43 lines
1.5 KiB
Text
43 lines
1.5 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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-- Authors: Jeremy Avigad
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-- logic.connectives.identities
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-- ============================
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-- Useful logical identities. In the absence of propositional extensionality, some of the
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-- calculations use the type class support provided by logic.connectives.instances
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import logic.connectives.instances
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using relation
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-- TODO: delete when calc bug is fixed
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calc_subst subst_iff
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theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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calc
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(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
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... ↔ a ∨ (c ∨ b) : {or_comm b c}
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... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
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theorem or_left_comm (a b c : Prop) : a ∨ (b ∨ c)↔ b ∨ (a ∨ c) :=
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calc
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a ∨ (b ∨ c) ↔ (a ∨ b) ∨ c : iff_symm (or_assoc _ _ _)
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... ↔ (b ∨ a) ∨ c : {or_comm a b}
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... ↔ b ∨ (a ∨ c) : or_assoc _ _ _
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theorem and_right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=
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calc
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(a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and_assoc _ _ _
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... ↔ a ∧ (c ∧ b) : {and_comm b c}
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... ↔ (a ∧ c) ∧ b : iff_symm (and_assoc _ _ _)
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theorem and_left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
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calc
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a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff_symm (and_assoc _ _ _)
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... ↔ (b ∧ a) ∧ c : {and_comm a b}
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... ↔ b ∧ (a ∧ c) : and_assoc _ _ _
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-- TODO: delete when calc bug is fixed
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calc_subst subst
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