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LOGICS Some first-order theories

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Bordeaux university

Master Computer Science, 2015/2016

LOGICS

Some first-order theories

EG (Equality theory) REF : ∀ x x = x

SYM : ∀ x, y (x = y → y = x)

TRANS : ∀ x, y, z (x = y ∧ y = z) → x = z) COMPF : ∀ ~ x, ~ y (~ x = ~ y → f (~ x) = f (~ y)) COMPR : ∀ ~ x, ~ y (~ x = ~ y → R(~ x) → R(~ y))

P0 (Elementary arithmetics) All the axioms from EG ; A1 : ∀ x ¬ S(x) = 0

A2 : ∀ x (x = 0 ∨ ∃ y, x = S(y) A3 : ∀ x, y (S(x) = S(y) → x = y) A4 : ∀ x (x + 0 = x)

A5 : ∀ x, y (x + S(y) = S(x + y)) A6 : ∀ x (x × 0 = 0)

A7 : ∀ x, y (x × S(y) = x × y + x)

PA (Peano arithmetics) All the axioms from P

0

;

REC

Φ

: (Φ(0) ∧ ( ∀ x(Φ(x) → Φ(S(x)))) → ∀ xΦ(x) for every formula Φ(x) ;

MO (Monoıd theory) All the axioms from EG ;

ASS : ∀ x, y, z x ∗ (y ∗ z) = (x ∗ y) ∗ z NE : ∀ x (x ∗ e = x ∧ e ∗ x = x)

GR (Group theory) All the axioms from M O ;

INV : ∀ x (x ∗ I(x) = e ∧ I (x) ∗ x = e)

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