Theory: unless

Parents


Type constants


Term constants


Axioms


Definitions

UNLESS_STMT
|- !p q st.
     (p UNLESS_STMT q) st = (\s. p s /\ ~(q s) ==> p (st s) \/ q (st s))
UNLESS
|- (!p q. (p UNLESS q) [] = T) /\
   (!p q Pr st.
     (p UNLESS q) (CONS st Pr) =
     (!s. (p UNLESS_STMT q) st s) /\ (p UNLESS q) Pr)
STABLE
|- !p Pr. p STABLE Pr = (p UNLESS False) Pr
INVARIANT
|- !p p0 Pr. p INVARIANT (p0,Pr) = (!s. p0 s ==> p s) /\ p STABLE Pr

Theorems

UNLESS_STMT_thm0
|- !p st s. (p UNLESS_STMT p) st s
UNLESS_STMT_thm1
|- !p q r st.
     (!s. (p UNLESS_STMT q) st s) /\ (!s. q s ==> r s) ==>
     (!s. (p UNLESS_STMT r) st s)
UNLESS_STMT_thm2
|- !p q p' q' st.
     (!s. (p UNLESS_STMT q) st s) /\ (!s. (p' UNLESS_STMT q') st s) ==>
     (!s. (p \/* p' UNLESS_STMT q \/* q') st s)
UNLESS_STMT_thm3
|- !p q p' q' st.
     (!s. (p UNLESS_STMT q) st s) /\ (!s. (p' UNLESS_STMT q') st s) ==>
     (!s. (p /\* p' UNLESS_STMT (p /\* q' \/* p' /\* q) \/* q /\* q') st s)
UNLESS_STMT_thm4
|- !p q p' q' st.
     (!s. (p UNLESS_STMT q) st s) /\ (!s. (p' UNLESS_STMT q') st s) ==>
     (!s.
       (p \/* p' UNLESS_STMT (~* p /\* q' \/* ~* p' /\* q) \/* q /\* q') st s)
UNLESS_STMT_thm5
|- !P q st.
     (!m s. (P m UNLESS_STMT q) st s) ==>
     (!s. ((\s. ?n. P n s) UNLESS_STMT q) st s)
UNLESS_thm1
|- !p Pr. (p UNLESS p) Pr
UNLESS_thm2
|- !p Pr. (p UNLESS ~* p) Pr
UNLESS_thm3
|- !p q r Pr. (p UNLESS q) Pr /\ (!s. q s ==> r s) ==> (p UNLESS r) Pr
UNLESS_thm4
|- !p q p' q' Pr.
     (p UNLESS q) Pr /\ (p' UNLESS q') Pr ==>
     (p /\* p' UNLESS (p /\* q' \/* p' /\* q) \/* q /\* q') Pr
UNLESS_thm5
|- !p q p' q' Pr.
     (p UNLESS q) Pr /\ (p' UNLESS q') Pr ==>
     (p \/* p' UNLESS (~* p /\* q' \/* ~* p' /\* q) \/* q /\* q') Pr
UNLESS_thm6
|- !p q p' q' Pr.
     (p UNLESS q) Pr /\ (p' UNLESS q') Pr ==> (p /\* p' UNLESS q \/* q') Pr
UNLESS_thm7
|- !p q p' q' Pr.
     (p UNLESS q) Pr /\ (p' UNLESS q') Pr ==> (p \/* p' UNLESS q \/* q') Pr
UNLESS_thm8
|- !p q r Pr. (p UNLESS q) Pr /\ (q UNLESS r) Pr ==> (p \/* q UNLESS r) Pr
UNLESS_cor1
|- !p q Pr. (!s. p s ==> q s) ==> (p UNLESS q) Pr
UNLESS_cor2
|- !p q Pr. (!s. ~* p s ==> q s) ==> (p UNLESS q) Pr
UNLESS_cor3
|- !p q r Pr. (p /\* ~* q UNLESS q \/* r) Pr = (p UNLESS q \/* r) Pr
UNLESS_cor4
|- !p q r Pr. (p \/* q UNLESS r) Pr ==> (p UNLESS q \/* r) Pr
UNLESS_cor5
|- !p Pr. (p UNLESS True) Pr
UNLESS_cor6
|- !p Pr. (True UNLESS p) Pr
UNLESS_cor7
|- !p Pr. (False UNLESS p) Pr
UNLESS_cor8
|- !p q p' Pr.
     (!s. p s /\ ~(q s)) ==>
     (!s. p' s) ==>
     (!s. p s \/ q s) ==>
     ((p /\* ~* q UNLESS q) Pr = (p' /\* ~* q UNLESS q) Pr)
UNLESS_cor9
|- !p q p' q' r r' Pr.
     (p \/* p' UNLESS q \/* r) Pr /\ (q \/* q' UNLESS p \/* r') Pr ==>
     (p \/* p' \/* q \/* q' UNLESS p /\* q \/* r \/* r') Pr
UNLESS_cor10
|- !p q Pr. p \/* q STABLE Pr ==> (p UNLESS q) Pr
UNLESS_cor11
|- !p Pr. (!s. ~* p s) ==> p STABLE Pr
UNLESS_cor12
|- !p Pr. (!s. ~* p s) ==> ~* p STABLE Pr
UNLESS_cor13
|- !p q Pr.
     (p UNLESS q) Pr /\ (q UNLESS p) Pr /\ (!s. ~* (p /\* q) s) ==>
     p \/* q STABLE Pr
UNLESS_cor14
|- !p q Pr. (p UNLESS ~* q) Pr /\ q STABLE Pr ==> (p UNLESS p /\* ~* q) Pr
UNLESS_cor15
|- !P Q Pr.
     (!i. (P i UNLESS P i /\* Q i) Pr) ==> ($!* P UNLESS $!* P /\* $?* Q) Pr
UNLESS_cor16
|- !P Q Pr.
     (!i. (P i UNLESS Q i) Pr) ==> (!i. (/<=\* P i UNLESS \<=/* Q i) Pr)
UNLESS_cor17
|- !P q Pr. (!i. (P i UNLESS q) Pr) ==> (!i. (/<=\* P i UNLESS q) Pr)
UNLESS_cor18
|- !P q Pr. (!m. (P m UNLESS q) Pr) ==> ($?* P UNLESS q) Pr
UNLESS_cor19
|- !Pr. False STABLE Pr
UNLESS_cor20
|- !p q Pr. p STABLE Pr /\ q STABLE Pr ==> p /\* q STABLE Pr
UNLESS_cor21
|- !p q Pr. p STABLE Pr /\ q STABLE Pr ==> p \/* q STABLE Pr
UNLESS_cor22
|- !p q r Pr. (p UNLESS q) Pr /\ r STABLE Pr ==> (p /\* r UNLESS q /\* r) Pr