48 lines
1.2 KiB
Text
48 lines
1.2 KiB
Text
--- 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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--- Author: Jeremy Avigad
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import ..instances
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open relation
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open relation.general_operations
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open relation.iff_ops
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open eq_ops
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section
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theorem test1 (a b : Prop) (H : a ↔ b) (H1 : a) : b := mp H H1
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end
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section
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theorem test2 (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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theorem test3 (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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end
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theorem test4 (a b c d e : Prop) (H1 : a ↔ b) : (a ∨ c → ¬(d → a)) ↔ (b ∨ c → ¬(d → b)) :=
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congruence.infer iff iff (λa, (a ∨ c → ¬(d → a))) H1
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section
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theorem test5 (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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theorem test6 (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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theorem test7 (T : Type) (a b c d : T) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
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trans H1 (trans (symm H2) H3)
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theorem test8 (a b c d : Prop) (H1 : a ↔ b) (H2 : c ↔ b) (H3 : c ↔ d) : a ↔ d :=
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trans H1 (trans (symm H2) H3)
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end
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