Theory: state_logic

Parents


Type constants


Term constants


Axioms


Definitions

False
|- False = (\s. F)
True
|- True = (\s. T)
~*
|- !p. ~* p = (\s. ~(p s))
/\*
|- !p q. p /\* q = (\s. p s /\ q s)
\/*
|- !p q. p \/* q = (\s. p s \/ q s)
!*
|- !P. $!* P = (\s. !x. P x s)
?*
|- !P. $?* P = (\s. ?x. P x s)
==>*
|- !p q. p ==>* q = (\s. p s ==> q s)
<*
|- !p q. p <* q = (\s. p s < q s)
>*
|- !p q. p >* q = (\s. p s > q s)
<=*
|- !p q. p <=* q = (\s. p s <= q s)
>=*
|- !p q. p >=* q = (\s. p s >= q s)
=*
|- !p q. p =* q = (\s. p s = q s)
=>*
|- !p r1 r2. (p =>* r1) r2 = (\s. (p s) => (r1 s) | (r2 s))
+*
|- !p q. p +* q = (\s. p s + q s)
-*
|- !p q. p -* q = (\s. p s - q s)
**
|- !p q. p ** q = (\s. p s * q s)
Suc
|- !p. Suc p = (\s. SUC (p s))
Pre
|- !p. Pre p = (\s. PRE (p s))
%*
|- !p q. p %* q = (\s. p s MOD q s)
/*
|- !p q. p /* q = (\s. p s DIV q s)
***
|- !p q. p *** q = (\s. p s EXP q s)
Ind
|- !a i. a Ind i = (\s. a s (i s))
!<=*
|- !P m. !<=* P m = (\s. !i. i <= m ==> P i s)
?<=*
|- !P m. ?<=* P m = (\s. ?i. i <= m /\ P i s)
?<*
|- !P m. ?<* P m = (\s. ?i. i < m /\ P i s)
/<=\*
|- (!P. /<=\* P 0 = P 0) /\ (!i P. /<=\* P (SUC i) = /<=\* P i /\* P (SUC i))
\<=/*
|- (!P. \<=/* P 0 = P 0) /\ (!i P. \<=/* P (SUC i) = \<=/* P i \/* P (SUC i))
/<\*
|- (!P. /<\* P 0 = False) /\ (!i P. /<\* P (SUC i) = /<\* P i /\* P i)
\
|- (!P. \


Theorems

IMPLY_WEAK_lemma1
|- !p q p' q' s. ((p /\* q' \/* p' /\* q) \/* q /\* q') s ==> (q \/* q') s
IMPLY_WEAK_lemma2
|- !p q p' q' s.
     ((~* p /\* q' \/* ~* p' /\* q) \/* q /\* q') s ==> (q \/* q') s
IMPLY_WEAK_lemma3
|- !p q r s. ((~* p /\* r \/* ~* q /\* q) \/* q /\* r) s ==> r s
IMPLY_WEAK_lemma4
|- !p q p' q' r r' s.
     ((~* (p \/* p') /\* (p \/* r') \/* ~* (q \/* q') /\* (q \/* r)) \/*
      (q \/* r) /\* (p \/* r'))
       s ==>
     (p /\* q \/* r \/* r') s
IMPLY_WEAK_lemma5
|- !p q r s. (p /\* r \/* (p \/* q) /\* (q \/* r) \/* r) s ==> (q \/* r) s
IMPLY_WEAK_lemma6
|- !p q b r s. (r /\* q \/* p /\* b \/* b /\* q) s ==> (q /\* r \/* b) s
IMPLY_WEAK_lemma7
|- !p q b r s.
     ((r /\* q \/* (r /\* p) /\* b) \/* b /\* q) s ==> (q /\* r \/* b) s
AND_COMM_OR_lemma
|- !p q r. r /\* q \/* p = q /\* r \/* p
AND_OR_COMM_lemma
|- !p q r. p /\* (r \/* q) = p /\* (q \/* r)
OR_COMM_AND_lemma
|- !p q r. (r \/* q) /\* p = (q \/* r) /\* p
OR_COMM_OR_lemma
|- !p q r. (r \/* q) \/* p = (q \/* r) \/* p
OR_OR_COMM_lemma
|- !p q r. p \/* r \/* q = p \/* q \/* r
AND_COMM_AND_lemma
|- !p q r. (r /\* q) /\* p = (q /\* r) /\* p
AND_AND_COMM_lemma
|- !p q r. p /\* r /\* q = p /\* q /\* r
OR_AND_COMM_lemma
|- !p q r. p \/* r /\* q = p \/* q /\* r
NOT_NOT_lemma
|- !p. ~* (~* p) = p
OR_COMM_lemma
|- !p q. p \/* q = q \/* p
OR_OR_lemma
|- !p. p \/* p = p
OR_ASSOC_lemma
|- !p q r. (p \/* q) \/* r = p \/* q \/* r
AND_IMPLY_WEAK_lemma
|- !p q s. (p /\* q) s ==> q s
SYM_AND_IMPLY_WEAK_lemma
|- !p q s. (p /\* q) s ==> p s
OR_IMPLY_WEAK_lemma
|- !p q s. p s ==> (p \/* q) s
SYM_OR_IMPLY_WEAK_lemma
|- !p q s. p s ==> (q \/* p) s
IMPLY_WEAK_AND_lemma
|- !p q r. (!s. p s ==> q s) ==> (!s. (p /\* r) s ==> (q /\* r) s)
IMPLY_WEAK_OR_lemma
|- !p q r. (!s. p s ==> q s) ==> (!s. (p \/* r) s ==> (q \/* r) s)
AND_AND_lemma
|- !p. p /\* p = p
AND_COMM_lemma
|- !p q. p /\* q = q /\* p
AND_ASSOC_lemma
|- !p q r. (p /\* q) /\* r = p /\* q /\* r
NOT_True_lemma
|- ~* True = False
NOT_False_lemma
|- ~* False = True
AND_True_lemma
|- !p. p /\* True = p
OR_True_lemma
|- !p. p \/* True = True
AND_False_lemma
|- !p. p /\* False = False
OR_False_lemma
|- !p. p \/* False = p
P_OR_NOT_P_lemma
|- !p. p \/* ~* p = True
P_AND_NOT_P_lemma
|- !p. p /\* ~* p = False
AND_COMPL_OR_lemma
|- !p q. p /\* ~* q \/* p /\* q = p
OR_NOT_AND_lemma
|- !p q. (p \/* q) /\* ~* q = p /\* ~* q
P_AND_Q_OR_Q_lemma
|- !p q. p /\* q \/* q = q
P_OR_Q_AND_Q_lemma
|- !p q. (p \/* q) /\* q = q
NOT_OR_AND_NOT_lemma
|- !p q. ~* (p \/* q) = ~* p /\* ~* q
NOT_AND_OR_NOT_lemma
|- !p q. ~* (p /\* q) = ~* p \/* ~* q
NOT_IMPLY_OR_lemma
|- !p q. (!s. ~* p s ==> q s) = (!s. (p \/* q) s)
IMPLY_OR_lemma
|- !p q. (!s. p s ==> q s) = (!s. (~* p \/* q) s)
OR_IMPLY_lemma
|- !p q. (!s. (p \/* q) s) = (!s. ~* p s ==> q s)
NOT_OR_IMPLY_lemma
|- !p q. (!s. (~* p \/* q) s) = (!s. p s ==> q s)
OR_AND_DISTR_lemma
|- !p q r. p \/* q /\* r = (p \/* q) /\* (p \/* r)
AND_OR_DISTR_lemma
|- !p q r. p /\* (q \/* r) = p /\* q \/* p /\* r
NOT_IMPLIES_False_lemma
|- !p. (!s. ~* p s) ==> (!s. p s = False s)
NOT_P_IMPLIES_P_EQ_False_lemma
|- !p. (!s. ~* p s) ==> (p = False)
NOT_AND_IMPLIES_lemma
|- !p q. (!s. ~* (p /\* q) s) = (!s. p s ==> ~* q s)
NOT_AND_IMPLIES_lemma1
|- !p q. (!s. ~* (p /\* q) s) ==> (!s. p s ==> ~* q s)
NOT_AND_IMPLIES_lemma2
|- !p q. (!s. ~* (p /\* q) s) ==> (!s. q s ==> ~* p s)
AND_OR_EQ_lemma
|- !p q. p /\* (p \/* q) = p
AND_OR_EQ_AND_COMM_OR_lemma
|- !p q. p /\* (q \/* p) = p /\* (p \/* q)
IMPLY_WEAK_lemma
|- !p q. (!s. p s) ==> (!s. (p \/* q) s)
IMPLY_WEAK_lemma_b
|- !p q s. p s ==> (p \/* q) s
ALL_OR_lemma
|- !P i. $?* P = P i \/* $?* P
ALL_i_OR_lemma
|- !P. (\s. ?i. \<=/* P i s) = $?* P