Theory: leadsto

Parents


Type constants


Term constants


Axioms


Definitions

LUB
|- !P. LUB P = (\s. ?p. P p /\ p s)
In
|- !p P. p In P = P p
LeadstoRel
|- !R Pr.
     LeadstoRel R Pr =
     (!p q.
       ((p ENSURES q) Pr ==> R p q Pr) /\
       (!r. R p r Pr /\ R r q Pr ==> R p q Pr) /\
       (!P. (p = LUB P) /\ (!p. p In P ==> R p q Pr) ==> R p q Pr))
LEADSTO
|- !p q Pr. (p LEADSTO q) Pr = (!R. LeadstoRel R Pr ==> R p q Pr)
LEADSTO2Fn
|- !R.
     LEADSTO2Fn R =
     (\p q Pr.
       (p ENSURES q) Pr \/
       (?r. (p ENSURES r) Pr /\ R r q Pr) \/
       (?P. (p = LUB P) /\ (!p. p In P ==> R p q Pr)))
LEADSTO2
|- !p q Pr.
     LEADSTO2 p q Pr =
     (!R. (!p q. LEADSTO2Fn R p q Pr ==> R p q Pr) ==> R p q Pr)
LEADSTO2Fam
|- !R Pr.
     LEADSTO2Fam R Pr =
     (!p q.
       ((p ENSURES q) Pr ==> R p q Pr) /\
       (!r. (p ENSURES r) Pr /\ R r q Pr ==> R p q Pr) /\
       (!P. (!p. p In P ==> R p q Pr) ==> R (LUB P) q Pr))
EQmetric
|- !M m. M EQmetric m = (\s. M s = m)
LESSmetric
|- !M m. M LESSmetric m = (\s. M s < m)

Theorems

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. (\

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))