set a = GBP ;
let s be State of SCMPDS ; :: thesis: for f, g being FinSequence of INT
for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),s,p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let f, g be FinSequence of INT ; :: thesis: for p0, n being Element of NAT st s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),s,p0 & len f = n & len g = n holds
( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let p0, n be Element of NAT ; :: thesis: ( s . GBP = 0 & s . (intpos 2) = n - 1 & s . (intpos 3) = p0 + 1 & s . (intpos 1) = 0 & p0 >= 6 & f is_FinSequence_on s,p0 & g is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),s,p0 & len f = n & len g = n implies ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 2) = n - 1 and
A3: s . (intpos 3) = p0 + 1 and
A4: s . (intpos 1) = 0 and
A5: p0 >= 6 and
A6: ( f is_FinSequence_on s,p0 & g is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),s,p0 ) and
A7: len f = n and
A8: len g = n ; :: thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
per cases ( n = 0 or n <> 0 ) ;
suppose A9: n = 0 ; :: thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
then g = {} by A8;
hence f,g are_fiberwise_equipotent by A7, A9; :: thesis: g is_non_decreasing_on 1,n
thus g is_non_decreasing_on 1,n by A9, Th1; :: thesis: verum
end;
suppose n <> 0 ; :: thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
then n >= 1 + 0 by INT_1:20;
then n - 1 >= 0 by XREAL_1:21;
then reconsider n1 = n - 1 as Element of NAT by INT_1:16;
defpred S1[ Nat] means for t being State of SCMPDS
for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = $1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n );
A10: ( (s . (intpos 2)) + (s . (intpos 1)) = (n - 1) + 0 & 1 = n - (s . (intpos 2)) ) by A2, A4;
A11: now
let k be Element of NAT ; :: thesis: ( S1[k] implies S1[k + 1] )
assume A12: S1[k] ; :: thesis: S1[k + 1]
now
let t be State of SCMPDS ; :: thesis: for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let f1, f2 be FinSequence of INT ; :: thesis: for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let m be Element of NAT ; :: thesis: ( t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = k + 1 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n ) )
assume that
A13: t . GBP = 0 and
A14: (t . (intpos 2)) + (t . (intpos 1)) = n - 1 and
A15: t . (intpos 2) = k + 1 and
A16: m = n - (t . (intpos 2)) and
A17: p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 and
A18: f1 is_FinSequence_on t,p0 and
A19: f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 and
A20: ( f1 is_non_decreasing_on 1,m & len f1 = n ) and
A21: len f2 = n ; :: thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
set t1 = IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t;
set Bt = IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t;
A22: (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 4) = (t . (intpos 3)) + 1 by A13, Lm11;
p0 + ((t . (intpos 1)) + 1) = t . (intpos 3) by A17;
then t . (intpos 3) >= 6 + ((t . (intpos 1)) + 1) by A5, XREAL_1:8;
then A23: t . (intpos 3) >= (6 + 1) + (t . (intpos 1)) ;
then A24: (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . GBP = 0 by A13, Lm12;
A25: (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 2) = (t . (intpos 2)) - 1 by A13, A23, Lm12;
then A26: n - ((IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 2)) = m + 1 by A16;
A27: (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 1) = (t . (intpos 1)) + 1 by A13, A23, Lm12;
then A28: ((IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 2)) + ((IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 1)) = n - 1 by A14, A25;
(IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 3) = (t . (intpos 3)) + 1 by A13, A23, Lm12;
then A29: (((IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 3)) - ((IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos 1))) - 1 = p0 by A17, A27;
A30: (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 6) = (t . (intpos 1)) + 1 by A13, Lm11;
then A31: p0 = (((IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 4)) - ((IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 6))) - 1 by A17, A22;
now
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),(IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t)) . (intpos (p0 + i)) )
assume ( 1 <= i & i <= len f2 ) ; :: thesis: f2 . i = (IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),(IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t)) . (intpos (p0 + i))
hence f2 . i = (IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t) . (intpos (p0 + i)) by A19, Def1
.= (IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),(IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t)) . (intpos (p0 + i)) by A13, A15, A23, Lm14 ;
:: thesis: verum
end;
then A32: f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),(IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t),p0 by Def1;
now
A33: p0 + 1 >= 6 + 1 by A5, XREAL_1:8;
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= len f1 implies f1 . i = (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos (p0 + i)) )
assume that
A34: 1 <= i and
A35: i <= len f1 ; :: thesis: f1 . i = (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos (p0 + i))
p0 + 1 <= p0 + i by A34, XREAL_1:8;
then A36: p0 + i >= 7 by A33, XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A18, A34, A35, Def1
.= (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos (p0 + i)) by A13, A36, Lm11 ; :: thesis: verum
end;
then A37: f1 is_FinSequence_on IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t,p0 by Def1;
(IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 4) = p0 + (((IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 6)) + 1) by A17, A22, A30;
then (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 4) >= 6 + (((IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 6)) + 1) by A5, XREAL_1:8;
then A38: (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 4) >= (6 + 1) + ((IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . (intpos 6)) ;
m + (k + 1) = n by A15, A16;
then A39: n > 0 + m by XREAL_1:8;
consider h being FinSequence of INT such that
A40: len h = n and
A41: for i being Element of NAT st 1 <= i & i <= len h holds
h . i = (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t)) . (intpos (p0 + i)) by Th2;
A42: h is_FinSequence_on IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t),p0 by A41, Def1;
now
A43: p0 + 1 >= 6 + 1 by A5, XREAL_1:8;
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= len h implies h . i = (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos (p0 + i)) )
assume that
A44: 1 <= i and
A45: i <= len h ; :: thesis: h . i = (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos (p0 + i))
p0 + 1 <= p0 + i by A44, XREAL_1:8;
then p0 + i >= 7 by A43, XXREAL_0:2;
then A46: p0 + i > 2 by XXREAL_0:2;
thus h . i = (IExec (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))),(IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t)) . (intpos (p0 + i)) by A41, A44, A45
.= (IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t) . (intpos (p0 + i)) by A13, A23, A46, Lm12 ; :: thesis: verum
end;
then A47: h is_FinSequence_on IExec ((((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 )))))) ';' (AddTo GBP ,2,(- 1))),t,p0 by Def1;
A48: (IExec ((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)),t) . GBP = 0 by A13, Lm11;
then A49: f1,h are_fiberwise_equipotent by A14, A16, A17, A20, A22, A40, A31, A38, A37, A42, A39, Th17;
A50: h is_non_decreasing_on 1,m + 1 by A14, A16, A17, A20, A48, A22, A40, A31, A38, A37, A42, A39, Th17;
then h,f2 are_fiberwise_equipotent by A12, A15, A16, A21, A40, A24, A28, A26, A29, A47, A32;
hence f1,f2 are_fiberwise_equipotent by A49, CLASSES1:84; :: thesis: f2 is_non_decreasing_on 1,n
thus f2 is_non_decreasing_on 1,n by A12, A15, A16, A21, A40, A50, A24, A28, A26, A29, A47, A32; :: thesis: verum
end;
hence S1[k + 1] ; :: thesis: verum
end;
A51: S1[ 0 ]
proof
let t be State of SCMPDS ; :: thesis: for f1, f2 being FinSequence of INT
for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let f1, f2 be FinSequence of INT ; :: thesis: for m being Element of NAT st t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n holds
( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
let m be Element of NAT ; :: thesis: ( t . GBP = 0 & (t . (intpos 2)) + (t . (intpos 1)) = n - 1 & t . (intpos 2) = 0 & m = n - (t . (intpos 2)) & p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 & f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 & f1 is_non_decreasing_on 1,m & len f1 = n & len f2 = n implies ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n ) )
assume that
A52: t . GBP = 0 and
(t . (intpos 2)) + (t . (intpos 1)) = n - 1 and
A53: t . (intpos 2) = 0 and
A54: m = n - (t . (intpos 2)) and
p0 = ((t . (intpos 3)) - (t . (intpos 1))) - 1 and
A55: f1 is_FinSequence_on t,p0 and
A56: f2 is_FinSequence_on IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t,p0 and
A57: f1 is_non_decreasing_on 1,m and
A58: ( len f1 = n & len f2 = n ) ; :: thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n )
A59: t . (DataLoc (t . GBP ),2) = 0 by A52, A53, SCMP_GCD:5;
A60: now
let i be Nat; :: thesis: ( 1 <= i & i <= len f2 implies f2 . i = f1 . i )
assume A61: ( 1 <= i & i <= len f2 ) ; :: thesis: f2 . i = f1 . i
A62: i in NAT by ORDINAL1:def 13;
hence f2 . i = (IExec (for-down GBP ,2,1,(((((AddTo GBP ,3,1) ';' (GBP ,4 := GBP ,3)) ';' (AddTo GBP ,1,1)) ';' (GBP ,6 := GBP ,1)) ';' (while>0 GBP ,6,(((GBP ,5 := (intpos 4),(- 1)) ';' (SubFrom GBP ,5,(intpos 4),0 )) ';' (if>0 GBP ,5,(((((GBP ,5 := (intpos 4),(- 1)) ';' ((intpos 4),(- 1) := (intpos 4),0 )) ';' ((intpos 4),0 := GBP ,5)) ';' (AddTo GBP ,4,(- 1))) ';' (AddTo GBP ,6,(- 1))),(Load (GBP ,6 := 0 ))))))),t) . (intpos (p0 + i)) by A56, A61, Def1
.= t . (intpos (p0 + i)) by A59, SCMPDS_7:66
.= f1 . i by A55, A58, A62, A61, Def1 ;
:: thesis: verum
end;
hence f1,f2 are_fiberwise_equipotent by A58, FINSEQ_1:18; :: thesis: f2 is_non_decreasing_on 1,n
thus f2 is_non_decreasing_on 1,n by A53, A54, A57, A58, A60, FINSEQ_1:18; :: thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch 1(A51, A11);
then A63: S1[n1] ;
 ( p0 = ((s . (intpos 3)) - (s . (intpos 1))) - 1 & f is_non_decreasing_on 1,1 ) by A3, A4, Th1;
hence ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A1, A6, A7, A8, A10, A63; :: thesis: verum
end;
end;