73 lines
2.1 KiB
Text
73 lines
2.1 KiB
Text
/-
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Copyright (c) 2017 Johannes Hölzl. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Johannes Hölzl
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Relator for functions, pairs, sums, and lists.
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-/
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prelude
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import init.core init.data.basic
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namespace relator
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universe variables u₁ u₂ v₁ v₂
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reserve infixr ` ⇒ `:40
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/- TODO(johoelzl):
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* should we introduce relators of datatypes as recursive function or as inductive
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predicate? For now we stick to the recursor approach.
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* relation lift for datatypes, Π, Σ, set, and subtype types
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* proof composition and identity laws
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* implement method to derive relators from datatype
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-/
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section
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variables {α : Type u₁} {β : Type u₂} {γ : Type v₁} {δ : Type v₂}
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variables (R : α → β → Prop) (S : γ → δ → Prop)
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def lift_fun (f : α → γ) (g : β → δ) : Prop :=
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∀{a b}, R a b → S (f a) (g b)
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infixr ⇒ := lift_fun
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end
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section
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variables {α : Type u₁} {β : out_param (Type u₂)} (R : out_param (α → β → Prop))
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@[class] def right_total := ∀b, ∃a, R a b
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@[class] def left_total := ∀a, ∃b, R a b
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@[class] def bi_total := left_total R ∧ right_total R
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end
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section
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variables {α : Type u₁} {β : Type u₂} (R : α → β → Prop)
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@[class] def left_unique := ∀{a b c}, R a b → R c b → a = c
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@[class] def right_unique := ∀{a b c}, R a b → R a c → b = c
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lemma rel_forall_of_total [t : bi_total R] : ((R ⇒ iff) ⇒ iff) (λp, ∀i, p i) (λq, ∀i, q i) :=
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take p q Hrel,
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⟨take H b, exists.elim (t^.right b) (take a Rab, (Hrel Rab)^.mp (H _)),
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take H a, exists.elim (t^.left a) (take b Rab, (Hrel Rab)^.mpr (H _))⟩
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lemma left_unique_of_rel_eq {eq' : β → β → Prop} (he : (R ⇒ (R ⇒ iff)) eq eq') : left_unique R
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| a b c (ab : R a b) (cb : R c b) :=
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have eq' b b,
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from iff.mp (he ab ab) rfl,
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iff.mpr (he ab cb) this
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end
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lemma rel_imp : (iff ⇒ (iff ⇒ iff)) implies implies :=
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take p q h r s l, imp_congr h l
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lemma rel_not : (iff ⇒ iff) not not :=
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take p q h, not_congr h
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instance bi_total_eq {α : Type u₁} : relator.bi_total (@eq α) :=
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⟨take a, ⟨a, rfl⟩, take a, ⟨a, rfl⟩⟩
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end relator
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