Package sum: Sum types

Information

name	sum
version	1.13
description	Sum types
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool pair natural
show	Data.Bool Data.Pair Data.Sum Number.Natural

Files

Package tarball sum-1.13.tgz
Theory file sum.thy (included in the package tarball)

Defined Type Operator

Data
- Sum
  - +

Defined Constants

Data
- Sum
  - destLeft
  - destRight
  - left
  - right

Theorems

⊦ ∀x. destLeft (left x) = x

⊦ ∀y. destRight (right y) = y

⊦ ∀P. (∀a. P (left a)) ∧ (∀a. P (right a)) ⇒ ∀x. P x

⊦ ∀INL' INR'. ∃fn. (∀a. fn (left a) = INL' a) ∧ ∀a. fn (right a) = INR' a

Input Type Operators

→
bool
Data
- Pair
  - ×
Number
- Natural
  - natural

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - F
  - T
- Pair
  - ,
  - fst
  - snd
Number
- Natural
  - *
  - +
  - <
  - ≤
  - bit0
  - bit1
  - even
  - exp
  - suc
  - zero

Assumptions

⊦ T

⊦ ∀n. 0 ≤ n

⊦ F ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀a. ∃!x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. even (2 * n)

⊦ ∀n. bit0 n = n + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. bit1 n = suc (n + n)

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀n. 2 * n = n + n

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

⊦ ∀P a. (∃x. a = x ∧ P x) ⇔ P a

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

⊦ (even 0 ⇔ T) ∧ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀P Q. (∃x. P ∧ Q x) ⇔ P ∧ ∃x. Q x

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

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀m n p. m + p = n + p ⇔ m = n

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

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

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n. exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀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 ⇒ Q x) ⇒ (∃x. P x) ⇒ ∃x. Q x

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

⊦ ∀e f. ∃fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∧ C ⇒ B ∧ D

⊦ ∀A B C D. (A ⇒ B) ∧ (C ⇒ D) ⇒ A ∨ C ⇒ B ∨ D

⊦ ∀P. (∀x. ∃!y. P x y) ⇔ ∃f. ∀x y. P x y ⇔ f x = y

⊦ (∀m. exp m 0 = 1) ∧ ∀m n. exp m (suc n) = m * exp m n

⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

⊦ ∀P. (∃!x. P x) ⇔ (∃x. P x) ∧ ∀x x'. P x ∧ P x' ⇒ x = x'

⊦ (∀m. m ≤ 0 ⇔ m = 0) ∧ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)

⊦ (∀n. 0 + n = n) ∧ (∀m. m + 0 = m) ∧ (∀m n. suc m + n = suc (m + n)) ∧
∀m n. m + suc n = suc (m + n)