let s be All-State of (ZeroTuring ';' SuccTuring); :: thesis: for t being Tape of ZeroTuring
for head, n being Element of NAT st s = [[0,0],head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,1*> )
let t be Tape of ZeroTuring; :: thesis: for head, n being Element of NAT st s = [[0,0],head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,1*> )
let h, n be Element of NAT ; :: thesis: ( s = [[0,0],h,t] & t storeData <*h,n*> implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> ) )
assume that
A1: s = [[0,0],h,t] and
A2: t storeData <*h,n*> ; :: thesis: ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> )
reconsider h1 = h as Element of INT by INT_1:def 2;
set s1 = [ the InitS of ZeroTuring,h1,t];
A3: 0 = the InitS of ZeroTuring by Def19;
then A4: ( [ the InitS of ZeroTuring,h1,t] is Accept-Halt & (Result [ the InitS of ZeroTuring,h1,t]) `2_3 = h ) by A2, Lm15;
the Symbols of ZeroTuring = {0,1} by Def19
.= the Symbols of SuccTuring by Def17 ;
then reconsider t2 = (Result [ the InitS of ZeroTuring,h1,t]) `3_3 as Tape of SuccTuring ;
set s2 = [ the InitS of SuccTuring,h1,t2];
A5: 0 = the InitS of SuccTuring by Def17;
then A6: s = [ the InitS of (ZeroTuring ';' SuccTuring),h,t] by A1, A3, Def31;
(Result [ the InitS of ZeroTuring,h1,t]) `3_3 storeData <*h,0*> by A2, A3, Lm15;
then A7: t2 storeData <*h,0*> by Th48;
then A8: (Result [ the InitS of SuccTuring,h1,t2]) `3_3 storeData <*h,(0 + 1)*> by A5, Th31;
A9: [ the InitS of SuccTuring,h1,t2] is Accept-Halt by A7, A5, Th31;
hence s is Accept-Halt by A4, A6, Th45; :: thesis: ( (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> )
(Result [ the InitS of SuccTuring,h1,t2]) `2_3 = h by A7, A5, Th31;
hence (Result s) `2_3 = h by A4, A9, A6, Th45; :: thesis: (Result s) `3_3 storeData <*h,1*>
(Result s) `3_3 = (Result [ the InitS of SuccTuring,h1,t2]) `3_3 by A4, A9, A6, Th45;
hence (Result s) `3_3 storeData <*h,1*> by A8, Th48; :: thesis: verum