09a5b34931
This PR adjusts the experimental module system to make `private` the default visibility modifier in `module`s, introducing `public` as a new modifier instead. `public section` can be used to revert the default for an entire section, though this is more intended to ease gradual adoption of the new semantics such as in `Init` (and soon `Std`) where they should be replaced by a future decl-by-decl re-review of visibilities.
54 lines
1.8 KiB
Text
54 lines
1.8 KiB
Text
/-
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Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Kim Morrison
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-/
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module
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prelude
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public import Init.PropLemmas
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public section
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/-!
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# Specializations of basic logic lemmas
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These are useful for `omega` while constructing proofs, but not considered generally useful
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so are hidden in the `Lean.Omega` namespace.
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If you find yourself needing them elsewhere, please move them first to another file.
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-/
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namespace Lean.Omega
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theorem and_not_not_of_not_or (h : ¬ (p ∨ q)) : ¬ p ∧ ¬ q := not_or.mp h
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theorem Decidable.or_not_not_of_not_and [Decidable p] [Decidable q]
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(h : ¬ (p ∧ q)) : ¬ p ∨ ¬ q :=
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Decidable.not_and_iff_or_not.mp h
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theorem Decidable.and_or_not_and_not_of_iff {p q : Prop} [Decidable q] (h : p ↔ q) :
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(p ∧ q) ∨ (¬p ∧ ¬q) := Decidable.iff_iff_and_or_not_and_not.mp h
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theorem Decidable.not_iff_iff_and_not_or_not_and [Decidable a] [Decidable b] :
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(¬ (a ↔ b)) ↔ (a ∧ ¬ b) ∨ ((¬ a) ∧ b) :=
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⟨fun e => if hb : b then
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.inr ⟨fun ha => e ⟨fun _ => hb, fun _ => ha⟩, hb⟩
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else
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.inl ⟨if ha : a then ha else False.elim (e ⟨fun ha' => absurd ha' ha, fun hb' => absurd hb' hb⟩), hb⟩,
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Or.rec (And.rec fun ha nb w => nb (w.mp ha)) (And.rec fun na hb w => na (w.mpr hb))⟩
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theorem Decidable.and_not_or_not_and_of_not_iff [Decidable a] [Decidable b]
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(h : ¬ (a ↔ b)) : a ∧ ¬b ∨ ¬a ∧ b :=
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Decidable.not_iff_iff_and_not_or_not_and.mp h
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theorem Decidable.and_not_of_not_imp [Decidable a] (h : ¬(a → b)) : a ∧ ¬b :=
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Decidable.not_imp_iff_and_not.mp h
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theorem ite_disjunction {α : Type u} {P : Prop} [Decidable P] {a b : α} :
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(P ∧ (if P then a else b) = a) ∨ (¬ P ∧ (if P then a else b) = b) :=
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if h : P then
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.inl ⟨h, if_pos h⟩
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else
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.inr ⟨h, if_neg h⟩
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end Lean.Omega
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