Package pair: Product types

Information

name	pair
version	1.11
description	Product types
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool
show	Data.Bool Data.Pair

Files

Package tarball pair-1.11.tgz
Theory file pair.thy (included in the package tarball)

Defined Type Operator

Data
- Pair
  - ×

Defined Constants

Data
- Pair
  - ,
  - fst
  - snd

Theorems

⊦ ∀x. (fst x, snd x) = x

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ ∀p. ∃x y. p = (x, y)

⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (p1, p2)

⊦ ∀P. (∃p. P p) ⇔ ∃p1 p2. P (p1, p2)

⊦ ∀P. (∀x y. P (x, y)) ⇒ ∀p. P p

⊦ ∀PAIR'. ∃fn. ∀a0 a1. fn (a0, a1) = PAIR' a0 a1

⊦ ∀t. (λp. t p) = λ(x, y). t (x, y)

⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = b

Input Type Operators

→
bool

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - F
  - T

Assumptions

⊦ T

⊦ ¬F ⇔ T

⊦ ¬T ⇔ F

⊦ ∀t. t ⇒ t

⊦ F ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. T

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (T ⇔ t) ⇔ t

⊦ ∀t. (t ⇔ T) ⇔ t

⊦ ∀t. F ∧ t ⇔ F

⊦ ∀t. T ∧ t ⇔ t

⊦ ∀t. t ∧ T ⇔ t

⊦ ∀t. F ⇒ t ⇔ T

⊦ ∀t. T ⇒ t ⇔ t

⊦ ∀t. t ⇒ T ⇔ T

⊦ ∀t. F ∨ t ⇔ t

⊦ ∀t. T ∨ t ⇔ T

⊦ ∀t. t ∨ F ⇔ t

⊦ ∀t. t ∨ T ⇔ T

⊦ ∀t. (F ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ F ⇔ ¬t

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀P Q. P ∧ (∃x. Q x) ⇔ ∃x. P ∧ Q x

⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x

⊦ ∀P Q. (∃x. P x) ∧ Q ⇔ ∃x. P x ∧ Q

⊦ ∀P Q. (∀x. P x) ∨ Q ⇔ ∀x. P x ∨ Q

⊦ ∀P Q. (∃x. P x) ∨ Q ⇔ ∃x. P x ∨ Q

⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3

⊦ ∀P x. (∀y. P y ⇔ y = x) ⇒ (select) P = x

⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)

⊦ ∀P Q. (∀x. P x ∧ Q x) ⇔ (∀x. P x) ∧ ∀x. Q x

⊦ ∀P Q. (∃x. P x ∨ Q x) ⇔ (∃x. P x) ∨ ∃x. Q x

⊦ ∀P Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x