84 lines
3.4 KiB
Text
84 lines
3.4 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: Leonardo de Moura, Jeremy Avigad
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import .connectives ..classes.nonempty
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open inhabited nonempty
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inductive Exists {A : Type} (P : A → Prop) : Prop :=
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intro : ∀ (a : A), P a → Exists P
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abbreviation exists_intro := @Exists.intro
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notation `exists` binders `,` r:(scoped P, Exists P) := r
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notation `∃` binders `,` r:(scoped P, Exists P) := r
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theorem exists_elim {A : Type} {p : A → Prop} {B : Prop} (H1 : ∃x, p x) (H2 : ∀ (a : A) (H : p a), B) : B :=
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Exists.rec H2 H1
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theorem exists_not_forall {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬∀x, ¬p x :=
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assume H1 : ∀x, ¬p x,
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obtain (w : A) (Hw : p w), from H,
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absurd Hw (H1 w)
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theorem forall_not_exists {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬∃x, ¬p x :=
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assume H1 : ∃x, ¬p x,
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obtain (w : A) (Hw : ¬p w), from H1,
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absurd (H2 w) Hw
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definition exists_unique {A : Type} (p : A → Prop) :=
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∃x, p x ∧ ∀y, y ≠ x → ¬p y
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notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
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theorem exists_unique_intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, y ≠ w → ¬p y) : ∃!x, p x :=
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exists_intro w (and.intro H1 H2)
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theorem exists_unique_elim {A : Type} {p : A → Prop} {b : Prop}
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(H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, y ≠ x → ¬p y) → b) : b :=
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obtain w Hw, from H2,
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H1 w (and_elim_left Hw) (and_elim_right Hw)
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theorem forall_congr {A : Type} {φ ψ : A → Prop} (H : ∀x, φ x ↔ ψ x) : (∀x, φ x) ↔ (∀x, ψ x) :=
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iff_intro
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(assume Hl, take x, iff_elim_left (H x) (Hl x))
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(assume Hr, take x, iff_elim_right (H x) (Hr x))
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theorem exists_congr {A : Type} {φ ψ : A → Prop} (H : ∀x, φ x ↔ ψ x) : (∃x, φ x) ↔ (∃x, ψ x) :=
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iff_intro
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(assume Hex, obtain w Pw, from Hex,
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exists_intro w (iff_elim_left (H w) Pw))
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(assume Hex, obtain w Qw, from Hex,
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exists_intro w (iff_elim_right (H w) Qw))
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theorem forall_true_iff_true (A : Type) : (∀x : A, true) ↔ true :=
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iff_intro (assume H, trivial) (assume H, take x, trivial)
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theorem forall_p_iff_p (A : Type) {H : inhabited A} (p : Prop) : (∀x : A, p) ↔ p :=
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iff_intro (assume Hl, inhabited.destruct H (take x, Hl x)) (assume Hr, take x, Hr)
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theorem exists_p_iff_p (A : Type) {H : inhabited A} (p : Prop) : (∃x : A, p) ↔ p :=
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iff_intro
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(assume Hl, obtain a Hp, from Hl, Hp)
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(assume Hr, inhabited.destruct H (take a, exists_intro a Hr))
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theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) : (∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
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iff_intro
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(assume H, and.intro (take x, and_elim_left (H x)) (take x, and_elim_right (H x)))
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(assume H, take x, and.intro (and_elim_left H x) (and_elim_right H x))
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theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) : (∃x, φ x ∨ ψ x) ↔ (∃x, φ x) ∨ (∃x, ψ x) :=
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iff_intro
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(assume H, obtain (w : A) (Hw : φ w ∨ ψ w), from H,
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or_elim Hw
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(assume Hw1 : φ w, or_inl (exists_intro w Hw1))
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(assume Hw2 : ψ w, or_inr (exists_intro w Hw2)))
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(assume H, or_elim H
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(assume H1, obtain (w : A) (Hw : φ w), from H1,
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exists_intro w (or_inl Hw))
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(assume H2, obtain (w : A) (Hw : ψ w), from H2,
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exists_intro w (or_inr Hw)))
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theorem exists_imp_nonempty {A : Type} {P : A → Prop} (H : ∃x, P x) : nonempty A :=
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obtain w Hw, from H, nonempty.intro w
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