62 lines
1.5 KiB
Text
62 lines
1.5 KiB
Text
/-
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Copyright (c) 2017 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad
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-/
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import types.trunc
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open funext eq trunc is_trunc prod sum
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--reserve prefix `¬`:40
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--reserve prefix `~`:40
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--reserve infixr ` ∧ `:35
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--reserve infixr ` /\ `:35
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--reserve infixr ` \/ `:30
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--reserve infixr ` ∨ `:30
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--reserve infix ` <-> `:20
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--reserve infix ` ↔ `:20
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namespace logic
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section
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open trunc_index
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definition propext {p q : Prop} (h : p ↔ q) : p = q :=
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tua (equiv_of_iff_of_is_prop h _ _)
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end
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definition false : Prop := trunctype.mk (lift empty) _
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definition false.elim {A : Type} (h : false) : A := lift.rec empty.elim h
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definition true : Prop := trunctype.mk (lift unit) _
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definition true.intro : true := lift.up unit.star
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definition trivial : true := lift.up unit.star
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definition and (p q : Prop) : Prop := tprod p q
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infixr ` ∧ ` := and
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infixr ` /\ ` := and
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definition and.intro {p q : Prop} (h₁ : p) (h₂ : q) : and p q := prod.mk h₁ h₂
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definition and.left {p q : Prop} (h : p ∧ q) : p := prod.pr1 h
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definition and.right {p q : Prop} (h : p ∧ q) : q := prod.pr2 h
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definition not (p : Prop) : Prop := trunctype.mk (p → empty) _
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definition or.inl := @or.intro_left
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definition or.inr := @or.intro_right
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definition or.elim {A B C : Type} [is_prop C] (h₀ : A ∨ B) (h₁ : (A → C)) (h₂ : B → C) : C :=
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begin
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apply trunc.elim_on h₀,
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intro h₃,
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apply sum.elim h₃ h₁ h₂
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end
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end logic
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