feat(init/wf): port from standard library to HoTT library
After this commit we need some more advanced theorems in init/wf, notably function extenstionality. For this reason I had to refactor the init folder a little bit. To keep the init folders in both libraries similar, I did the same refactorization in the standard library, even though that was not required for the standard library
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
c0abcc7722
commit
e14d4a4c0c
7 changed files with 242 additions and 222 deletions
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@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Leonardo de Moura
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-/
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prelude
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import init.wf init.tactic init.num init.types init.path
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import init.tactic init.num init.types init.path
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open eq eq.ops decidable
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open algebra sum
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set_option class.force_new true
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@ -185,25 +185,6 @@ namespace nat
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protected theorem le_of_eq_sum_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
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sum.rec_on H !nat.le_of_eq !nat.le_of_lt
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-- less-than is well-founded
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definition lt.wf [instance] : well_founded (lt : ℕ → ℕ → Type₀) :=
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begin
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constructor, intro n, induction n with n IH,
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{ constructor, intros n H, exfalso, exact !not_lt_zero H},
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{ constructor, intros m H,
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assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
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{ intros n₁ hlt, induction hlt,
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{ intro p, injection p with q, exact q ▸ IH},
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{ intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
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apply aux H rfl},
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end
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definition measure {A : Type} : (A → ℕ) → A → A → Type₀ :=
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inv_image lt
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definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
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inv_image.wf f lt.wf
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theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
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succ_le_succ
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@ -5,7 +5,7 @@ Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Jakob von Raumer
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-/
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prelude
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import init.num init.wf
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import init.num init.relation
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open iff
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-- Empty type
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@ -92,61 +92,4 @@ namespace prod
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definition flip [unfold 3] {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)
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open well_founded
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section
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variables {A B : Type}
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variable (Ra : A → A → Type)
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variable (Rb : B → B → Type)
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-- Lexicographical order based on Ra and Rb
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inductive lex : A × B → A × B → Type :=
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| left : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
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| right : ∀a {b₁ b₂}, Rb b₁ b₂ → lex (a, b₁) (a, b₂)
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-- Relational product based on Ra and Rb
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inductive rprod : A × B → A × B → Type :=
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intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
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end
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section
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parameters {A B : Type}
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parameters {Ra : A → A → Type} {Rb : B → B → Type}
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local infix `≺`:50 := lex Ra Rb
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definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
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acc.rec_on aca
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(λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
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λb, acc.rec_on (acb b)
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(λxb acb
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(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
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acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
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have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
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@prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
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p (xa, xb) lt
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(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
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show acc (lex Ra Rb) (a₁, b₁), from
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have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
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iHa a₁ Ra₁ b₁)
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(λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
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show acc (lex Ra Rb) (a, b₁), from
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have Rb₁ : Rb b₁ xb, from eq.rec_on eq₃ H,
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have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
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eq.rec_on eq₂' (iHb b₁ Rb₁)),
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aux rfl rfl)))
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-- The lexicographical order of well founded relations is well-founded
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definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
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well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
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-- Relational product is a subrelation of the lex
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definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
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λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
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-- The relational product of well founded relations is well-founded
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definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
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subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
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end
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end prod
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@ -3,9 +3,10 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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prelude
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import init.relation init.tactic
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import init.relation init.tactic init.funext
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open eq
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inductive acc.{l₁ l₂} {A : Type.{l₁}} (R : A → A → Type.{l₂}) : A → Type.{max l₁ l₂} :=
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intro : ∀x, (∀ y, R y x → acc R y) → acc R x
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@ -13,15 +14,38 @@ intro : ∀x, (∀ y, R y x → acc R y) → acc R x
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namespace acc
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variables {A : Type} {R : A → A → Type}
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definition acc_eq {a : A} (H₁ H₂ : acc R a) : H₁ = H₂ :=
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begin
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induction H₁ with a K₁ IH₁,
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induction H₂ with a K₂ IH₂,
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apply eq.ap (intro a),
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apply eq_of_homotopy, intro a,
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apply eq_of_homotopy, intro r,
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apply IH₁
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end
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definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
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acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
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-- dependent elimination for acc
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protected definition drec [recursor]
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{C : Π (a : A), acc R a → Type}
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(h₁ : Π (x : A) (acx : Π (y : A), R y x → acc R y),
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(Π (y : A) (ryx : R y x), C y (acx y ryx)) → C x (acc.intro x acx))
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{a : A} (h₂ : acc R a) : C a h₂ :=
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begin
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refine acc.rec _ h₂ h₂,
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intro x acx ih h₂,
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exact transport (C x) !acc_eq (h₁ x acx (λ y ryx, ih y ryx (acx y ryx)))
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end
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end acc
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inductive well_founded [class] {A : Type} (R : A → A → Type) : Type :=
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intro : (∀ a, acc R a) → well_founded R
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intro : (Π a, acc R a) → well_founded R
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namespace well_founded
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definition apply [coercion] {A : Type} {R : A → A → Type} (wf : well_founded R) : ∀a, acc R a :=
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definition apply [coercion] {A : Type} {R : A → A → Type} (wf : well_founded R) : Πa, acc R a :=
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take a, well_founded.rec_on wf (λp, p) a
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section
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@ -30,60 +54,42 @@ namespace well_founded
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hypothesis [Hwf : well_founded R]
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definition recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
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theorem recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
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acc.rec_on (Hwf a) (λ x₁ ac₁ iH, H x₁ iH)
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definition induction {C : A → Type} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
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theorem induction {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
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recursion a H
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parameter {C : A → Type}
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parameter F : Πx, (Πy, y ≺ x → C y) → C x
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variable {C : A → Type}
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variable F : Πx, (Πy, y ≺ x → C y) → C x
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definition fix_F (x : A) (a : acc R x) : C x :=
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acc.rec_on a (λ x₁ ac₁ iH, F x₁ iH)
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definition fix_F_eq (x : A) (r : acc R x) :
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fix_F x r = F x (λ (y : A) (p : y ≺ x), fix_F y (acc.inv r p)) :=
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acc.rec_on r (λ x H ih, rfl)
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theorem fix_F_eq (x : A) (r : acc R x) :
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fix_F F x r = F x (λ (y : A) (p : y ≺ x), fix_F F y (acc.inv r p)) :=
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begin
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induction r using acc.drec,
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reflexivity -- proof is star due to proof irrelevance
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end
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end
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-- Remark: after we prove function extensionality from univalence, we can drop this hypothesis
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hypothesis F_ext : Π (x : A) (f g : Π y, y ≺ x → C y),
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(Π (y : A) (p : y ≺ x), f y p = g y p) → F x f = F x g
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lemma fix_F_inv (x : A) (r : acc R x) : Π (s : acc R x), fix_F x r = fix_F x s :=
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acc.rec_on r (λ
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(x₁ : A)
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(h₁ : Π y, y ≺ x₁ → acc R y)
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(ih₁ : Π y (hlt : y ≺ x₁) (s : acc R y), fix_F y (h₁ y hlt) = fix_F y s)
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(s : acc R x₁),
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have aux₁ : Π (s : acc R x₁) (h₁ : Π y, y ≺ x₁ → acc R y) (ih₁ : Π y (hlt : y ≺ x₁) (s : acc R y),
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fix_F y (h₁ y hlt) = fix_F y s), fix_F x₁ (acc.intro x₁ h₁) = fix_F x₁ s, from
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λ s, acc.rec_on s (λ
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(x₂ : A)
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(h₂ : Π y, y ≺ x₂ → acc R y)
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(ih₂ : _)
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(h₁ : Π y, y ≺ x₂ → acc R y)
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(ih₁ : Π y (hlt : y ≺ x₂) (s : acc R y), fix_F y (h₁ y hlt) = fix_F y s),
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calc fix_F x₂ (acc.intro x₂ h₁)
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= F x₂ (λ (y : A) (p : y ≺ x₂), fix_F y (h₁ y p)) : rfl
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... = F x₂ (λ (y : A) (p : y ≺ x₂), fix_F y (h₂ y p)) : F_ext x₂ _ _ (λ (y : A) (p : y ≺ x₂), ih₁ y p (h₂ y p))
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... = fix_F x₂ (acc.intro x₂ h₂) : rfl),
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show fix_F x₁ (acc.intro x₁ h₁) = fix_F x₁ s, from
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aux₁ s h₁ ih₁)
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variables {A : Type} {C : A → Type} {R : A → A → Type}
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-- Well-founded fixpoint
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definition fix (x : A) : C x :=
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fix_F x (Hwf x)
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definition fix [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) : C x :=
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fix_F F x (Hwf x)
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-- Well-founded fixpoint satisfies fixpoint equation
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definition fix_eq (x : A) : fix x = F x (λy h, fix y) :=
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calc
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fix x
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= fix_F x (Hwf x) : rfl
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... = F x (λy h, fix_F y (acc.inv (Hwf x) h)) : fix_F_eq x (Hwf x)
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... = F x (λy h, fix_F y (Hwf y)) : F_ext x _ _ (λ y h, fix_F_inv y _ _)
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... = F x (λy h, fix y) : rfl
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theorem fix_eq [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) :
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fix F x = F x (λy h, fix F y) :=
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begin
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refine fix_F_eq F x (Hwf x) ⬝ _,
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apply ap (F x),
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apply eq_of_homotopy, intro a,
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apply eq_of_homotopy, intro r,
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apply ap (fix_F F a),
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apply acc.acc_eq
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end
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end well_founded
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@ -97,17 +103,20 @@ well_founded.intro (λ (a : A),
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-- Subrelation of a well-founded relation is well-founded
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namespace subrelation
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section
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universe variable u
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parameters {A : Type} {R Q : A → A → Type}
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parameters (H₁ : subrelation Q R)
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parameters (H₂ : well_founded R)
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definition accessible {a : A} (ac : acc R a) : acc Q a :=
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acc.rec_on ac
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(λ (x : A) (ax : _) (iH : ∀ (y : A), R y x → acc Q y),
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acc.intro x (λ (y : A) (lt : Q y x), iH y (H₁ lt)))
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using H₁,
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begin
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induction ac with x ax ih, constructor,
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exact λ (y : A) (lt : Q y x), ih y (H₁ lt)
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end
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definition wf : well_founded Q :=
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well_founded.intro (λ a, accessible (H₂ a))
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well_founded.intro (λ a, accessible proof (@apply A R H₂ a) qed)
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end
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end subrelation
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@ -118,14 +127,16 @@ section
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parameters (f : A → B)
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parameters (H : well_founded R)
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private definition acc_aux {b : B} (ac : acc R b) : Π x, f x = b → acc (inv_image R f) x :=
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begin
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induction ac with x acx ih,
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intro z e, constructor,
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intro y lt, subst x,
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exact ih (f y) lt y rfl
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end
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definition accessible {a : A} (ac : acc R (f a)) : acc (inv_image R f) a :=
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have gen : ∀x, f x = f a → acc (inv_image R f) x, from
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acc.rec_on ac
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(λx acx (iH : ∀y, R y x → (∀z, f z = y → acc (inv_image R f) z))
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(z : A) (eq₁ : f z = x),
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acc.intro z (λ (y : A) (lt : R (f y) (f z)),
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iH (f y) (eq.rec_on eq₁ lt) y rfl)),
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gen a rfl
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acc_aux ac a rfl
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definition wf : well_founded (inv_image R f) :=
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well_founded.intro (λ a, accessible (H (f a)))
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@ -139,22 +150,99 @@ section
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local notation `R⁺` := tc R
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definition accessible {z} (ac: acc R z) : acc R⁺ z :=
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acc.rec_on ac
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(λ x acx (iH : ∀y, R y x → acc R⁺ y),
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acc.intro x (λ (y : A) (lt : R⁺ y x),
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have gen : x = x → acc R⁺ y, from
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tc.rec_on lt
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(λa b (H : R a b) (Heq : b = x),
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iH a (eq.rec_on Heq H))
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(λa b c (H₁ : R⁺ a b) (H₂ : R⁺ b c)
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(iH₁ : b = x → acc R⁺ a)
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(iH₂ : c = x → acc R⁺ b)
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(Heq : c = x),
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acc.inv (iH₂ Heq) H₁),
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gen rfl))
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begin
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induction ac with x acx ih,
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constructor, intro y rel,
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induction rel with a b rab a b c rab rbc ih₁ ih₂,
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{exact ih a rab},
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{exact acc.inv (ih₂ acx ih) rab}
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end
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definition wf (H : well_founded R) : well_founded R⁺ :=
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well_founded.intro (λ a, accessible (H a))
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end
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end tc
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namespace nat
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-- less-than is well-founded
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definition lt.wf [instance] : well_founded (lt : ℕ → ℕ → Type₀) :=
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begin
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constructor, intro n, induction n with n IH,
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{ constructor, intros n H, exfalso, exact !not_lt_zero H},
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{ constructor, intros m H,
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assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
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{ intros n₁ hlt, induction hlt,
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{ intro p, injection p with q, exact q ▸ IH},
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{ intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
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apply aux H rfl},
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end
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definition measure {A : Type} : (A → ℕ) → A → A → Type₀ :=
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inv_image lt
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definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
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inv_image.wf f lt.wf
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end nat
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namespace prod
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open well_founded prod.ops
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section
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variables {A B : Type}
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variable (Ra : A → A → Type)
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variable (Rb : B → B → Type)
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-- Lexicographical order based on Ra and Rb
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inductive lex : A × B → A × B → Type :=
|
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| left : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
|
||||
| right : ∀a {b₁ b₂}, Rb b₁ b₂ → lex (a, b₁) (a, b₂)
|
||||
|
||||
-- Relational product based on Ra and Rb
|
||||
inductive rprod : A × B → A × B → Type :=
|
||||
intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
|
||||
end
|
||||
|
||||
section
|
||||
parameters {A B : Type}
|
||||
parameters {Ra : A → A → Type} {Rb : B → B → Type}
|
||||
local infix `≺`:50 := lex Ra Rb
|
||||
|
||||
definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
|
||||
acc.rec_on aca
|
||||
(λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
|
||||
λb, acc.rec_on (acb b)
|
||||
(λxb acb
|
||||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a₁, b₁), from
|
||||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||||
iHa a₁ Ra₁ b₁)
|
||||
(λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a, b₁), from
|
||||
have Rb₁ : Rb b₁ xb, from eq.rec_on eq₃ H,
|
||||
have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
|
||||
eq.rec_on eq₂' (iHb b₁ Rb₁)),
|
||||
aux rfl rfl)))
|
||||
|
||||
-- The lexicographical order of well founded relations is well-founded
|
||||
definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
|
||||
well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
|
||||
|
||||
-- Relational product is a subrelation of the lex
|
||||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
|
||||
-- The relational product of well founded relations is well-founded
|
||||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||||
subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
|
||||
|
||||
end
|
||||
|
||||
end prod
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Authors: Floris van Doorn, Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import init.wf init.tactic init.num
|
||||
import init.relation init.tactic init.num
|
||||
open eq.ops decidable or
|
||||
|
||||
notation `ℕ` := nat
|
||||
|
|
@ -189,20 +189,6 @@ namespace nat
|
|||
protected theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ∨ a < b) : a ≤ b :=
|
||||
or.elim H !nat.le_of_eq !nat.le_of_lt
|
||||
|
||||
-- less-than is well-founded
|
||||
definition lt.wf [instance] : well_founded lt :=
|
||||
well_founded.intro (nat.rec
|
||||
(!acc.intro (λn H, absurd H (not_lt_zero n)))
|
||||
(λn IH, !acc.intro (λm H,
|
||||
or.elim (nat.eq_or_lt_of_le (le_of_succ_le_succ H))
|
||||
(λe, eq.substr e IH) (acc.inv IH))))
|
||||
|
||||
definition measure {A : Type} : (A → ℕ) → A → A → Prop :=
|
||||
inv_image lt
|
||||
|
||||
definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
|
||||
inv_image.wf f lt.wf
|
||||
|
||||
theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
|
||||
succ_le_succ
|
||||
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Author: Leonardo de Moura, Jeremy Avigad
|
||||
-/
|
||||
prelude
|
||||
import init.num init.wf
|
||||
import init.num init.relation
|
||||
|
||||
definition pair [constructor] := @prod.mk
|
||||
notation A × B := prod A B
|
||||
|
|
@ -30,60 +30,4 @@ namespace prod
|
|||
| (a, b) := rfl
|
||||
end
|
||||
|
||||
open well_founded
|
||||
|
||||
section
|
||||
variables {A B : Type}
|
||||
variable (Ra : A → A → Prop)
|
||||
variable (Rb : B → B → Prop)
|
||||
|
||||
-- Lexicographical order based on Ra and Rb
|
||||
inductive lex : A × B → A × B → Prop :=
|
||||
| left : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
|
||||
| right : ∀a {b₁ b₂}, Rb b₁ b₂ → lex (a, b₁) (a, b₂)
|
||||
|
||||
-- Relational product based on Ra and Rb
|
||||
inductive rprod : A × B → A × B → Prop :=
|
||||
intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
|
||||
end
|
||||
|
||||
section
|
||||
parameters {A B : Type}
|
||||
parameters {Ra : A → A → Prop} {Rb : B → B → Prop}
|
||||
local infix `≺`:50 := lex Ra Rb
|
||||
|
||||
definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
|
||||
acc.rec_on aca
|
||||
(λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
|
||||
λb, acc.rec_on (acb b)
|
||||
(λxb acb
|
||||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a₁, b₁), from
|
||||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||||
iHa a₁ Ra₁ b₁)
|
||||
(λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a, b₁), from
|
||||
have Rb₁ : Rb b₁ xb, from eq.rec_on eq₃ H,
|
||||
have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
|
||||
eq.rec_on eq₂' (iHb b₁ Rb₁)),
|
||||
aux rfl rfl)))
|
||||
|
||||
-- The lexicographical order of well founded relations is well-founded
|
||||
definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
|
||||
well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
|
||||
|
||||
-- Relational product is a subrelation of the lex
|
||||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
|
||||
-- The relational product of well founded relations is well-founded
|
||||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||||
subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
|
||||
|
||||
end
|
||||
end prod
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
|
|||
Author: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import init.relation init.tactic
|
||||
import init.relation init.tactic init.nat init.prod
|
||||
|
||||
inductive acc {A : Type} (R : A → A → Prop) : A → Prop :=
|
||||
intro : ∀x, (∀ y, R y x → acc R y) → acc R x
|
||||
|
|
@ -141,3 +141,81 @@ section
|
|||
well_founded.intro (λ a, accessible (H a))
|
||||
end
|
||||
end tc
|
||||
|
||||
namespace nat
|
||||
|
||||
-- less-than is well-founded
|
||||
definition lt.wf [instance] : well_founded lt :=
|
||||
well_founded.intro (nat.rec
|
||||
(!acc.intro (λn H, absurd H (not_lt_zero n)))
|
||||
(λn IH, !acc.intro (λm H,
|
||||
or.elim (nat.eq_or_lt_of_le (le_of_succ_le_succ H))
|
||||
(λe, eq.substr e IH) (acc.inv IH))))
|
||||
|
||||
definition measure {A : Type} : (A → ℕ) → A → A → Prop :=
|
||||
inv_image lt
|
||||
|
||||
definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
|
||||
inv_image.wf f lt.wf
|
||||
|
||||
end nat
|
||||
|
||||
namespace prod
|
||||
open well_founded
|
||||
|
||||
section
|
||||
variables {A B : Type}
|
||||
variable (Ra : A → A → Prop)
|
||||
variable (Rb : B → B → Prop)
|
||||
|
||||
-- Lexicographical order based on Ra and Rb
|
||||
inductive lex : A × B → A × B → Prop :=
|
||||
| left : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
|
||||
| right : ∀a {b₁ b₂}, Rb b₁ b₂ → lex (a, b₁) (a, b₂)
|
||||
|
||||
-- Relational product based on Ra and Rb
|
||||
inductive rprod : A × B → A × B → Prop :=
|
||||
intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
|
||||
end
|
||||
|
||||
section
|
||||
parameters {A B : Type}
|
||||
parameters {Ra : A → A → Prop} {Rb : B → B → Prop}
|
||||
local infix `≺`:50 := lex Ra Rb
|
||||
|
||||
definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
|
||||
acc.rec_on aca
|
||||
(λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
|
||||
λb, acc.rec_on (acb b)
|
||||
(λxb acb
|
||||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||||
p (xa, xb) lt
|
||||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a₁, b₁), from
|
||||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||||
iHa a₁ Ra₁ b₁)
|
||||
(λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
|
||||
show acc (lex Ra Rb) (a, b₁), from
|
||||
have Rb₁ : Rb b₁ xb, from eq.rec_on eq₃ H,
|
||||
have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
|
||||
eq.rec_on eq₂' (iHb b₁ Rb₁)),
|
||||
aux rfl rfl)))
|
||||
|
||||
-- The lexicographical order of well founded relations is well-founded
|
||||
definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
|
||||
well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
|
||||
|
||||
-- Relational product is a subrelation of the lex
|
||||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||||
|
||||
-- The relational product of well founded relations is well-founded
|
||||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||||
subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
|
||||
|
||||
end
|
||||
|
||||
end prod
|
||||
|
|
|
|||
|
|
@ -2,7 +2,7 @@
|
|||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Leonardo de Moura
|
||||
prelude
|
||||
import init.nat
|
||||
import init.wf
|
||||
|
||||
namespace well_founded
|
||||
-- This is an auxiliary definition that useful for generating a new "proof" for (well_founded R)
|
||||
|
|
|
|||
Loading…
Reference in a new issue