95 lines
3.6 KiB
Text
95 lines
3.6 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
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-/
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prelude
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import init.num init.wf
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definition pair [constructor] := @prod.mk
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notation A × B := prod A B
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-- notation for n-ary tuples
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notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
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namespace prod
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notation [parsing-only] A * B := prod A B
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namespace low_precedence_times
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reserve infixr [parsing-only] `*`:30 -- conflicts with notation for multiplication
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infixr `*` := prod
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end low_precedence_times
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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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definition destruct [reducible] := @prod.cases_on
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section
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variables {A B : Type}
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lemma pr1.mk (a : A) (b : B) : pr1 (mk a b) = a := rfl
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lemma pr2.mk (a : A) (b : B) : pr2 (mk a b) = b := rfl
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lemma eta : ∀ (p : A × B), mk (pr1 p) (pr2 p) = p
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| (a, b) := rfl
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end
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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 → Prop)
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variable (Rb : B → B → Prop)
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-- Lexicographical order based on Ra and Rb
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inductive lex : A × B → A × 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 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 → Prop :=
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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 → Prop} {Rb : B → B → Prop}
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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, 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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