val := val
+-- ofNatCore := ofNatCore
+-- end $typeName
+-- ))
+
+-- -- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- -- Lean to infer `src`.
+
+-- def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
+-- if h: MachineInteger.val x < MachineInteger.size dst then
+-- .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
+-- else
+-- .fail integerOverflow
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Vec.lean b/backends/lean/Base/Primitives/Vec.lean
new file mode 100644
index 00000000..a09d6ac2
--- /dev/null
+++ b/backends/lean/Base/Primitives/Vec.lean
@@ -0,0 +1,145 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.IList
+import Base.Primitives.Scalar
+import Base.Arith
+import Base.Progress.Base
+
+namespace Primitives
+
+open Result Error
+
+-------------
+-- VECTORS --
+-------------
+
+def Vec (α : Type u) := { l : List α // l.length ≤ Usize.max }
+
+-- TODO: do we really need it? It should be with Subtype by default
+instance Vec.cast (a : Type u): Coe (Vec a) (List a) where coe := λ v => v.val
+
+instance (a : Type u) : Arith.HasIntProp (Vec a) where
+ prop_ty := λ v => 0 ≤ v.val.len ∧ v.val.len ≤ Scalar.max ScalarTy.Usize
+ prop := λ ⟨ _, l ⟩ => by simp[Scalar.max, List.len_eq_length, *]
+
+@[simp]
+abbrev Vec.length {α : Type u} (v : Vec α) : Int := v.val.len
+
+@[simp]
+abbrev Vec.v {α : Type u} (v : Vec α) : List α := v.val
+
+example {a: Type u} (v : Vec a) : v.length ≤ Scalar.max ScalarTy.Usize := by
+ scalar_tac
+
+def Vec.new (α : Type u): Vec α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
+
+-- TODO: very annoying that the α is an explicit parameter
+def Vec.len (α : Type u) (v : Vec α) : Usize :=
+ Usize.ofIntCore v.val.len (by scalar_tac) (by scalar_tac)
+
+@[simp]
+theorem Vec.len_val {α : Type u} (v : Vec α) : (Vec.len α v).val = v.length :=
+ by rfl
+
+-- This shouldn't be used
+def Vec.push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
+
+-- This is actually the backward function
+def Vec.push (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
+ :=
+ let nlen := List.length v.val + 1
+ if h : nlen ≤ U32.max || nlen ≤ Usize.max then
+ have h : nlen ≤ Usize.max := by
+ simp [Usize.max] at *
+ have hm := Usize.refined_max.property
+ cases h <;> cases hm <;> simp [U32.max, U64.max] at * <;> try linarith
+ return ⟨ List.concat v.val x, by simp at *; assumption ⟩
+ else
+ fail maximumSizeExceeded
+
+-- This shouldn't be used
+def Vec.insert_fwd (α : Type u) (v: Vec α) (i: Usize) (_: α) : Result Unit :=
+ if i.val < v.length then
+ .ret ()
+ else
+ .fail arrayOutOfBounds
+
+-- This is actually the backward function
+def Vec.insert (α : Type u) (v: Vec α) (i: Usize) (x: α) : Result (Vec α) :=
+ if i.val < v.length then
+ .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+ else
+ .fail arrayOutOfBounds
+
+@[pspec]
+theorem Vec.insert_spec {α : Type u} (v: Vec α) (i: Usize) (x: α)
+ (hbound : i.val < v.length) :
+ ∃ nv, v.insert α i x = ret nv ∧ nv.val = v.val.update i.val x := by
+ simp [insert, *]
+
+def Vec.index (α : Type u) (v: Vec α) (i: Usize) : Result α :=
+ match v.val.indexOpt i.val with
+ | none => fail .arrayOutOfBounds
+ | some x => ret x
+
+/- In the theorems below: we don't always need the `∃ ..`, but we use one
+ so that `progress` introduces an opaque variable and an equality. This
+ helps control the context.
+ -/
+
+@[pspec]
+theorem Vec.index_spec {α : Type u} [Inhabited α] (v: Vec α) (i: Usize)
+ (hbound : i.val < v.length) :
+ ∃ x, v.index α i = ret x ∧ x = v.val.index i.val := by
+ simp only [index]
+ -- TODO: dependent rewrite
+ have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+ simp [*]
+
+-- This shouldn't be used
+def Vec.index_back (α : Type u) (v: Vec α) (i: Usize) (_: α) : Result Unit :=
+ if i.val < List.length v.val then
+ .ret ()
+ else
+ .fail arrayOutOfBounds
+
+def Vec.index_mut (α : Type u) (v: Vec α) (i: Usize) : Result α :=
+ match v.val.indexOpt i.val with
+ | none => fail .arrayOutOfBounds
+ | some x => ret x
+
+@[pspec]
+theorem Vec.index_mut_spec {α : Type u} [Inhabited α] (v: Vec α) (i: Usize)
+ (hbound : i.val < v.length) :
+ ∃ x, v.index_mut α i = ret x ∧ x = v.val.index i.val := by
+ simp only [index_mut]
+ -- TODO: dependent rewrite
+ have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+ simp [*]
+
+instance {α : Type u} (p : Vec α → Prop) : Arith.HasIntProp (Subtype p) where
+ prop_ty := λ x => p x
+ prop := λ x => x.property
+
+def Vec.index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α) : Result (Vec α) :=
+ match v.val.indexOpt i.val with
+ | none => fail .arrayOutOfBounds
+ | some _ =>
+ .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+
+@[pspec]
+theorem Vec.index_mut_back_spec {α : Type u} (v: Vec α) (i: Usize) (x : α)
+ (hbound : i.val < v.length) :
+ ∃ nv, v.index_mut_back α i x = ret nv ∧
+ nv.val = v.val.update i.val x
+ := by
+ simp only [index_mut_back]
+ have h := List.indexOpt_bounds v.val i.val
+ split
+ . simp_all [length]; cases h <;> scalar_tac
+ . simp_all
+
+end Primitives
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..6f820a84
--- /dev/null
+++ b/backends/lean/Base/Progress/Base.lean
@@ -0,0 +1,316 @@
+import Lean
+import Std.Lean.HashSet
+import Base.Utils
+import Base.Primitives.Base
+
+namespace Progress
+
+open Lean Elab Term Meta
+open Utils
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Progress
+
+/- # Progress tactic -/
+
+structure PSpecDesc where
+ -- The universally quantified variables
+ fvars : Array Expr
+ -- The existentially quantified variables
+ evars : Array Expr
+ -- The function
+ fExpr : Expr
+ 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] "Proposition: {th}"
+ -- Dive into the quantified variables and the assumptions
+ forallTelescope th.consumeMData fun fvars th => do
+ trace[Progress] "Universally quantified arguments and assumptions: {fvars}"
+ -- Dive into the existentials
+ existsTelescope th.consumeMData fun evars th => do
+ trace[Progress] "Existentials: {evars}"
+ trace[Progress] "Proposition after stripping the quantifiers: {th}"
+ -- Take the first conjunct
+ let (th, post) ← optSplitConj th.consumeMData
+ trace[Progress] "After splitting the conjunction:\n- eq: {th}\n- post: {post}"
+ -- Destruct the equality
+ let (mExpr, ret) ← destEq th.consumeMData
+ trace[Progress] "After splitting the equality:\n- lhs: {th}\n- rhs: {ret}"
+ -- Destruct the monadic application to dive into the bind, if necessary (this
+ -- is for when we use `withPSpec` inside of the `progress` tactic), and
+ -- destruct the application to get the function name
+ mExpr.consumeMData.withApp fun mf margs => do
+ trace[Progress] "After stripping the arguments of the monad expression:\n- mf: {mf}\n- margs: {margs}"
+ let (fExpr, f, args) ← do
+ if mf.isConst ∧ mf.constName = ``Bind.bind then do
+ -- Dive into the bind
+ let fExpr := (margs.get! 4).consumeMData
+ fExpr.withApp fun f args => pure (fExpr, f, args)
+ else pure (mExpr, mf, margs)
+ trace[Progress] "After stripping the arguments of the function call:\n- f: {f}\n- args: {args}"
+ 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
+ -- All the variables which appear in the inputs given to the function are
+ -- universally quantified (in particular, they are not *existentially* quantified)
+ 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
+ fExpr
+ 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
+
+def getPSpecClassFunNames (th : Expr) : MetaM (Name × Name) :=
+ withPSpec th (fun d => do
+ let arg0 := d.args.get! 0
+ arg0.withApp fun f _ => do
+ if ¬ f.isConst then throwError "Not a constant: {f}"
+ pure (d.fName, f.constName)
+ ) true
+
+def getPSpecClassFunNameArg (th : Expr) : MetaM (Name × Expr) :=
+ withPSpec th (fun d => do
+ let arg0 := d.args.get! 0
+ pure (d.fName, arg0)
+ ) true
+
+-- "Regular" pspec attribute
+structure PSpecAttr where
+ attr : AttributeImpl
+ ext : MapDeclarationExtension Name
+ deriving Inhabited
+
+/- pspec attribute for type classes: we use the name of the type class to
+ lookup another map. We use the *first* argument of the type class to lookup
+ into this second map.
+
+ Example:
+ ========
+ We use type classes for addition. For instance, the addition between two
+ U32 is written (without syntactic sugar) as `HAdd.add (Scalar ty) x y`. As a consequence,
+ we store the theorem through the bindings: HAdd.add → Scalar → ...
+
+ SH: TODO: this (and `PSpecClassExprAttr`) is a bit ad-hoc. For now it works for the
+ specs of the scalar operations, which is what I really need, but I'm not sure it
+ applies well to other situations. A better way would probably to use type classes, but
+ I couldn't get them to work on those cases. It is worth retrying.
+-/
+structure PSpecClassAttr where
+ attr : AttributeImpl
+ ext : MapDeclarationExtension (NameMap Name)
+ deriving Inhabited
+
+/- Same as `PSpecClassAttr` but we use the full first argument (it works when it
+ is a constant). -/
+structure PSpecClassExprAttr where
+ attr : AttributeImpl
+ ext : MapDeclarationExtension (HashMap Expr Name)
+ deriving Inhabited
+
+-- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
+def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) : IO (MapDeclarationExtension α) :=
+ registerSimplePersistentEnvExtension {
+ name := name,
+ addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
+ addEntryFn := fun s n => s.insert n.1 n.2 ,
+ toArrayFn := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
+ }
+
+/- The persistent map from function to pspec theorems. -/
+initialize pspecAttr : PSpecAttr ← do
+ let ext ← mkMapDeclarationExtension `pspecMap
+ let attrImpl : AttributeImpl := {
+ 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 }
+
+/- The persistent map from type classes to pspec theorems -/
+initialize pspecClassAttr : PSpecClassAttr ← do
+ let ext : MapDeclarationExtension (NameMap Name) ← mkMapDeclarationExtension `pspecClassMap
+ let attrImpl : AttributeImpl := {
+ name := `cpspec
+ descr := "Marks theorems to use for type classes 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 'cpspec', must be global"
+ -- Lookup the theorem
+ let env ← getEnv
+ let thDecl := env.constants.find! thName
+ let (fName, argName) ← MetaM.run' (getPSpecClassFunNames thDecl.type)
+ trace[Progress] "Registering class spec theorem for ({fName}, {argName})"
+ -- Update the entry if there is one, add an entry if there is none
+ let env :=
+ match (ext.getState (← getEnv)).find? fName with
+ | none =>
+ let m := RBMap.ofList [(argName, thName)]
+ ext.addEntry env (fName, m)
+ | some m =>
+ let m := m.insert argName thName
+ ext.addEntry env (fName, m)
+ setEnv env
+ pure ()
+ }
+ registerBuiltinAttribute attrImpl
+ pure { attr := attrImpl, ext := ext }
+
+/- The 2nd persistent map from type classes to pspec theorems -/
+initialize pspecClassExprAttr : PSpecClassExprAttr ← do
+ let ext : MapDeclarationExtension (HashMap Expr Name) ← mkMapDeclarationExtension `pspecClassExprMap
+ let attrImpl : AttributeImpl := {
+ name := `cepspec
+ descr := "Marks theorems to use for type classes 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 'cpspec', must be global"
+ -- Lookup the theorem
+ let env ← getEnv
+ let thDecl := env.constants.find! thName
+ let (fName, arg) ← MetaM.run' (getPSpecClassFunNameArg thDecl.type)
+ -- Sanity check: no variables appear in the argument
+ MetaM.run' do
+ let fvars ← getFVarIds arg
+ if ¬ fvars.isEmpty then throwError "The first argument ({arg}) contains variables"
+ -- We store two bindings:
+ -- - arg to theorem name
+ -- - reduced arg to theorem name
+ let rarg ← MetaM.run' (reduceAll arg)
+ trace[Progress] "Registering class spec theorem for ({fName}, {arg}) and ({fName}, {rarg})"
+ -- Update the entry if there is one, add an entry if there is none
+ let env :=
+ match (ext.getState (← getEnv)).find? fName with
+ | none =>
+ let m := HashMap.ofList [(arg, thName), (rarg, thName)]
+ ext.addEntry env (fName, m)
+ | some m =>
+ let m := m.insert arg thName
+ let m := m.insert rarg thName
+ ext.addEntry env (fName, m)
+ 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
+
+def PSpecClassAttr.find? (s : PSpecClassAttr) (className argName : Name) : MetaM (Option Name) := do
+ match (s.ext.getState (← getEnv)).find? className with
+ | none => return none
+ | some map => return map.find? argName
+
+def PSpecClassExprAttr.find? (s : PSpecClassExprAttr) (className : Name) (arg : Expr) : MetaM (Option Name) := do
+ match (s.ext.getState (← getEnv)).find? className with
+ | none => return none
+ | some map => return map.find? arg
+
+def PSpecAttr.getState (s : PSpecAttr) : MetaM (NameMap Name) := do
+ pure (s.ext.getState (← getEnv))
+
+def PSpecClassAttr.getState (s : PSpecClassAttr) : MetaM (NameMap (NameMap Name)) := do
+ pure (s.ext.getState (← getEnv))
+
+def PSpecClassExprAttr.getState (s : PSpecClassExprAttr) : MetaM (NameMap (HashMap Expr Name)) := do
+ pure (s.ext.getState (← getEnv))
+
+def showStoredPSpec : MetaM Unit := do
+ let st ← pspecAttr.getState
+ let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+ IO.println s
+
+def showStoredPSpecClass : MetaM Unit := do
+ let st ← pspecClassAttr.getState
+ let s := st.toList.foldl (fun s (f, m) =>
+ let ms := m.toList.foldl (fun s (f, th) =>
+ f!"{s}\n {f} → {th}") f!""
+ f!"{s}\n{f} → [{ms}]") f!""
+ IO.println s
+
+def showStoredPSpecExprClass : MetaM Unit := do
+ let st ← pspecClassExprAttr.getState
+ let s := st.toList.foldl (fun s (f, m) =>
+ let ms := m.toList.foldl (fun s (f, th) =>
+ f!"{s}\n {f} → {th}") f!""
+ f!"{s}\n{f} → [{ms}]") f!""
+ IO.println s
+
+end Progress
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
new file mode 100644
index 00000000..6a4729dc
--- /dev/null
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -0,0 +1,377 @@
+import Lean
+import Base.Arith
+import Base.Progress.Base
+import Base.Primitives -- TODO: remove?
+
+namespace Progress
+
+open Lean Elab Term Meta Tactic
+open Utils
+
+inductive TheoremOrLocal where
+| Theorem (thName : Name)
+| Local (asm : LocalDecl)
+
+instance : ToMessageData TheoremOrLocal where
+ toMessageData := λ x => match x with | .Theorem thName => m!"{thName}" | .Local asm => m!"{asm.userName}"
+
+/- Type to propagate the errors of `progressWith`.
+ We need this because we use the exceptions to backtrack, when trying to
+ use the assumptions for instance. When there is actually an error we want
+ to propagate to the user, we return it. -/
+inductive ProgressError
+| Ok
+| Error (msg : MessageData)
+deriving Inhabited
+
+def progressWith (fExpr : Expr) (th : TheoremOrLocal)
+ (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
+ (asmTac : TacticM Unit) : TacticM ProgressError := do
+ /- 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 thTy ← do
+ match th with
+ | .Theorem thName =>
+ let thDecl := env.constants.find! thName
+ -- We have to introduce fresh meta-variables for the universes already
+ let ul : List (Name × Level) ←
+ thDecl.levelParams.mapM (λ x => do pure (x, ← mkFreshLevelMVar))
+ let ulMap : HashMap Name Level := HashMap.ofList ul
+ let thTy := thDecl.type.instantiateLevelParamsCore (λ x => ulMap.find! x)
+ pure thTy
+ | .Local asmDecl => pure asmDecl.type
+ trace[Progress] "Looked up theorem/assumption type: {thTy}"
+ -- TODO: the tactic fails if we uncomment withNewMCtxDepth
+ -- withNewMCtxDepth do
+ let (mvars, binders, thExBody) ← forallMetaTelescope thTy
+ trace[Progress] "After stripping foralls: {thExBody}"
+ -- Introduce the existentially quantified variables and the post-condition
+ -- in the context
+ let thBody ←
+ existsTelescope thExBody.consumeMData fun _evars thBody => do
+ trace[Progress] "After stripping existentials: {thBody}"
+ let (thBody, _) ← optSplitConj thBody
+ trace[Progress] "After splitting the conjunction: {thBody}"
+ let (thBody, _) ← destEq thBody
+ trace[Progress] "After splitting equality: {thBody}"
+ -- There shouldn't be any existential variables in thBody
+ pure thBody.consumeMData
+ -- Match the body with the target
+ trace[Progress] "Matching:\n- body:\n{thBody}\n- target:\n{fExpr}"
+ let ok ← isDefEq thBody fExpr
+ if ¬ ok then throwError "Could not unify the theorem with the target:\n- theorem: {thBody}\n- target: {fExpr}"
+ let mgoal ← Tactic.getMainGoal
+ 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 ← do
+ match th with
+ | .Theorem thName => mkAppOptM thName (mvars.map some)
+ | .Local decl => mkAppOptM' (mkFVar decl.fvarId) (mvars.map some)
+ let asmName ← do match keep with | none => mkFreshAnonPropUserName | some n => do pure n
+ let thTy ← inferType th
+ let thAsm ← Utils.addDeclTac asmName th thTy (asLet := false)
+ withMainContext do -- The context changed - TODO: remove once addDeclTac is updated
+ let ngoal ← getMainGoal
+ trace[Progress] "current goal: {ngoal}"
+ trace[Progress] "current goal: {← ngoal.isAssigned}"
+ -- The assumption should be of the shape:
+ -- `∃ x1 ... xn, f args = ... ∧ ...`
+ -- We introduce the existentially quantified variables and split the top-most
+ -- conjunction if there is one. We use the provided `ids` list to name the
+ -- introduced variables.
+ let res ← splitAllExistsTac thAsm ids.toList fun h ids => do
+ -- Split the conjunctions.
+ -- For the conjunctions, we split according once to separate the equality `f ... = .ret ...`
+ -- from the postcondition, if there is, then continue to split the postcondition if there
+ -- are remaining ids.
+ let splitEqAndPost (k : Expr → Option Expr → List (Option Name) → TacticM ProgressError) : TacticM ProgressError := do
+ if ← isConj (← inferType h) then do
+ let hName := (← h.fvarId!.getDecl).userName
+ let (optIds, ids) ← do
+ match ids with
+ | [] => do pure (some (hName, ← mkFreshAnonPropUserName), [])
+ | none :: ids => do pure (some (hName, ← mkFreshAnonPropUserName), ids)
+ | some id :: ids => do pure (some (hName, id), ids)
+ splitConjTac h optIds (fun hEq hPost => k hEq (some hPost) ids)
+ else k h none ids
+ -- Simplify the target by using the equality and some monad simplifications,
+ -- then continue splitting the post-condition
+ splitEqAndPost fun hEq hPost ids => do
+ trace[Progress] "eq and post:\n{hEq} : {← inferType hEq}\n{hPost}"
+ simpAt [] [``Primitives.bind_tc_ret, ``Primitives.bind_tc_fail, ``Primitives.bind_tc_div]
+ [hEq.fvarId!] (.targets #[] true)
+ -- Clear the equality, unless the user requests not to do so
+ let mgoal ← do
+ if keep.isSome then getMainGoal
+ else do
+ let mgoal ← getMainGoal
+ mgoal.tryClearMany #[hEq.fvarId!]
+ setGoals (mgoal :: (← getUnsolvedGoals))
+ trace[Progress] "Goal after splitting eq and post and simplifying the target: {mgoal}"
+ -- Continue splitting following the post following the user's instructions
+ match hPost with
+ | none =>
+ -- Sanity check
+ if ¬ ids.isEmpty then
+ return (.Error m!"Too many ids provided ({ids}): there is no postcondition to split")
+ else return .Ok
+ | some hPost => do
+ let rec splitPostWithIds (prevId : Name) (hPost : Expr) (ids0 : List (Option Name)) : TacticM ProgressError := do
+ match ids0 with
+ | [] =>
+ /- We used all the user provided ids.
+ Split the remaining conjunctions by using fresh ids if the user
+ instructed to fully split the post-condition, otherwise stop -/
+ if splitPost then
+ splitFullConjTac true hPost (λ _ => pure .Ok)
+ else pure .Ok
+ | nid :: ids => do
+ trace[Progress] "Splitting post: {← inferType hPost}"
+ -- Split
+ let nid ← do
+ match nid with
+ | none => mkFreshAnonPropUserName
+ | some nid => pure nid
+ trace[Progress] "\n- prevId: {prevId}\n- nid: {nid}\n- remaining ids: {ids}"
+ if ← isConj (← inferType hPost) then
+ splitConjTac hPost (some (prevId, nid)) (λ _ nhPost => splitPostWithIds nid nhPost ids)
+ else return (.Error m!"Too many ids provided ({ids0}) not enough conjuncts to split in the postcondition")
+ let curPostId := (← hPost.fvarId!.getDecl).userName
+ splitPostWithIds curPostId hPost ids
+ match res with
+ | .Error _ => return res -- Can we get there? We're using "return"
+ | .Ok =>
+ -- 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)
+ trace[Progress] "progress: replaced the goals"
+ --
+ pure .Ok
+
+-- Small utility: if `args` is not empty, return the name of the app in the first
+-- arg, if it is a const.
+def getFirstArgAppName (args : Array Expr) : MetaM (Option Name) := do
+ if args.size = 0 then pure none
+ else
+ (args.get! 0).withApp fun f _ => do
+ if f.isConst then pure (some f.constName)
+ else pure none
+
+def getFirstArg (args : Array Expr) : Option Expr := do
+ if args.size = 0 then none
+ else some (args.get! 0)
+
+/- Helper: try to lookup a theorem and apply it, or continue with another tactic
+ if it fails -/
+def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
+ (asmTac : TacticM Unit) (fExpr : Expr)
+ (kind : String) (th : Option TheoremOrLocal) (x : TacticM Unit) : TacticM Unit := do
+ let res ← do
+ match th with
+ | none =>
+ trace[Progress] "Could not find a {kind}"
+ pure none
+ | some th => do
+ trace[Progress] "Lookuped up {kind}: {th}"
+ -- Apply the theorem
+ let res ← do
+ try
+ let res ← progressWith fExpr th keep ids splitPost asmTac
+ pure (some res)
+ catch _ => none
+ match res with
+ | some .Ok => return ()
+ | some (.Error msg) => throwError msg
+ | none => x
+
+-- The array of ids are identifiers to use when introducing fresh variables
+def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrLocal)
+ (ids : Array (Option Name)) (splitPost : Bool) (asmTac : TacticM Unit) : TacticM Unit := do
+ withMainContext do
+ -- Retrieve the goal
+ let mgoal ← Tactic.getMainGoal
+ let goalTy ← mgoal.getType
+ trace[Progress] "goal: {goalTy}"
+ -- Dive into the goal to lookup the theorem
+ let (fExpr, fName, args) ← do
+ withPSpec goalTy fun desc =>
+ -- TODO: check that no quantified variables in the arguments
+ pure (desc.fExpr, desc.fName, desc.args)
+ trace[Progress] "Function: {fName}"
+ -- If the user provided a theorem/assumption: use it.
+ -- Otherwise, lookup one.
+ match withTh with
+ | some th => do
+ match ← progressWith fExpr th keep ids splitPost asmTac with
+ | .Ok => return ()
+ | .Error msg => throwError msg
+ | none =>
+ -- Try all the assumptions one by one and if it fails try to lookup a theorem.
+ let ctx ← Lean.MonadLCtx.getLCtx
+ let decls ← ctx.getDecls
+ for decl in decls.reverse do
+ trace[Progress] "Trying assumption: {decl.userName} : {decl.type}"
+ let res ← do try progressWith fExpr (.Local decl) keep ids splitPost asmTac catch _ => continue
+ match res with
+ | .Ok => return ()
+ | .Error msg => throwError msg
+ -- It failed: try to lookup a theorem
+ -- TODO: use a list of theorems, and try them one by one?
+ trace[Progress] "No assumption succeeded: trying to lookup a theorem"
+ let pspec ← do
+ let thName ← pspecAttr.find? fName
+ pure (thName.map fun th => .Theorem th)
+ tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec do
+ -- It failed: try to lookup a *class* expr spec theorem (those are more
+ -- specific than class spec theorems)
+ let pspecClassExpr ← do
+ match getFirstArg args with
+ | none => pure none
+ | some arg => do
+ let thName ← pspecClassExprAttr.find? fName arg
+ pure (thName.map fun th => .Theorem th)
+ tryLookupApply keep ids splitPost asmTac fExpr "pspec class expr theorem" pspecClassExpr do
+ -- It failed: try to lookup a *class* spec theorem
+ let pspecClass ← do
+ match ← getFirstArgAppName args with
+ | none => pure none
+ | some argName => do
+ let thName ← pspecClassAttr.find? fName argName
+ pure (thName.map fun th => .Theorem th)
+ tryLookupApply keep ids splitPost asmTac fExpr "pspec class theorem" pspecClass do
+ -- Try a recursive call - we try the assumptions of kind "auxDecl"
+ let ctx ← Lean.MonadLCtx.getLCtx
+ let decls ← ctx.getAllDecls
+ let decls := decls.filter (λ decl => match decl.kind with
+ | .default | .implDetail => false | .auxDecl => true)
+ for decl in decls.reverse do
+ trace[Progress] "Trying recursive assumption: {decl.userName} : {decl.type}"
+ let res ← do try progressWith fExpr (.Local decl) keep ids splitPost asmTac catch _ => continue
+ match res with
+ | .Ok => return ()
+ | .Error msg => throwError msg
+ -- Nothing worked: failed
+ throwError "Progress failed"
+
+syntax progressArgs := ("keep" (ident <|> "_"))? ("with" ident)? ("as" " ⟨ " (ident <|> "_"),* " .."? " ⟩")?
+
+def evalProgress (args : TSyntax `Progress.progressArgs) : TacticM Unit := do
+ let args := args.raw
+ -- Process the arguments to retrieve the identifiers to use
+ trace[Progress] "Progress arguments: {args}"
+ let (keepArg, withArg, asArgs) ←
+ match args.getArgs.toList with
+ | [keepArg, withArg, asArgs] => do pure (keepArg, withArg, asArgs)
+ | _ => throwError "Unexpected: invalid arguments"
+ let keep : Option Name ← do
+ trace[Progress] "Keep arg: {keepArg}"
+ let args := keepArg.getArgs
+ if args.size > 0 then do
+ trace[Progress] "Keep args: {args}"
+ let arg := args.get! 1
+ trace[Progress] "Keep arg: {arg}"
+ if arg.isIdent then pure (some arg.getId)
+ else do pure (some (← mkFreshAnonPropUserName))
+ else do pure none
+ trace[Progress] "Keep: {keep}"
+ let withArg ← do
+ let withArg := withArg.getArgs
+ if withArg.size > 0 then
+ let id := withArg.get! 1
+ trace[Progress] "With arg: {id}"
+ -- Attempt to lookup a local declaration
+ match (← getLCtx).findFromUserName? id.getId with
+ | some decl => do
+ trace[Progress] "With arg: local decl"
+ pure (some (.Local decl))
+ | none => do
+ -- Not a local declaration: should be a theorem
+ trace[Progress] "With arg: theorem"
+ addCompletionInfo <| CompletionInfo.id id id.getId (danglingDot := false) {} none
+ let cs ← resolveGlobalConstWithInfos id
+ match cs with
+ | [] => throwError "Could not find theorem {id}"
+ | id :: _ =>
+ pure (some (.Theorem id))
+ else pure none
+ let ids :=
+ let args := asArgs.getArgs
+ let args := (args.get! 2).getSepArgs
+ args.map (λ s => if s.isIdent then some s.getId else none)
+ trace[Progress] "User-provided ids: {ids}"
+ let splitPost : Bool :=
+ let args := asArgs.getArgs
+ (args.get! 3).getArgs.size > 0
+ trace[Progress] "Split post: {splitPost}"
+ /- For scalarTac we have a fast track: if the goal is not a linear
+ arithmetic goal, we skip (note that otherwise, scalarTac would try
+ to prove a contradiction) -/
+ let scalarTac : TacticM Unit := do
+ if ← Arith.goalIsLinearInt then
+ -- Also: we don't try to split the goal if it is a conjunction
+ -- (it shouldn't be)
+ Arith.scalarTac false
+ else
+ throwError "Not a linear arithmetic goal"
+ progressAsmsOrLookupTheorem keep withArg ids splitPost (
+ withMainContext do
+ trace[Progress] "trying to solve assumption: {← getMainGoal}"
+ firstTac [assumptionTac, scalarTac])
+ trace[Diverge] "Progress done"
+
+elab "progress" args:progressArgs : tactic =>
+ evalProgress args
+
+namespace Test
+ open Primitives Result
+
+ set_option trace.Progress true
+ set_option pp.rawOnError true
+
+ #eval showStoredPSpec
+ #eval showStoredPSpecClass
+
+ example {ty} {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val + y.val)
+ (hmax : x.val + y.val ≤ Scalar.max ty) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ progress keep _ as ⟨ z, h1 .. ⟩
+ simp [*, h1]
+
+ example {ty} {x y : Scalar ty}
+ (hmin : Scalar.min ty ≤ x.val + y.val)
+ (hmax : x.val + y.val ≤ Scalar.max ty) :
+ ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+ progress keep h with Scalar.add_spec as ⟨ z ⟩
+ simp [*, h]
+
+ /- Checking that universe instantiation works: the original spec uses
+ `α : Type u` where u is quantified, while here we use `α : Type 0` -/
+ example {α : Type} (v: Vec α) (i: Usize) (x : α)
+ (hbounds : i.val < v.length) :
+ ∃ nv, v.index_mut_back α i x = ret nv ∧
+ nv.val = v.val.update i.val x := by
+ progress
+ simp [*]
+
+end Test
+
+end Progress
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
new file mode 100644
index 00000000..1f8f1455
--- /dev/null
+++ b/backends/lean/Base/Utils.lean
@@ -0,0 +1,640 @@
+import Lean
+import Mathlib.Tactic.Core
+import Mathlib.Tactic.LeftRight
+import Base.UtilsBase
+
+/-
+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
+-/
+
+/-
+TODO:
+- we want an easier to use cases:
+ - keeps in the goal an equation of the shape: `t = case`
+ - if called on Prop terms, uses Classical.em
+ Actually, the cases from mathlib seems already quite powerful
+ (https://leanprover-community.github.io/mathlib_docs/tactics.html#cases)
+ For instance: cases h : e
+ Also: **casesm**
+- better split tactic
+- we need conversions to operate on the head of applications.
+ Actually, something like this works:
+ ```
+ conv at Hl =>
+ apply congr_fun
+ simp [fix_fuel_P]
+ ```
+ Maybe we need a rpt ... ; focus?
+- simplifier/rewriter have a strange behavior sometimes
+-/
+
+
+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
+
+-- TODO: move?
+@[simp]
+theorem neq_imp {α : Type u} {x y : α} (h : ¬ x = y) : ¬ y = x := by intro; simp_all
+
+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 Utils
+
+open Lean Elab Term Meta Tactic
+
+-- 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
+
+def printDecls (decls : List LocalDecl) : MetaM Unit := do
+ let decls ← decls.foldrM (λ decl msg => do
+ pure (m!"\n{decl.toExpr} : {← inferType decl.toExpr}" ++ msg)) m!""
+ logInfo m!"# Ctx decls:{decls}"
+
+-- Small utility: print all the declarations in the context (including the "implementation details")
+elab "print_all_ctx_decls" : tactic => do
+ let ctx ← Lean.MonadLCtx.getLCtx
+ let decls ← ctx.getAllDecls
+ printDecls decls
+
+-- Small utility: print all declarations in the context
+elab "print_ctx_decls" : tactic => do
+ let ctx ← Lean.MonadLCtx.getLCtx
+ let decls ← ctx.getDecls
+ printDecls decls
+
+-- 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
+
+-- Generate a fresh user name for an anonymous proposition to introduce in the
+-- assumptions
+def mkFreshAnonPropUserName := mkFreshUserName `_
+
+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
+
+-- TODO: this should take a continuation
+def addDeclTac (name : Name) (val : Expr) (type : Expr) (asLet : Bool) : TacticM Expr :=
+ -- I don't think we need that
+ 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 ← 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 ← getUnsolvedGoals
+ 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 addDeclTacSyntax (name : Name) (val : Syntax) (asLet : Bool) : TacticM Unit :=
+ -- I don't think we need that
+ withMainContext do
+ --
+ let val ← Term.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 _ ← addDeclTac name val type asLet
+
+elab "custom_let " n:ident " := " v:term : tactic => do
+ addDeclTacSyntax n.getId v (asLet := true)
+
+elab "custom_have " n:ident " := " v:term : tactic =>
+ addDeclTacSyntax 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
+
+-- Repeatedly apply a tactic
+partial def repeatTac (tac : TacticM Unit) : TacticM Unit := do
+ try
+ tac
+ allGoals (focus (repeatTac tac))
+ -- TODO: does this restore the state?
+ catch _ => pure ()
+
+def firstTac (tacl : List (TacticM Unit)) : TacticM Unit := do
+ match tacl with
+ | [] => pure ()
+ | tac :: tacl =>
+ -- Should use try ... catch or Lean.observing?
+ -- Generally speaking we should use Lean.observing? to restore the state,
+ -- but with tactics the try ... catch variant seems to work
+ try do
+ tac
+ -- Check that there are no remaining goals
+ let gl ← Tactic.getUnsolvedGoals
+ if ¬ gl.isEmpty then throwError "tactic failed"
+ catch _ => firstTac tacl
+/- let res ← Lean.observing? do
+ tac
+ -- Check that there are no remaining goals
+ let gl ← Tactic.getUnsolvedGoals
+ if ¬ gl.isEmpty then throwError "tactic failed"
+ match res with
+ | some _ => pure ()
+ | none => firstTac tacl -/
+
+-- Taken from Lean.Elab.evalAssumption
+def assumptionTac : TacticM Unit :=
+ liftMetaTactic fun mvarId => do mvarId.assumption; pure []
+
+def isConj (e : Expr) : MetaM Bool :=
+ e.consumeMData.withApp fun f args => pure (f.isConstOf ``And ∧ args.size = 2)
+
+-- 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 optSplitConj (e : Expr) : MetaM (Expr × Option Expr) := do
+ e.consumeMData.withApp fun f args =>
+ if f.isConstOf ``And ∧ args.size = 2 then pure (args.get! 0, some (args.get! 1))
+ else pure (e, none)
+
+-- Split the goal if it is a conjunction
+def splitConjTarget : TacticM Unit := do
+ withMainContext do
+ let g ← getMainTarget
+ trace[Utils] "splitConjTarget: goal: {g}"
+ -- The tactic was initially implemened with `_root_.Lean.MVarId.apply`
+ -- but it tended to mess the goal by unfolding terms, even when it failed
+ let (l, r) ← optSplitConj g
+ match r with
+ | none => do throwError "The goal is not a conjunction"
+ | some r => do
+ let lmvar ← mkFreshExprSyntheticOpaqueMVar l
+ let rmvar ← mkFreshExprSyntheticOpaqueMVar r
+ let and_intro ← mkAppM ``And.intro #[lmvar, rmvar]
+ let g ← getMainGoal
+ g.assign and_intro
+ let goals ← getUnsolvedGoals
+ setGoals (lmvar.mvarId! :: rmvar.mvarId! :: goals)
+
+-- 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
+
+-- Tactic to split on a disjunction.
+-- The expression `h` should be an fvar.
+-- TODO: there must be simpler. Use use _root_.Lean.MVarId.cases for instance
+def splitDisjTac (h : Expr) (kleft kright : TacticM Unit) : TacticM Unit := do
+ trace[Arith] "assumption on which to split: {h}"
+ -- Retrieve the main goal
+ withMainContext do
+ let goalType ← 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 splitDisjTac"
+ 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 (← mkFreshAnonPropUserName) hTy fun hVar => do
+ let motive ← mkLambdaFVars #[hVar] goalType
+ let casesExpr ← mkAppOptM ``Or.casesOn #[a, b, motive, h, inl, inr]
+ let mgoal ← getMainGoal
+ trace[Arith] "goals: {← getUnsolvedGoals}"
+ trace[Arith] "main goal: {mgoal}"
+ mgoal.assign casesExpr
+ let goals ← getUnsolvedGoals
+ -- Focus on the left
+ setGoals [mleft]
+ withMainContext kleft
+ let leftGoals ← getUnsolvedGoals
+ -- Focus on the right
+ setGoals [mright]
+ withMainContext kright
+ let rightGoals ← getUnsolvedGoals
+ -- Put all the goals back
+ setGoals (leftGoals ++ rightGoals ++ goals)
+ trace[Arith] "new goals: {← getUnsolvedGoals}"
+
+elab "split_disj " n:ident : tactic => do
+ withMainContext do
+ let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
+ let fvar := mkFVar decl.fvarId
+ splitDisjTac 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
+
+
+-- Tactic to split on an exists.
+-- `h` must be an FVar
+def splitExistsTac (h : Expr) (optId : Option Name) (k : Expr → Expr → TacticM α) : TacticM α := do
+ withMainContext do
+ let goal ← getMainGoal
+ let hTy ← inferType h
+ if isExists hTy then do
+ -- Try to use the user-provided names
+ let altVarNames ← do
+ let hDecl ← h.fvarId!.getDecl
+ let id ← do
+ match optId with
+ | none => mkFreshUserName `x
+ | some id => pure id
+ pure #[{ varNames := [id, hDecl.userName] }]
+ let newGoals ← goal.cases h.fvarId! altVarNames
+ -- There should be exactly one goal
+ match newGoals.toList with
+ | [ newGoal ] =>
+ -- Set the new goal
+ let goals ← getUnsolvedGoals
+ setGoals (newGoal.mvarId :: goals)
+ -- There should be exactly two fields
+ let fields := newGoal.fields
+ withMainContext do
+ k (fields.get! 0) (fields.get! 1)
+ | _ =>
+ throwError "Unreachable"
+ else
+ throwError "Not a conjunction"
+
+-- TODO: move
+def listTryPopHead (ls : List α) : Option α × List α :=
+ match ls with
+ | [] => (none, ls)
+ | hd :: tl => (some hd, tl)
+
+/- Destruct all the existentials appearing in `h`, and introduce them as variables
+ in the context.
+
+ If `ids` is not empty, we use it to name the introduced variables. We
+ transmit the stripped expression and the remaining ids to the continuation.
+ -/
+partial def splitAllExistsTac [Inhabited α] (h : Expr) (ids : List (Option Name)) (k : Expr → List (Option Name) → TacticM α) : TacticM α := do
+ try
+ let (optId, ids) :=
+ match ids with
+ | [] => (none, [])
+ | x :: ids => (x, ids)
+ splitExistsTac h optId (fun _ body => splitAllExistsTac body ids k)
+ catch _ => k h ids
+
+-- Tactic to split on a conjunction.
+def splitConjTac (h : Expr) (optIds : Option (Name × Name)) (k : Expr → Expr → TacticM α) : TacticM α := do
+ withMainContext do
+ let goal ← getMainGoal
+ let hTy ← inferType h
+ if ← isConj hTy then do
+ -- Try to use the user-provided names
+ let altVarNames ←
+ match optIds with
+ | none => do
+ let id0 ← mkFreshAnonPropUserName
+ let id1 ← mkFreshAnonPropUserName
+ pure #[{ varNames := [id0, id1] }]
+ | some (id0, id1) => do
+ pure #[{ varNames := [id0, id1] }]
+ let newGoals ← goal.cases h.fvarId! altVarNames
+ -- There should be exactly one goal
+ match newGoals.toList with
+ | [ newGoal ] =>
+ -- Set the new goal
+ let goals ← getUnsolvedGoals
+ setGoals (newGoal.mvarId :: goals)
+ -- There should be exactly two fields
+ let fields := newGoal.fields
+ withMainContext do
+ k (fields.get! 0) (fields.get! 1)
+ | _ =>
+ throwError "Unreachable"
+ else
+ throwError "Not a conjunction"
+
+-- Tactic to fully split a conjunction
+partial def splitFullConjTacAux [Inhabited α] [Nonempty α] (keepCurrentName : Bool) (l : List Expr) (h : Expr) (k : List Expr → TacticM α) : TacticM α := do
+ try
+ let ids ← do
+ if keepCurrentName then do
+ let cur := (← h.fvarId!.getDecl).userName
+ let nid ← mkFreshAnonPropUserName
+ pure (some (cur, nid))
+ else
+ pure none
+ splitConjTac h ids (λ h1 h2 =>
+ splitFullConjTacAux keepCurrentName l h1 (λ l1 =>
+ splitFullConjTacAux keepCurrentName l1 h2 (λ l2 =>
+ k l2)))
+ catch _ =>
+ k (h :: l)
+
+-- Tactic to fully split a conjunction
+-- `keepCurrentName`: if `true`, then the first conjunct has the name of the original assumption
+def splitFullConjTac [Inhabited α] [Nonempty α] (keepCurrentName : Bool) (h : Expr) (k : List Expr → TacticM α) : TacticM α := do
+ splitFullConjTacAux keepCurrentName [] h (λ l => k l.reverse)
+
+syntax optAtArgs := ("at" ident)?
+def elabOptAtArgs (args : TSyntax `Utils.optAtArgs) : TacticM (Option Expr) := do
+ withMainContext do
+ let args := (args.raw.getArgs.get! 0).getArgs
+ if args.size > 0 then do
+ let n := (args.get! 1).getId
+ let decl ← Lean.Meta.getLocalDeclFromUserName n
+ let fvar := mkFVar decl.fvarId
+ pure (some fvar)
+ else
+ pure none
+
+elab "split_conj" args:optAtArgs : tactic => do
+ withMainContext do
+ match ← elabOptAtArgs args with
+ | some fvar => do
+ trace[Utils] "split at {fvar}"
+ splitConjTac fvar none (fun _ _ => pure ())
+ | none => do
+ trace[Utils] "split goal"
+ splitConjTarget
+
+elab "split_conjs" args:optAtArgs : tactic => do
+ withMainContext do
+ match ← elabOptAtArgs args with
+ | some fvar =>
+ trace[Utils] "split at {fvar}"
+ splitFullConjTac false fvar (fun _ => pure ())
+ | none =>
+ trace[Utils] "split goal"
+ repeatTac splitConjTarget
+
+elab "split_existsl" " at " n:ident : tactic => do
+ withMainContext do
+ let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
+ let fvar := mkFVar decl.fvarId
+ splitAllExistsTac fvar [] (fun _ _ => pure ())
+
+example (h : a ∧ b) : a := by
+ split_existsl at h
+ split_conj at h
+ assumption
+
+example (h : ∃ x y z, x + y + z ≥ 0) : ∃ x, x ≥ 0 := by
+ split_existsl at h
+ rename_i x y z
+ exists x + y + z
+
+/- Call the simp tactic.
+ The initialization of the context is adapted from Tactic.elabSimpArgs.
+ Something very annoying is that there is no function which allows to
+ initialize a simp context without doing an elaboration - as a consequence
+ we write our own here. -/
+def simpAt (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVarId)
+ (loc : Tactic.Location) :
+ Tactic.TacticM Unit := do
+ -- Initialize with the builtin simp theorems
+ let simpThms ← Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) ({} : SimpTheorems)
+ -- Add the equational theorem for the declarations to unfold
+ let simpThms ←
+ declsToUnfold.foldlM (fun thms decl => thms.addDeclToUnfold decl) simpThms
+ -- Add the hypotheses and the rewriting theorems
+ let simpThms ←
+ hypsToUse.foldlM (fun thms fvarId =>
+ -- post: TODO: don't know what that is
+ -- inv: invert the equality
+ thms.add (.fvar fvarId) #[] (mkFVar fvarId) (post := false) (inv := false)
+ -- thms.eraseCore (.fvar fvar)
+ ) simpThms
+ -- Add the rewriting theorems to use
+ let simpThms ←
+ thms.foldlM (fun thms thmName => do
+ let info ← getConstInfo thmName
+ if (← isProp info.type) then
+ -- post: TODO: don't know what that is
+ -- inv: invert the equality
+ thms.addConst thmName (post := false) (inv := false)
+ else
+ throwError "Not a proposition: {thmName}"
+ ) simpThms
+ let congrTheorems ← getSimpCongrTheorems
+ let ctx : Simp.Context := { simpTheorems := #[simpThms], congrTheorems }
+ -- Apply the simplifier
+ let _ ← Tactic.simpLocation ctx (discharge? := .none) loc
+
+end Utils
diff --git a/backends/lean/Primitives.lean b/backends/lean/Primitives.lean
deleted file mode 100644
index 4a66a453..00000000
--- a/backends/lean/Primitives.lean
+++ /dev/null
@@ -1,583 +0,0 @@
-import Lean
-import Lean.Meta.Tactic.Simp
-import Init.Data.List.Basic
-import Mathlib.Tactic.RunCmd
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
- runTermElabM (fun _ => do
- let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
- if not r then
- logInfo "Assertion failed for: "
- logInfo _stx[1]
- logError "Expression reduced to false"
- pure ())
-
-#eval 2 == 2
-#assert (2 == 2)
-
--------------
--- PRELUDE --
--------------
-
--- Results & monadic combinators
-
-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
-deriving Repr, BEq
-
-open Result
-
-instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
- Inhabited.mk (fail panic)
-
-/- HELPERS -/
-
-def ret? {α: Type} (r: Result α): Bool :=
- match r with
- | Result.ret _ => true
- | Result.fail _ => false
-
-def massert (b:Bool) : Result Unit :=
- if b then .ret () else fail assertionFailure
-
-def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
- match x with
- | Result.fail _ => by contradiction
- | Result.ret x => x
-
-/- DO-DSL SUPPORT -/
-
-def bind (x: Result α) (f: α -> Result β) : Result β :=
- match x with
- | ret v => f v
- | fail v => fail v
-
--- 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
-
-/- CUSTOM-DSL SUPPORT -/
-
--- Let-binding the Result of a monadic operation is oftentimes not sufficient,
--- because we may need a hypothesis for equational reasoning in the scope. We
--- rely on subtype, and a custom let-binding operator, in effect recreating our
--- own variant of the do-dsl
-
-def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
- match o with
- | .ret x => .ret ⟨x, rfl⟩
- | .fail e => .fail e
-
-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
-
-----------------------
--- MACHINE INTEGERS --
-----------------------
-
--- We redefine our machine integers types.
-
--- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits`
--- using the simplifier, meaning that proofs do not depend on the compile-time value of
--- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at
--- least officially, 16-bit microcontrollers, so this seems like a fine design decision
--- for now.)
-
--- Note from Chris Bailey: "If there's more than one salient property of your
--- definition then the subtyping strategy might get messy, and the property part
--- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." So, we will not add refinements on the return values of the
--- operations defined on Primitives, but will rather rely on custom lemmas to
--- invert on possible return values of the primitive operations.
-
--- Machine integer constants, done via `ofNatCore`, which requires a proof that
--- the `Nat` fits within the desired integer type. We provide a custom tactic.
-
-open System.Platform.getNumBits
-
--- TODO: is there a way of only importing System.Platform.getNumBits?
---
-@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
-
--- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
--- We keep the F* convention for now.
-@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
-@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
-@[simp] def I8.min : Int := - (HPow.hPow 2 7)
-@[simp] def I8.max : Int := HPow.hPow 2 7 - 1
-@[simp] def I16.min : Int := - (HPow.hPow 2 15)
-@[simp] def I16.max : Int := HPow.hPow 2 15 - 1
-@[simp] def I32.min : Int := -(HPow.hPow 2 31)
-@[simp] def I32.max : Int := HPow.hPow 2 31 - 1
-@[simp] def I64.min : Int := -(HPow.hPow 2 63)
-@[simp] def I64.max : Int := HPow.hPow 2 63 - 1
-@[simp] def I128.min : Int := -(HPow.hPow 2 127)
-@[simp] def I128.max : Int := HPow.hPow 2 127 - 1
-@[simp] def Usize.min : Int := 0
-@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
-@[simp] def U8.min : Int := 0
-@[simp] def U8.max : Int := HPow.hPow 2 8 - 1
-@[simp] def U16.min : Int := 0
-@[simp] def U16.max : Int := HPow.hPow 2 16 - 1
-@[simp] def U32.min : Int := 0
-@[simp] def U32.max : Int := HPow.hPow 2 32 - 1
-@[simp] def U64.min : Int := 0
-@[simp] def U64.max : Int := HPow.hPow 2 64 - 1
-@[simp] def U128.min : Int := 0
-@[simp] def U128.max : Int := HPow.hPow 2 128 - 1
-
-#assert (I8.min == -128)
-#assert (I8.max == 127)
-#assert (I16.min == -32768)
-#assert (I16.max == 32767)
-#assert (I32.min == -2147483648)
-#assert (I32.max == 2147483647)
-#assert (I64.min == -9223372036854775808)
-#assert (I64.max == 9223372036854775807)
-#assert (I128.min == -170141183460469231731687303715884105728)
-#assert (I128.max == 170141183460469231731687303715884105727)
-#assert (U8.min == 0)
-#assert (U8.max == 255)
-#assert (U16.min == 0)
-#assert (U16.max == 65535)
-#assert (U32.min == 0)
-#assert (U32.max == 4294967295)
-#assert (U64.min == 0)
-#assert (U64.max == 18446744073709551615)
-#assert (U128.min == 0)
-#assert (U128.max == 340282366920938463463374607431768211455)
-
-inductive ScalarTy :=
-| Isize
-| I8
-| I16
-| I32
-| I64
-| I128
-| Usize
-| U8
-| U16
-| U32
-| U64
-| U128
-
-def Scalar.min (ty : ScalarTy) : Int :=
- match ty with
- | .Isize => Isize.min
- | .I8 => I8.min
- | .I16 => I16.min
- | .I32 => I32.min
- | .I64 => I64.min
- | .I128 => I128.min
- | .Usize => Usize.min
- | .U8 => U8.min
- | .U16 => U16.min
- | .U32 => U32.min
- | .U64 => U64.min
- | .U128 => U128.min
-
-def Scalar.max (ty : ScalarTy) : Int :=
- match ty with
- | .Isize => Isize.max
- | .I8 => I8.max
- | .I16 => I16.max
- | .I32 => I32.max
- | .I64 => I64.max
- | .I128 => I128.max
- | .Usize => Usize.max
- | .U8 => U8.max
- | .U16 => U16.max
- | .U32 => U32.max
- | .U64 => U64.max
- | .U128 => U128.max
-
--- "Conservative" bounds
--- We use those because we can't compare to the isize bounds (which can't
--- reduce at compile-time). Whenever we perform an arithmetic operation like
--- addition we need to check that the result is in bounds: we first compare
--- to the conservative bounds, which reduce, then compare to the real bounds.
--- This is useful for the various #asserts that we want to reduce at
--- type-checking time.
-def Scalar.cMin (ty : ScalarTy) : Int :=
- match ty with
- | .Isize => I32.min
- | _ => Scalar.min ty
-
-def Scalar.cMax (ty : ScalarTy) : Int :=
- match ty with
- | .Isize => I32.max
- | .Usize => U32.max
- | _ => Scalar.max ty
-
-theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
-theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
-
-structure Scalar (ty : ScalarTy) where
- val : Int
- hmin : Scalar.min ty <= val
- hmax : val <= Scalar.max ty
-
-theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
- Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
- (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
- := by sorry
-
-def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
- (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
- { val := x, hmin := hmin, hmax := hmax }
-
-def Scalar.ofInt {ty : ScalarTy} (x : Int)
- (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
- let hmin: Scalar.min ty <= x := by sorry
- let hmax: x <= Scalar.max ty := by sorry
- Scalar.ofIntCore x hmin hmax
-
--- Further thoughts: look at what has been done here:
--- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
--- and
--- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
--- which both contain a fair amount of reasoning already!
-def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
- -- TODO: write this with only one if then else
- if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
- if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
- let hmin: Scalar.min ty <= x := by sorry
- let hmax: x <= Scalar.max ty := by sorry
- return Scalar.ofIntCore x hmin hmax
- else fail integerOverflow
- else fail integerOverflow
-
-def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
-
-def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
- if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
-
--- Checking that the % operation in Lean computes the same as the remainder operation in Rust
-#assert 1 % 2 = (1:Int)
-#assert (-1) % 2 = -1
-#assert 1 % (-2) = 1
-#assert (-1) % (-2) = -1
-
-def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
- if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
-
-def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
- Scalar.tryMk ty (x.val + y.val)
-
-def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
- Scalar.tryMk ty (x.val - y.val)
-
-def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
- Scalar.tryMk ty (x.val * y.val)
-
--- TODO: instances of +, -, * etc. for scalars
-
--- Cast an integer from a [src_ty] to a [tgt_ty]
--- TODO: check the semantics of casts in Rust
-def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
- Scalar.tryMk tgt_ty x.val
-
--- The scalar types
--- We declare the definitions as reducible so that Lean can unfold them (useful
--- for type class resolution for instance).
-@[reducible] def Isize := Scalar .Isize
-@[reducible] def I8 := Scalar .I8
-@[reducible] def I16 := Scalar .I16
-@[reducible] def I32 := Scalar .I32
-@[reducible] def I64 := Scalar .I64
-@[reducible] def I128 := Scalar .I128
-@[reducible] def Usize := Scalar .Usize
-@[reducible] def U8 := Scalar .U8
-@[reducible] def U16 := Scalar .U16
-@[reducible] def U32 := Scalar .U32
-@[reducible] def U64 := Scalar .U64
-@[reducible] def U128 := Scalar .U128
-
--- TODO: below: not sure this is the best way.
--- Should we rather overload operations like +, -, etc.?
--- Also, it is possible to automate the generation of those definitions
--- with macros (but would it be a good idea? It would be less easy to
--- read the file, which is not supposed to change a lot)
-
--- Negation
-
-/--
-Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
-one here.
-
-The notation typeclass for heterogeneous addition.
-This enables the notation `- a : β` where `a : α`.
--/
-class HNeg (α : Type u) (β : outParam (Type v)) where
- /-- `- a` computes the negation of `a`.
- The meaning of this notation is type-dependent. -/
- hNeg : α → β
-
-prefix:75 "-" => HNeg.hNeg
-
-instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
-instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
-instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
-instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
-instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
-instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
-
--- Addition
-instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
- hAdd x y := Scalar.add x y
-
--- Substraction
-instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
- hSub x y := Scalar.sub x y
-
--- Multiplication
-instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
- hMul x y := Scalar.mul x y
-
--- Division
-instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
- hDiv x y := Scalar.div x y
-
--- Remainder
-instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
- hMod x y := Scalar.rem x y
-
--- ofIntCore
--- TODO: typeclass?
-def Isize.ofIntCore := @Scalar.ofIntCore .Isize
-def I8.ofIntCore := @Scalar.ofIntCore .I8
-def I16.ofIntCore := @Scalar.ofIntCore .I16
-def I32.ofIntCore := @Scalar.ofIntCore .I32
-def I64.ofIntCore := @Scalar.ofIntCore .I64
-def I128.ofIntCore := @Scalar.ofIntCore .I128
-def Usize.ofIntCore := @Scalar.ofIntCore .Usize
-def U8.ofIntCore := @Scalar.ofIntCore .U8
-def U16.ofIntCore := @Scalar.ofIntCore .U16
-def U32.ofIntCore := @Scalar.ofIntCore .U32
-def U64.ofIntCore := @Scalar.ofIntCore .U64
-def U128.ofIntCore := @Scalar.ofIntCore .U128
-
--- ofInt
--- TODO: typeclass?
-def Isize.ofInt := @Scalar.ofInt .Isize
-def I8.ofInt := @Scalar.ofInt .I8
-def I16.ofInt := @Scalar.ofInt .I16
-def I32.ofInt := @Scalar.ofInt .I32
-def I64.ofInt := @Scalar.ofInt .I64
-def I128.ofInt := @Scalar.ofInt .I128
-def Usize.ofInt := @Scalar.ofInt .Usize
-def U8.ofInt := @Scalar.ofInt .U8
-def U16.ofInt := @Scalar.ofInt .U16
-def U32.ofInt := @Scalar.ofInt .U32
-def U64.ofInt := @Scalar.ofInt .U64
-def U128.ofInt := @Scalar.ofInt .U128
-
--- Comparisons
-instance {ty} : LT (Scalar ty) where
- lt a b := LT.lt a.val b.val
-
-instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
-
-instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
-instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
-
-theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
- | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
-
-theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
- h ▸ rfl
-
-theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
- fun h' => absurd (val_eq_of_eq h') h
-
-instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
- fun i j =>
- match decEq i.val j.val with
- | isTrue h => isTrue (Scalar.eq_of_val_eq h)
- | isFalse h => isFalse (Scalar.ne_of_val_ne h)
-
-def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
-
--- Tactic to prove that integers are in bounds
-syntax "intlit" : tactic
-
-macro_rules
- | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
-
--- -- We now define a type class that subsumes the various machine integer types, so
--- -- as to write a concise definition for scalar_cast, rather than exhaustively
--- -- enumerating all of the possible pairs. We remark that Rust has sane semantics
--- -- and fails if a cast operation would involve a truncation or modulo.
-
--- class MachineInteger (t: Type) where
--- size: Nat
--- val: t -> Fin size
--- ofNatCore: (n:Nat) -> LT.lt n size -> t
-
--- set_option hygiene false in
--- run_cmd
--- for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
--- Lean.Elab.Command.elabCommand (← `(
--- namespace $typeName
--- instance: MachineInteger $typeName where
--- size := size
--- val := val
--- ofNatCore := ofNatCore
--- end $typeName
--- ))
-
--- -- Aeneas only instantiates the destination type (`src` is implicit). We rely on
--- -- Lean to infer `src`.
-
--- def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
--- if h: MachineInteger.val x < MachineInteger.size dst then
--- .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
--- else
--- .fail integerOverflow
-
--------------
--- VECTORS --
--------------
-
-def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
-
-def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
-
-def vec_len (α : Type u) (v : Vec α) : Usize :=
- let ⟨ v, l ⟩ := v
- Usize.ofIntCore (List.length v) (by sorry) l
-
-def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
-
-def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
- :=
- if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
- return ⟨ List.concat v.val x, by sorry ⟩
- else
- fail maximumSizeExceeded
-
-def vec_insert_fwd (α : Type u) (v: Vec α) (i: Usize) (_: α): Result Unit :=
- if i.val < List.length v.val then
- .ret ()
- else
- .fail arrayOutOfBounds
-
-def vec_insert_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
- if i.val < List.length v.val then
- -- TODO: maybe we should redefine a list library which uses integers
- -- (instead of natural numbers)
- let i : Nat :=
- match i.val with
- | .ofNat n => n
- | .negSucc n => by sorry -- TODO: we can't get here
- let isLt: i < USize.size := by sorry
- let i : Fin USize.size := { val := i, isLt := isLt }
- .ret ⟨ List.set v.val i.val x, by
- have h: List.length v.val <= Usize.max := v.property
- rewrite [ List.length_set v.val i.val x ]
- assumption
- ⟩
- else
- .fail arrayOutOfBounds
-
-def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
- if i.val < List.length v.val then
- let i : Nat :=
- match i.val with
- | .ofNat n => n
- | .negSucc n => by sorry -- TODO: we can't get here
- let isLt: i < USize.size := by sorry
- let i : Fin USize.size := { val := i, isLt := isLt }
- let h: i < List.length v.val := by sorry
- .ret (List.get v.val ⟨i.val, h⟩)
- else
- .fail arrayOutOfBounds
-
-def vec_index_back (α : Type u) (v: Vec α) (i: Usize) (_: α): Result Unit :=
- if i.val < List.length v.val then
- .ret ()
- else
- .fail arrayOutOfBounds
-
-def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
- if i.val < List.length v.val then
- let i : Nat :=
- match i.val with
- | .ofNat n => n
- | .negSucc n => by sorry -- TODO: we can't get here
- let isLt: i < USize.size := by sorry
- let i : Fin USize.size := { val := i, isLt := isLt }
- let h: i < List.length v.val := by sorry
- .ret (List.get v.val ⟨i.val, h⟩)
- else
- .fail arrayOutOfBounds
-
-def vec_index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
- if i.val < List.length v.val then
- let i : Nat :=
- match i.val with
- | .ofNat n => n
- | .negSucc n => by sorry -- TODO: we can't get here
- let isLt: i < USize.size := by sorry
- let i : Fin USize.size := { val := i, isLt := isLt }
- .ret ⟨ List.set v.val i.val x, by
- have h: List.length v.val <= Usize.max := v.property
- rewrite [ List.length_set v.val i.val x ]
- assumption
- ⟩
- else
- .fail arrayOutOfBounds
-
-----------
--- MISC --
-----------
-
-def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
- x
-
-def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
- y
-
-/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
- Use with `simp [ aeneas ]` -/
-register_simp_attr aeneas
diff --git a/backends/lean/lake-manifest.json b/backends/lean/lake-manifest.json
index 7b23fc19..5a089838 100644
--- a/backends/lean/lake-manifest.json
+++ b/backends/lean/lake-manifest.json
@@ -2,26 +2,38 @@
"packagesDir": "lake-packages",
"packages":
[{"git":
+ {"url": "https://github.com/EdAyers/ProofWidgets4",
+ "subDir?": null,
+ "rev": "c43db94a8f495dad37829e9d7ad65483d68c86b8",
+ "name": "proofwidgets",
+ "inputRev?": "v0.0.11"}},
+ {"git":
+ {"url": "https://github.com/mhuisi/lean4-cli.git",
+ "subDir?": null,
+ "rev": "5a858c32963b6b19be0d477a30a1f4b6c120be7e",
+ "name": "Cli",
+ "inputRev?": "nightly"}},
+ {"git":
{"url": "https://github.com/leanprover-community/mathlib4.git",
"subDir?": null,
- "rev": "f89ee53085b8aad0bacd3bc94d7ef4b8d9aba643",
+ "rev": "fa05951a270fef2873666c46f138e90338cd48d6",
"name": "mathlib",
"inputRev?": null}},
{"git":
{"url": "https://github.com/gebner/quote4",
"subDir?": null,
- "rev": "2412c4fdf4a8b689f4467618e5e7b371ae5014aa",
+ "rev": "c0d9516f44d07feee01c1103c8f2f7c24a822b55",
"name": "Qq",
"inputRev?": "master"}},
{"git":
{"url": "https://github.com/JLimperg/aesop",
"subDir?": null,
- "rev": "7fe9ecd9339b0e1796e89d243b776849c305c690",
+ "rev": "f04538ab6ad07642368cf11d2702acc0a9b4bcee",
"name": "aesop",
"inputRev?": "master"}},
{"git":
{"url": "https://github.com/leanprover/std4",
"subDir?": null,
- "rev": "24897887905b3a1254b244369f5dd2cf6174b0ee",
+ "rev": "dff883c55395438ae2a5c65ad5ddba084b600feb",
"name": "std",
"inputRev?": "main"}}]}
diff --git a/backends/lean/lakefile.lean b/backends/lean/lakefile.lean
index 9633e1e8..21a4a332 100644
--- a/backends/lean/lakefile.lean
+++ b/backends/lean/lakefile.lean
@@ -1,10 +1,11 @@
import Lake
open Lake DSL
+-- Important: mathlib imports std4 and quote4: we mustn't add a `require std4` line
require mathlib from git
"https://github.com/leanprover-community/mathlib4.git"
package «base» {}
@[default_target]
-lean_lib «Primitives» {}
+lean_lib «Base» {}
diff --git a/backends/lean/lean-toolchain b/backends/lean/lean-toolchain
new file mode 100644
index 00000000..334c5053
--- /dev/null
+++ b/backends/lean/lean-toolchain
@@ -0,0 +1 @@
+leanprover/lean4:nightly-2023-07-12 \ No newline at end of file