diff options
Diffstat (limited to 'backends/lean')
| -rw-r--r-- | backends/lean/Base/Diverge.lean | 95 |
1 files changed, 47 insertions, 48 deletions
diff --git a/backends/lean/Base/Diverge.lean b/backends/lean/Base/Diverge.lean index a5cf3459..d65e77a1 100644 --- a/backends/lean/Base/Diverge.lean +++ b/backends/lean/Base/Diverge.lean @@ -88,7 +88,8 @@ namespace Fix open Primitives open Result - variable {a b c d : Type} + variable {a : Type} {b : a → Type} + variable {c d : Type} /-! # The least fixed point definition and its properties -/ @@ -132,19 +133,23 @@ namespace Fix /-! # The fixed point definitions -/ - def fix_fuel (n : Nat) (f : (a → Result b) → a → Result b) (x : a) : Result b := + 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 : (a → Result b) → a → Result b) (x : a) (n : Nat) := + @[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 : (a → Result b) → a → Result b) (x : a) (n : Nat) : Prop := + 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 - noncomputable def fix (f : (a → Result b) → a → Result b) (x : a) : Result b := + noncomputable + 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 -/ @@ -158,11 +163,11 @@ namespace Fix | ret _ => x2 = x1 -- TODO: generalize -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows) - def karrow_rel (k1 k2 : a → Result b) : Prop := + 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 : (a → Result b) → a → Result b) : Prop := + 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. @@ -170,11 +175,12 @@ namespace Fix -- 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 : (a → Result b) → a → Result b) : Prop := + 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 : (a → Result b) → a → Result b} (Hmono : is_mono f) : + 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 @@ -195,12 +201,13 @@ namespace Fix simp [result_rel] at Hmono apply Hmono - @[simp] theorem neg_fix_fuel_P {f : (a → Result b) → a → Result b} {x : a} {n : Nat} : + @[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 : (a → Result b) → a → Result b} (Hmono : is_mono f) : + 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] @@ -234,7 +241,7 @@ namespace Fix cases Heq':fix_fuel n f x <;> simp_all - theorem fix_fuel_P_least {f : (a → Result b) → a → Result b} (Hmono : is_mono f) : + 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 @@ -247,7 +254,7 @@ namespace Fix -- Prove the fixed point equation in the case there exists some fuel for which -- the execution terminates - theorem fix_fixed_eq_terminates (f : (a → Result b) → a → Result b) (Hmono : is_mono f) + 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 @@ -271,7 +278,7 @@ namespace Fix revert Hvm split <;> simp [*] <;> intros <;> simp [*] - theorem fix_fixed_eq_forall {{f : (a → Result b) → a → Result b}} + 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 @@ -290,7 +297,7 @@ namespace Fix | .inl ⟨ n, He ⟩ => apply fix_fixed_eq_terminates f Hmono x n He -- The final fixed point equation - theorem fix_fixed_eq {{f : (a → Result b) → a → Result b}} + 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 @@ -301,7 +308,7 @@ namespace Fix /-! # Making the proofs of validity manageable (and automatable) -/ -- Monotonicity property for expressions - def is_mono_p (e : (a → Result b) → Result c) : Prop := + 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) : @@ -310,16 +317,17 @@ namespace Fix split <;> simp theorem is_mono_p_rec (x : a) : - @is_mono_p a b b (λ f => f x) := by + @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` + -- TODO: generalize d? theorem is_mono_p_bind - (g : (a → Result b) → Result c) - (h : c → (a → Result b) → Result d) : + (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 (λ k => do let y ← g k; h y k) := by + @is_mono_p a b d (λ k => do let y ← g k; h y k) := by intro hg hh simp [is_mono_p] intro fg fh Hrgh @@ -332,25 +340,26 @@ namespace Fix -- Continuity property for expressions - note that we take the continuation -- as parameter - def is_cont_p (k : (a → Result b) → a → Result b) - (e : (a → Result b) → Result c) : Prop := + 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 : (a → Result b) → a → Result b) (x : Result c) : + 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 : (a → Result b) → a → Result b) (x : a) : + 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 : (a → Result b) → a → Result b) + (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (Hkmono : is_mono k) - (g : (a → Result b) → Result c) - (h : c → (a → Result b) → Result d) : + (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)) → @@ -402,16 +411,18 @@ namespace Fix apply Hhcont; assumption -- The validity property for an expression - def is_valid_p (k : (a → Result b) → a → Result b) - (e : (a → Result b) → Result c) : Prop := + 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 : (a → Result b) → a → Result b) (x : Result c) : + @[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 : (a → Result b) → a → Result b) (x : a) : + @[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] @@ -419,9 +430,9 @@ namespace Fix -- (in particular, it will manage to figure out `g` and `h` when we -- apply the lemma) theorem is_valid_p_bind - {{k : (a → Result b) → a → Result b}} - {{g : (a → Result b) → Result c}} - {{h : c → (a → Result b) → Result d}} + {{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 @@ -438,7 +449,7 @@ namespace Fix have Hhcont := Hhcont Hkmono apply is_cont_p_bind <;> assumption - theorem is_valid_p_imp_is_valid {{e : (a → Result b) → a → Result b}} + theorem is_valid_p_imp_is_valid {{e : ((x:a) → Result (b x)) → (x:a) → Result (b x)}} (Hvalid : ∀ k x, is_valid_p k (λ k => e k x)) : is_mono e ∧ is_cont e := by have Hmono : is_mono e := by @@ -457,7 +468,7 @@ namespace Fix simp [*] simp [*] - theorem is_valid_p_fix_fixed_eq {{e : (a → Result b) → a → Result b}} + theorem is_valid_p_fix_fixed_eq {{e : ((x:a) → Result (b x)) → (x:a) → Result (b x)}} (Hvalid : ∀ k x, is_valid_p k (λ k => e k x)) : fix e = e (fix e) := by have ⟨ Hmono, Hcont ⟩ := is_valid_p_imp_is_valid Hvalid @@ -625,18 +636,6 @@ namespace Ex3 | .inl _ => .fail .panic | .inr b => .ret b - -- TODO: move - -- TODO: this is not enough - theorem swap_if_bind {a b : Type} (e : Prop) [Decidable e] - (x y : Result a) (f : a → Result b) : - (do - let z ← (if e then x else y) - f z) - = - (if e then do let z ← x; f z - else do let z ← y; f z) := by - split <;> simp - -- 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)) |