/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Module: data.prod.decl Author: Leonardo de Moura, Jeremy Avigad -/ prelude import init.wf definition pair := @prod.mk namespace prod notation A * B := prod A B notation A × B := prod A B namespace low_precedence_times reserve infixr `*`:30 -- conflicts with notation for multiplication infixr `*` := prod end low_precedence_times notation `pr₁` := pr1 notation `pr₂` := pr2 -- notation for n-ary tuples notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t 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 context parameters {A B : Type} parameters {Ra : A → A → Prop} {Rb : B → B → Prop} 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 @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, 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