71 lines
3.1 KiB
Text
71 lines
3.1 KiB
Text
/-
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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, Jeremy Avigad, Floris van Doorn
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-/
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prelude
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import init.datatypes init.num init.wf init.logic init.tactic
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definition dpair := @sigma.mk
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notation `Σ` binders `,` r:(scoped P, sigma P) := r
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-- notation for n-ary tuples; input ⟨ ⟩ as \< \>
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notation `⟨`:max t:(foldr `,` (e r, sigma.mk e r)) `⟩`:0 := t
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lemma ex_of_sig {A : Type} {P : A → Prop} : (Σ x, P x) → ∃ x, P x :=
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assume h, obtain x hx, from h, exists.intro x hx
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namespace sigma
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notation `pr₁` := pr1
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notation `pr₂` := pr2
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namespace ops
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postfix `.1`:(max+1) := pr1
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postfix `.2`:(max+1) := pr2
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end ops
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open ops well_founded
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section
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variables {A : Type} {B : A → Type}
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variable (Ra : A → A → Prop)
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variable (Rb : ∀ a, B a → B a → Prop)
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theorem dpair_eq : ∀ {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂), eq.rec_on H₁ b₁ = b₂ → ⟨a₁, b₁⟩ = ⟨a₂, b₂⟩
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| a₁ a₁ b₁ b₁ (eq.refl a₁) (eq.refl b₁) := rfl
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protected theorem eq {p₁ p₂ : Σa : A, B a} :
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∀(H₁ : p₁.1 = p₂.1) (H₂ : eq.rec_on H₁ p₁.2 = p₂.2), p₁ = p₂ :=
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destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))
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-- Lexicographical order based on Ra and Rb
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inductive lex : sigma B → sigma B → Prop :=
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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 a b₁ b₂ → lex ⟨a, b₁⟩ ⟨a, b₂⟩
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end
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section
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parameters {A : Type} {B : A → Type}
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parameters {Ra : A → A → Prop} {Rb : Π a : A, B a → B a → Prop}
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local infix `≺`:50 := lex Ra Rb
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definition lex.accessible {a} (aca : acc Ra a) (acb : ∀a, well_founded (Rb a)) : ∀ (b : B a), 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 : B y, acc (lex Ra Rb) ⟨y, b⟩),
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λb : B xa, acc.rec_on (acb xa b)
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(λxb acb
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(iHb : ∀ (y : B xa), Rb xa 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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@sigma.lex.rec_on A B Ra Rb (λp₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex Ra Rb) p₁)
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p ⟨xa, xb⟩ lt
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(λ (a₁ : A) (b₁ : B a₁) (a₂ : A) (b₂ : B a₂) (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
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begin cases eq₂, exact (iHa a₁ H b₁) end)
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(λ (a : A) (b₁ b₂ : B a) (H : Rb a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
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begin cases eq₂, cases eq₃, exact (iHb b₁ H) end),
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aux rfl !heq.refl)))
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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 : ∀ x, well_founded (Rb x)) : well_founded (lex Ra Rb) :=
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well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) Hb b))
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end
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end sigma
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