- LEADSTO_thm0
-
|- !p q Pr. (p ENSURES q) Pr ==> (p LEADSTO q) Pr
- LEADSTO_thm1
-
|- !p r q Pr. (p LEADSTO r) Pr /\ (r LEADSTO q) Pr ==> (p LEADSTO q) Pr
- LEADSTO_thm2
-
|- !p r q Pr. (p ENSURES r) Pr /\ (r LEADSTO q) Pr ==> (p LEADSTO q) Pr
- LEADSTO_thm2a
-
|- !p r q Pr. (p ENSURES r) Pr /\ (r ENSURES q) Pr ==> (p LEADSTO q) Pr
- LEADSTO_thm3
-
|- !p P q Pr.
(p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr) ==> (p LEADSTO q) Pr
- LEADSTO_thm3a
-
|- !P q Pr. (!p. p In P ==> (p LEADSTO q) Pr) ==> (LUB P LEADSTO q) Pr
- LEADSTO_thm3c
-
|- !P q Pr. (!i. (P i LEADSTO q) Pr) ==> ($?* P LEADSTO q) Pr
- LEADSTO_thm4
-
|- !p1 p2 q Pr.
(p1 LEADSTO q) Pr /\ (p2 LEADSTO q) Pr ==> (p1 \/* p2 LEADSTO q) Pr
- LEADSTO_thm5
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?r. (p LEADSTO r) Pr /\ (r LEADSTO q) Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm6
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?r. (p ENSURES r) Pr /\ (r LEADSTO q) Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm7
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?r. (p ENSURES r) Pr /\ (r ENSURES q) Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm8
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm9
-
|- !p q Pr.
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) = (p LEADSTO q) Pr
- LEADSTO_thm11
-
|- !p q st Pr.
(?r. (p ENSURES r) (CONS st Pr) /\ (r LEADSTO q) (CONS st Pr)) =
(p LEADSTO q) (CONS st Pr)
- LEADSTO_thm12
-
|- !p st Pr. (p LEADSTO p) (CONS st Pr)
- LEADSTO_thm13
-
|- !p q st Pr.
(?r. (p LEADSTO r) (CONS st Pr) /\ (r LEADSTO q) (CONS st Pr)) =
(p LEADSTO q) (CONS st Pr)
- LEADSTO_thm14
-
|- !p q st Pr.
(?r. (p LEADSTO r) (CONS st Pr) /\ (r LEADSTO q) (CONS st Pr)) =
(?r. (p ENSURES r) (CONS st Pr) /\ (r LEADSTO q) (CONS st Pr))
- LEADSTO_thm15
-
|- !p q Pr.
(p ENSURES q) Pr \/
(!r. (p ENSURES r) Pr /\ (r LEADSTO q) Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm16
-
|- !p q Pr.
(!r. (p ENSURES r) Pr /\ (r LEADSTO q) Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> (p LEADSTO q) Pr)) =
(p LEADSTO q) Pr
- LEADSTO_thm17
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p LEADSTO r) Pr /\
((p LEADSTO r) Pr ==> X p r Pr) /\
(r LEADSTO q) Pr /\
((r LEADSTO q) Pr ==> X r q Pr) ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> (p LEADSTO q) Pr ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr)) ==>
(p LEADSTO q) Pr ==>
X p q Pr
- LEADSTO_thm18
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr.
(p LEADSTO r) Pr /\
((p LEADSTO r) Pr ==> X p r Pr) /\
(r LEADSTO q) Pr /\
((r LEADSTO q) Pr ==> X r q Pr) ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!p P q Pr.
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> (p LEADSTO q) Pr ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm19
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p LEADSTO r) Pr /\ X p r Pr /\ (r LEADSTO q) Pr /\ X r q Pr ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr)) ==>
(p LEADSTO q) Pr ==>
X p q Pr
- LEADSTO_thm20
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr.
(p LEADSTO r) Pr /\ X p r Pr /\ (r LEADSTO q) Pr /\ X r q Pr ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!p P q Pr.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm21
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r. X p r Pr /\ X r q Pr ==> X p q Pr) /\
(!P. (p = LUB P) /\ (!p. p In P ==> X p q Pr) ==> X p q Pr)) ==>
(p LEADSTO q) Pr ==>
X p q Pr
- LEADSTO_thm22
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr. X p r Pr /\ X r q Pr ==> X p q Pr) /\
(!p P q Pr. (p = LUB P) /\ (!p. p In P ==> X p q Pr) ==> X p q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm23
-
|- !X Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p LEADSTO r) Pr /\ (r LEADSTO q) Pr /\ X p r Pr /\ X r q Pr ==>
X p q Pr) /\
(!P.
(p = LUB P) /\
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> X p q Pr) ==>
X p q Pr)) ==>
(!p q. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm24
-
|- !X Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p LEADSTO r) Pr /\ (r LEADSTO q) Pr /\ X p r Pr /\ X r q Pr ==>
X p q Pr) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q Pr) ==>
X (LUB P) q Pr)) ==>
(!p q. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm25
-
|- !p q st Pr. (!s. p s ==> q s) ==> (p LEADSTO q) (CONS st Pr)
- LEADSTO_thm26
-
|- !p q q' st Pr.
(p LEADSTO q) (CONS st Pr) ==> (p LEADSTO q \/* q') (CONS st Pr)
- LEADSTO_thm27
-
|- !p q p' q' st Pr.
(p LEADSTO q) (CONS st Pr) /\ (p' LEADSTO q') (CONS st Pr) ==>
(p \/* p' LEADSTO q \/* q') (CONS st Pr)
- LEADSTO_thm28
-
|- !p q b r st Pr.
(p LEADSTO q \/* b) (CONS st Pr) /\ (b LEADSTO r) (CONS st Pr) ==>
(p LEADSTO q \/* r) (CONS st Pr)
- LEADSTO_thm29
-
|- !p q r b st Pr.
(p LEADSTO q) (CONS st Pr) /\ (r UNLESS b) (CONS st Pr) ==>
(p /\* r LEADSTO q /\* r \/* b) (CONS st Pr)
- LEADSTO_thm30
-
|- !p st Pr. (p LEADSTO False) (CONS st Pr) ==> (!s. ~* p s)
- LEADSTO_cor1
-
|- !p b q Pr.
(p /\* b LEADSTO q) Pr /\ (p /\* ~* b LEADSTO q) Pr ==> (p LEADSTO q) Pr
- LEADSTO_cor2
-
|- !p q r st Pr.
(p LEADSTO q) (CONS st Pr) /\ r STABLE CONS st Pr ==>
(p /\* r LEADSTO q /\* r) (CONS st Pr)
- LEADSTO_cor3
-
|- !p q st Pr.
(p LEADSTO q) (CONS st Pr) = (p /\* ~* q LEADSTO q) (CONS st Pr)
- LEADSTO_cor4
-
|- !p b q st Pr.
(p /\* b LEADSTO q) (CONS st Pr) /\
(p /\* ~* b LEADSTO p /\* b \/* q) (CONS st Pr) ==>
(p LEADSTO q) (CONS st Pr)
- LEADSTO_cor5
-
|- !p q r st Pr.
(p /\* q LEADSTO r) (CONS st Pr) ==> (p LEADSTO ~* q \/* r) (CONS st Pr)
- LEADSTO_cor6
-
|- !p q r st Pr.
(p LEADSTO q) (CONS st Pr) /\ (r UNLESS q /\* r) (CONS st Pr) ==>
(p /\* r LEADSTO q /\* r) (CONS st Pr)
- LEADSTO_cor7
-
|- !p q r st Pr.
(p LEADSTO q) (CONS st Pr) /\ r /\* ~* q STABLE CONS st Pr ==>
(!s. (p /\* r) s ==> q s)
- LEADSTO_cor8
-
|- !p r q st Pr.
(p LEADSTO r) (CONS st Pr) ==> (p /\* q LEADSTO r) (CONS st Pr)
- LEADSTO_cor9
-
|- !p q r st Pr.
(p LEADSTO q) (CONS st Pr) /\ (!s. q s ==> r s) ==>
(p LEADSTO r) (CONS st Pr)
- LEADSTO_cor10
-
|- !P q Pr. (!i. (P i LEADSTO q) Pr) ==> (!i. (\<=/* P i LEADSTO q) Pr)
- LEADSTO_cor11
-
|- !p st Pr. (False LEADSTO p) (CONS st Pr)
- LEADSTO_cor12
-
|- !P q st Pr.
(!i. (P i LEADSTO q) (CONS st Pr)) ==>
(!i. (\* P i LEADSTO q) (CONS st Pr))
- LEADSTO2_thm0
-
|- !p q Pr. (p ENSURES q) Pr ==> LEADSTO2 p q Pr
- LEADSTO2_thm1
-
|- !p r q Pr. (p ENSURES r) Pr /\ LEADSTO2 r q Pr ==> LEADSTO2 p q Pr
- LEADSTO2_thm3
-
|- !P q Pr. (!p. p In P ==> LEADSTO2 p q Pr) ==> LEADSTO2 (LUB P) q Pr
- LEADSTO2_thm3a
-
|- !P q Pr.
(p = LUB P) /\ (!p. p In P ==> LEADSTO2 p q Pr) ==> LEADSTO2 p q Pr
- LEADSTO2_thm4
-
|- !p1 p2 q Pr.
LEADSTO2 p1 q Pr /\ LEADSTO2 p2 q Pr ==> LEADSTO2 (p1 \/* p2) q Pr
- LEADSTO2_thm8
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r. (p ENSURES r) Pr /\ X r q Pr ==> X p q Pr) /\
(!P. (!p. p In P ==> X p q Pr) ==> X (LUB P) q Pr)) ==>
LEADSTO2 p q Pr ==>
X p q Pr
- LEADSTO2_thm2
-
|- !p r q Pr. LEADSTO2 p r Pr /\ LEADSTO2 r q Pr ==> LEADSTO2 p q Pr
- LEADSTO2_thm5
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?r. LEADSTO2 p r Pr /\ LEADSTO2 r q Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> LEADSTO2 p q Pr)) =
LEADSTO2 p q Pr
- LEADSTO2_thm6
-
|- !p q Pr.
(p ENSURES q) Pr \/
(?r. (p ENSURES r) Pr /\ LEADSTO2 r q Pr) \/
(?P. (p = LUB P) /\ (!p. p In P ==> LEADSTO2 p q Pr)) =
LEADSTO2 p q Pr
- LEADSTO2_thm7
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p ENSURES r) Pr /\
LEADSTO2 r q Pr /\
(LEADSTO2 r q Pr ==> X r q Pr) ==>
LEADSTO2 p q Pr ==>
X p q Pr) /\
(!P.
(!p. p In P ==> LEADSTO2 p q Pr) /\
(!p. p In P ==> LEADSTO2 p q Pr ==> X p q Pr) ==>
LEADSTO2 (LUB P) q Pr ==>
X (LUB P) q Pr)) ==>
LEADSTO2 p q Pr ==>
X p q Pr
- LEADSTO_EQ_LEADSTO2
-
|- !p q Pr. (p LEADSTO q) Pr = LEADSTO2 p q Pr
- LEADSTO_thm31
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r. (p ENSURES r) Pr /\ X r q Pr ==> X p q Pr) /\
(!P. (!p. p In P ==> X p q Pr) ==> X (LUB P) q Pr)) ==>
(p LEADSTO q) Pr ==>
X p q Pr
- LEADSTO_thm32
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr. (p ENSURES r) Pr /\ X r q Pr ==> X p q Pr) /\
(!P q Pr. (!p. p In P ==> X p q Pr) ==> X (LUB P) q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm33
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p ENSURES r) Pr /\
(r LEADSTO q) Pr /\
((r LEADSTO q) Pr ==> X r q Pr) ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> (p LEADSTO q) Pr ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr)) ==>
(p LEADSTO q) Pr ==>
X p q Pr
- LEADSTO_thm34
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr.
(p ENSURES r) Pr /\
(r LEADSTO q) Pr /\
((r LEADSTO q) Pr ==> X r q Pr) ==>
(p LEADSTO q) Pr ==>
X p q Pr) /\
(!P q Pr.
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> (p LEADSTO q) Pr ==> X p q Pr) ==>
(LUB P LEADSTO q) Pr ==>
X (LUB P) q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm34a
-
|- !X Pr.
(!p q. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q.
(p ENSURES r) Pr /\ (r LEADSTO q) Pr /\ X r q Pr ==> X p q Pr) /\
(!P q.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q Pr) ==>
X (LUB P) q Pr) ==>
(!p q. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm34b
-
|- !X.
(!p q st Pr. (p ENSURES q) (CONS st Pr) ==> X p q (CONS st Pr)) /\
(!p r q st Pr.
(p ENSURES r) (CONS st Pr) /\
(r LEADSTO q) (CONS st Pr) /\
X r q (CONS st Pr) ==>
X p q (CONS st Pr)) /\
(!P q st Pr.
(!p. p In P ==> (p LEADSTO q) (CONS st Pr)) /\
(!p. p In P ==> X p q (CONS st Pr)) ==>
X (LUB P) q (CONS st Pr)) ==>
(!p q st Pr. (p LEADSTO q) (CONS st Pr) ==> X p q (CONS st Pr))
- LEADSTO_thm35
-
|- !p q p' q' r st Pr.
(p LEADSTO q) (CONS st Pr) /\
(p' LEADSTO q') (CONS st Pr) /\
(q UNLESS r) (CONS st Pr) /\
(q' UNLESS r) (CONS st Pr) ==>
(p /\* p' LEADSTO q /\* q' \/* r) (CONS st Pr)
- LEADSTO_thm36
-
|- !p q st Pr M.
(!m.
(p /\* (M EQmetric m) LEADSTO p /\* (M LESSmetric m) \/* q)
(CONS st Pr)) ==>
(p LEADSTO q) (CONS st Pr)
- LEADSTO_thm37
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q) /\
(!r.
(p LEADSTO r) Pr /\ X p r /\ (r LEADSTO q) Pr /\ X r q ==> X p q) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q) ==>
X (LUB P) q)) ==>
(p LEADSTO q) Pr ==>
X p q
- LEADSTO_thm38
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q) /\
(!p r q Pr.
(p LEADSTO r) Pr /\ X p r /\ (r LEADSTO q) Pr /\ X r q ==> X p q) /\
(!P q Pr.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q) ==>
X (LUB P) q) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q)
- LEADSTO_thm39
-
|- !X p q Pr.
(!p q.
((p ENSURES q) Pr ==> X p q) /\
(!r. (p ENSURES r) Pr /\ (r LEADSTO q) Pr /\ X r q ==> X p q) /\
(!P.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q) ==>
X (LUB P) q)) ==>
(p LEADSTO q) Pr ==>
X p q
- LEADSTO_thm40
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q) /\
(!p r q Pr. (p ENSURES r) Pr /\ (r LEADSTO q) Pr /\ X r q ==> X p q) /\
(!P q Pr.
(!p. p In P ==> (p LEADSTO q) Pr) /\ (!p. p In P ==> X p q) ==>
X (LUB P) q) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q)
- LEADSTO_thm41
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr.
(p LEADSTO r) Pr /\ (r LEADSTO q) Pr /\ X p r Pr /\ X r q Pr ==>
X p q Pr) /\
(!p P q Pr.
(p = LUB P) /\
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> X p q Pr) ==>
X p q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm42
-
|- !X Pr.
(!p q.
((p ENSURES q) Pr ==> X p q Pr) /\
(!r.
(p ENSURES r) Pr /\ (r LEADSTO q) Pr /\ X p r Pr /\ X r q Pr ==>
X p q Pr) /\
(!P.
(p = LUB P) /\
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> X p q Pr) ==>
X p q Pr)) ==>
(!p q. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_thm43
-
|- !X.
(!p q Pr. (p ENSURES q) Pr ==> X p q Pr) /\
(!p r q Pr.
(p ENSURES r) Pr /\ (r LEADSTO q) Pr /\ X p r Pr /\ X r q Pr ==>
X p q Pr) /\
(!p P q Pr.
(p = LUB P) /\
(!p. p In P ==> (p LEADSTO q) Pr) /\
(!p. p In P ==> X p q Pr) ==>
X p q Pr) ==>
(!p q Pr. (p LEADSTO q) Pr ==> X p q Pr)
- LEADSTO_cor13
-
|- !P Q r st Pr.
(!i. (P i LEADSTO Q i \/* r) (CONS st Pr)) /\
(!i. (Q i UNLESS r) (CONS st Pr)) ==>
(!i. (/<=\* P i LEADSTO /<=\* Q i \/* r) (CONS st Pr))
- LEADSTO_cor14
-
|- !p q r p' q' st Pr.
(p LEADSTO q \/* r) (CONS st Pr) /\
(q UNLESS r) (CONS st Pr) /\
(p' LEADSTO q' \/* r) (CONS st Pr) /\
(q' UNLESS r) (CONS st Pr) ==>
(p /\* p' LEADSTO q /\* q' \/* r) (CONS st Pr)
- LEADSTO_cor15
-
|- !p q r b p' q' r' b' st Pr.
(p LEADSTO q \/* r) (CONS st Pr) /\
(q UNLESS b) (CONS st Pr) /\
(p' LEADSTO q' \/* r') (CONS st Pr) /\
(q' UNLESS b') (CONS st Pr) ==>
(p /\* p' LEADSTO q /\* q' \/* (r \/* b) \/* r' \/* b') (CONS st Pr)
- LEADSTO_cor16
-
|- !P Q R B st Pr.
(!i. (P i LEADSTO Q i \/* R i) (CONS st Pr)) /\
(!i. (Q i UNLESS B i) (CONS st Pr)) ==>
(!i.
(/<=\* P i LEADSTO /<=\* Q i \/* \<=/* R i \/* \<=/* B i) (CONS st Pr))