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