- 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. \* P 0 = False) /\ (!i P. \* P (SUC i) = \* P i \/* P i)
- 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