62 lines
2.9 KiB
Text
62 lines
2.9 KiB
Text
/-
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Copyright (c) 2016 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.sigma init.meta init.combinator
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namespace sigma
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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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section
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open ops well_founded tactic
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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 (well_founded.apply (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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by do
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/- TODO(Leo): cleanup using quotations -/
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get_local `eq₂ >>= subst,
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iHa : expr ← get_local `iHa, a₁ ← get_local `a₁, H ← get_local `H, b₁ ← get_local `b₁,
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exact (expr.app_of_list iHa [a₁, H, b₁]))
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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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by do
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/- TODO(Leo): cleanup using quotations -/
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get_local `eq₂ >>= subst,
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eq₃ ← get_local `eq₃,
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new_eq₃ ← mk_app `eq_of_heq [eq₃],
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note `new_eq₃ new_eq₃,
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get_local `new_eq₃ >>= subst,
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iHb : expr ← get_local `iHb, b₁ ← get_local `b₁, H ← get_local `H,
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exact (expr.app_of_list iHb [b₁, H])),
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-- begin cases eq₂, cases eq₃, exact (iHb b₁ H) end),
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aux rfl (heq.refl xb))))
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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 (well_founded.apply Ha a) Hb b))
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end
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end sigma
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