93 lines
4.2 KiB
Text
93 lines
4.2 KiB
Text
-- 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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prelude
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import init.num init.wf init.logic init.tactic
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structure sigma {A : Type} (B : A → Type) :=
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dpair :: (dpr1 : A) (dpr2 : B dpr1)
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notation `Σ` binders `,` r:(scoped P, sigma P) := r
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namespace sigma
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notation `dpr₁` := dpr1
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notation `dpr₂` := dpr2
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namespace ops
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postfix `.1`:(max+1) := dpr1
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postfix `.2`:(max+1) := dpr2
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notation `⟨`:max t:(foldr `,` (e r, dpair e r)) `⟩`:0 := t --input ⟨ ⟩ as \< \>
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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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-- 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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context
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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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infix `≺`:50 := lex Ra Rb
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set_option pp.beta true
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variables {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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variables {a a' : A}
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{b : B a} {b' : B a'}
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{c : C a b} {c' : C a' b'}
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{d : D a b c} {d' : D a' b' c'}
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private theorem hcongr_arg2 (f : Πa b, C a b) (Ha : a = a') (Hb : b == b') : f a b == f a' b' :=
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hcongr (hcongr_arg f Ha) (hcongr_arg C Ha) Hb
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private theorem hcongr_arg3 (f : Πa b c, D a b c) (Ha : a = a') (Hb : b == b') (Hc : c == c')
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: f a b c == f a' b' c' :=
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hcongr (hcongr_arg2 f Ha Hb) (hcongr_arg2 D Ha Hb) Hc
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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, 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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@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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(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 a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
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-- TODO(Leo): cleanup this proof
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show acc (lex Ra Rb) ⟨a, b₁⟩, from
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let b₁' : B xa := eq.rec_on eq₂ b₁ in
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have aux₁ : b₁ == b₁', from
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heq.symm (eq_rec_heq eq₂ b₁),
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have aux₂ : Rb a b₁ b₂ = Rb xa b₁' xb, from
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heq.to_eq (hcongr_arg3 Rb eq₂ aux₁ eq₃),
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have aux₃ : Rb xa b₁' xb, from
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eq.rec_on aux₂ H,
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have aux₄ : acc (lex Ra Rb) ⟨xa, b₁'⟩, from
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iHb b₁' aux₃,
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have aux₅ : ∀ (b₁ b₂ : B a) (H₁ : a = a) (H₂ : b₁ == b₂), acc (lex Ra Rb) ⟨a, b₁⟩ → acc (lex Ra Rb) ⟨a, b₂⟩, from
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λ b₁ b₂ H₁ H₂ Ha, eq.rec_on (heq.to_eq H₂) Ha,
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have aux₆ : ∀ (b₁ : B xa) (b₂ : B a) (H₁ : a = xa) (H₂ : b₁ == b₂), acc (lex Ra Rb) ⟨xa, b₁⟩ → acc (lex Ra Rb) ⟨a, b₂⟩, from
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eq.rec_on eq₂ aux₅,
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aux₆ b₁' b₁ eq₂ (heq.symm aux₁) aux₄),
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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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