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authorSon HO2023-07-31 16:15:58 +0200
committerGitHub2023-07-31 16:15:58 +0200
commit887d0ef1efc8912c6273b5ebcf979384e9d7fa97 (patch)
tree92d6021eb549f7cc25501856edd58859786b7e90 /backends/lean/Base/Diverge
parent53adf30fe440eb8b6f58ba89f4a4c0acc7877498 (diff)
parent9b3a58e423333fc9a4a5a264c3beb0a3d951e86b (diff)
Merge pull request #31 from AeneasVerif/son_lean_backend
Improve the Lean backend
Diffstat (limited to 'backends/lean/Base/Diverge')
-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
3 files changed, 2315 insertions, 0 deletions
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