41 lines
1.2 KiB
Text
41 lines
1.2 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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Module: logic.examples.instances_test
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Author: Jeremy Avigad
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Illustrates substitution and congruence with iff.
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-/
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import ..instances
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open relation
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open relation.general_subst
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open relation.iff_ops
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open eq.ops
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example (a b : Prop) (H : a ↔ b) (H1 : a) : b := mp H H1
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set_option class.conservative false
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example (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
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subst iff H1 H2
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exit
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example (a b c d e : Prop) (H1 : a ↔ b) (H2 : a ∨ c → ¬(d → a)) : b ∨ c → ¬(d → b) :=
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H1 ▸ H2
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example (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) :=
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is_congruence.congr iff (λa, (a ∨ c → ¬(d → a))) H1
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example (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
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H1 ⬝ H2⁻¹ ⬝ H3
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example (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
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H1 ⬝ (H2⁻¹ ⬝ H3)
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example (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
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H1 ⬝ H2⁻¹ ⬝ H3
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example (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
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H1 ⬝ H2⁻¹ ⬝ H3
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