:: Formalization of the Advanced Encryption Standard -- Part {I}
::  by Kenichi Arai and Hiroyuki Okazaki

environ

 vocabularies AESCIP_1, DESCIP_1, TARSKI, XBOOLE_0, FINSEQ_1, RELAT_1, FUNCT_1,
      ARYTM_1, FUNCT_2, FINSEQ_2, NAT_1, MARGREL1, ZFMISC_1, SUBSET_1, NUMBERS,
      INT_1, CARD_1, JORDAN3, XXREAL_0, ARYTM_3, ORDINAL4, FINSEQ_5, XCMPLX_0;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1,
      RELSET_1, PARTFUN1, FUNCT_2, BINOP_1, CARD_1, NUMBERS, XCMPLX_0,
      XXREAL_0, NAT_1, INT_1, NAT_D, FINSEQ_1, FINSEQ_2, MARGREL1, FINSEQ_4,
      FINSEQ_6, DESCIP_1;
 constructors RELSET_1, FINSEQ_4, NAT_D, FINSEQ_6, DESCIP_1, BINOP_1;
 registrations FINSEQ_1, RELSET_1, FINSEQ_2, FUNCT_2, ORDINAL1, MARGREL1,
      INT_1, NAT_1, XXREAL_0, XBOOLE_0, FUNCT_1, XREAL_0, FINSEQ_4;
 requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
 equalities FINSEQ_1, FINSEQ_2;
 theorems TARSKI, FUNCT_1, FUNCT_2, XXREAL_0, XREAL_1, FINSEQ_6, XREAL_0,
      NAT_1, INT_1, FINSEQ_1, CARD_1, BINOP_1, NAT_D, FINSEQ_2, FINSEQ_4,
      RFINSEQ, FINSEQ_5, XBOOLE_0, ORDINAL1, DESCIP_1, MARGREL1, NAT_2,
      WSIERP_1, XCMPLX_1, FINSEQ_3;
 schemes FUNCT_2, FINSEQ_1, NAT_1, BINOP_1, RECDEF_1;

begin :: Preliminaries

theorem XLMOD02:
  for k,m be Nat st m <> 0 & (k+1) mod m <> 0
  holds (k+1) mod m = (k mod m)+1
proof
  let k,m be Nat;
  assume
C1: m <> 0 & (k+1) mod m <> 0;
  (k mod m)+1 <= m by NAT_D:1,C1,NAT_1:13;
  then
P1: (k mod m)+1-1 <= m-1 by XREAL_1:9;
P2: (k+1) mod m = ((k mod m)+1) mod m by NAT_D:22;
  k mod m < m-1
  proof
    assume not k mod m < m-1;
    then (k+1) mod m = (m-1+1) mod m by XXREAL_0:1,P1,P2
      .= 0 by INT_1:50;
    hence contradiction by C1;
  end;
  then (k mod m)+1 < m-1+1 by XREAL_1:8;
  hence (k+1) mod m = (k mod m)+1 by NAT_D:24,P2;
end;

theorem XLMOD01:
  for k,m be Nat st m <> 0 & (k+1) mod m <> 0 holds (k+1) div m = k div m
proof
  let k,m be Nat;
  assume
C1: m <> 0 & (k+1) mod m <> 0;
  k+1 = ((k+1) div m )*m+((k+1) mod m) by INT_1:59,C1
    .= ((k+1) div m )*m+((k mod m)+1) by XLMOD02,C1;
  then
P1: ((k+1) div m)*m+(k mod m)-(k mod m) = (k div m )*m+(k mod m)-(k mod m)
  by INT_1:59,C1;
  thus ((k+1) div m) = (k div m)*m / m by XCMPLX_1:89,C1,P1
    .= (k div m) by XCMPLX_1:89,C1;
end;

theorem XLMOD02X:
  for k,m be Nat st m <> 0 & (k+1) mod m = 0 holds m-1 = (k mod m)
proof
  let k,m be Nat;
  assume
C1: m <> 0 & (k+1) mod m = 0;
  then (k mod m)+1 <= m by NAT_D:1,NAT_1:13;
  then
P1: (k mod m)+1-1 <= m-1 by XREAL_1:9;
P2: (k+1) mod m = ((k mod m)+1) mod m by NAT_D:22;
  assume not k mod m = m-1;
  then k mod m < m-1 by XXREAL_0:1,P1;
  then k mod m+1 < m-1+1 by XREAL_1:8;
  hence contradiction by P2,NAT_D:24,C1;
end;

theorem XLMOD01X:
  for k,m be Nat st m <> 0 & (k+1) mod m = 0
  holds (k+1) div m = (k div m)+1
proof
  let k,m be Nat;
  assume
C1: m <> 0 & (k+1) mod m = 0;
  then
P3: k mod m = m-1 by XLMOD02X;
P4: k+1 = ((k+1) div m)*m+((k+1) mod m) by INT_1:59,C1
    .= ((k+1) div m)*m by C1;
P5: k = (k div m )*m+(k mod m) by INT_1:59,C1
    .= (k div m)*m+m-1 by P3;
  thus ((k+1) div m) = ((k div m)+1)*m / m by XCMPLX_1:89,C1,P4,P5
    .= ((k div m)+1) by XCMPLX_1:89,C1;
end;

theorem XLMOD03:
  for k,m be Nat holds (k-m) mod m = k mod m
proof
  let k,m be Nat;
  thus (k-m) mod m = (k+m*(-1)) mod m
    .= k mod m by NAT_D:61;
end;

theorem XLMOD04:
  for k,m be Nat st m <> 0 holds (k-m) div m = (k div m)-1
proof
  let k,m be Nat;
  assume
AS: m <> 0;
  thus (k-m) div m = (k+m*(-1)) div m
    .= (k div m)+-1 by AS,NAT_D:61
    .= (k div m)-1;
end;

definition
  let m,n be Nat, X,D be non empty set;
  let F be Function of X, m-tuples_on(n-tuples_on D);
  let x be Element of X;
  redefine func F.x -> Element of m-tuples_on(n-tuples_on D);
  coherence
  proof
    F.x in m-tuples_on(n-tuples_on D);
    hence thesis;
  end;
end;

definition
  let m be Nat, X,Y,D be non empty set;
  let F be Function of [:X,Y:], m-tuples_on D;
  let x be Element of X,y be Element of Y;
  redefine func F.(x,y) -> Element of m-tuples_on D;
  coherence
  proof
    F.(x,y) in m-tuples_on D;
    hence thesis;
  end;
end;

theorem LM01:
  for m,n be Nat, D be non empty set,
  F1,F2 be Element of m-tuples_on (n-tuples_on D) st for i,j be Nat
  st i in Seg m & j in Seg n
  holds (F1.i).j = (F2.i).j holds F1 = F2
  proof
    let m,n be Nat, D be non empty set,
    F1,F2 be Element of m-tuples_on (n-tuples_on D);
    assume
AS: for i,j be Nat st i in Seg m & j in Seg n
    holds (F1.i).j = (F2.i).j;
    F1 in m-tuples_on (n-tuples_on D);
    then
P1: ex s be Element of (n-tuples_on D)* st F1 = s & len s = m;
    F2 in m-tuples_on (n-tuples_on D);
    then
P2: ex s be Element of (n-tuples_on D)* st F2 = s & len s = m;
    now let i be Nat;
      assume 1 <= i & i <= len F1;
      then
P4:   i in Seg m by P1;
      then i in dom F1 by FINSEQ_1:def 3,P1;
      then F1.i in rng F1 by FUNCT_1:3;
      then F1.i in n-tuples_on D;
      then
P6:   ex s be Element of D* st F1.i = s & len s = n;
      then reconsider F1i = F1.i as Element of D*;
      i in dom F2 by FINSEQ_1:def 3,P2,P4;
      then F2.i in rng F2 by FUNCT_1:3;
      then F2.i in n-tuples_on D;
      then
R6:   ex s be Element of D* st F2.i = s & len s = n;
      then reconsider F2i = F2.i as Element of D*;
      now let j be Nat;
      assume 1 <= j & j <= len F1i;
      then j in Seg n by P6;
      hence F1i.j = F2i.j by AS,P4;
    end;
    hence
    F1.i = F2.i by P6,R6,FINSEQ_1:14;
  end;
  hence F1 = F2 by P1,P2,FINSEQ_1:14;
end;

theorem LMGSEQ4:
  for D be non empty set, x1,x2,x3,x4 be Element of D
  holds <* x1,x2,x3,x4 *> is Element of (4-tuples_on D)
proof
  let D be non empty set, x1,x2,x3,x4 be Element of D;
  reconsider x1234 = <* x1,x2,x3,x4 *> as FinSequence of D;
P1: len x1234 = 4 by FINSEQ_4:76;
  x1234 in D* by FINSEQ_1:def 11;
  then x1234 in 4-tuples_on D by P1;
  hence thesis;
end;

theorem LMGSEQ5:
  for D be non empty set,x1,x2,x3,x4,x5 be Element of D
  holds <* x1,x2,x3,x4,x5 *> is Element of (5-tuples_on D)
proof
  let D be non empty set, x1,x2,x3,x4,x5 be Element of D;
  reconsider x12345 = <* x1,x2,x3,x4,x5 *> as FinSequence of D;
P1: len x12345 = 5 by FINSEQ_4:78;
  x12345 in D* by FINSEQ_1:def 11;
  then x12345 in 5-tuples_on D by P1;
  hence thesis;
end;

theorem
  for D be non empty set, x1,x2,x3,x4,x5,x6,x7,x8 be Element of D
  holds <* x1,x2,x3,x4 *>^<* x5,x6,x7,x8 *> is Element of (8-tuples_on D)
proof
  let D be non empty set,x1,x2,x3,x4,x5,x6,x7,x8 be Element of D;
  reconsider x1234 = <* x1,x2,x3,x4 *> as Element of 4-tuples_on D
  by LMGSEQ4;
  reconsider x5678 = <* x5,x6,x7,x8 *> as Element of 4-tuples_on D
  by LMGSEQ4;
  D c= D;
  hence thesis by FINSEQ_2:109;
end;

theorem LMGSEQ10:
  for D be non empty set, x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 be Element of D
  holds <* x1,x2,x3,x4,x5 *>^<* x6,x7,x8,x9,x10 *> is
  Element of (10-tuples_on D)
proof
  let D be non empty set, x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 be Element of D;
  reconsider x12345 = <* x1,x2,x3,x4,x5 *> as Element of 5-tuples_on D
  by LMGSEQ5;
  reconsider x67890 = <* x6,x7,x8,x9,x10 *> as Element of 5-tuples_on D
  by LMGSEQ5;
  D c= D;
  hence thesis by FINSEQ_2:109;
end;

theorem LMGSEQ16:
  for D be non empty set, x1,x2,x3,x4,x5,x6,x7,x8 be Element of (4-tuples_on D)
  holds <* x1^x5,x2^x6,x3^x7,x4^x8 *> is
  Element of 4-tuples_on (8-tuples_on D)
proof
  let D be non empty set,
  x1,x2,x3,x4,x5,x6,x7,x8 be Element of (4-tuples_on D);
X1: D c= D;
  then
P1: x1^x5 is Element of 8-tuples_on D by FINSEQ_2:109;
P2: x2^x6 is Element of 8-tuples_on D by X1,FINSEQ_2:109;
P3: x3^x7 is Element of 8-tuples_on D by X1,FINSEQ_2:109;
  x4^x8 is Element of 8-tuples_on D by X1,FINSEQ_2:109;
  hence thesis by P1,P2,P3,LMGSEQ4;
end;

theorem
  for D be non empty set, x be Element of 4-tuples_on(4-tuples_on D),
  k be Element of NAT st k in Seg 4 holds
  ex x1,x2,x3,x4 be Element of D st x1 = (x.k).1 & x2 = (x.k).2 &
  x3 = (x.k).3 & x4 = (x.k).4
proof
  let D be non empty set, x be Element of 4-tuples_on(4-tuples_on D),
  k be Element of NAT;
  assume
AS: k in Seg 4;
  x in 4-tuples_on(4-tuples_on D);
  then ex s be Element of (4-tuples_on D)* st x = s & len s = 4;
  then k in dom x by AS,FINSEQ_1:def 3;
  then x.k in rng x by FUNCT_1:3;
  then x.k in 4-tuples_on (D);
  then
Q13: ex s be Element of D* st x.k = s & len s = 4;
  then reconsider xk = x.k as Element of D*;
  1 in Seg 4;
  then 1 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.1 in rng xk by FUNCT_1:3;
  then reconsider x1 = xk.1 as Element of D;
  2 in Seg 4;
  then 2 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.2 in rng xk by FUNCT_1:3;
  then reconsider x2 = xk.2 as Element of D;
  3 in Seg 4;
  then 3 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.3 in rng xk by FUNCT_1:3;
  then reconsider x3 = xk.3 as Element of D;
  4 in Seg 4;
  then 4 in dom xk by Q13,FINSEQ_1:def 3;
  then xk.4 in rng xk by FUNCT_1:3;
  then reconsider x4 = xk.4 as Element of D;
  take x1,x2,x3,x4;
  thus thesis;
end;

theorem INV00:
  for X,Y be non empty set, f be Function of X,Y, g be Function of Y,X
  st (for x be Element of X holds g.(f.x) = x) &
  (for y be Element of Y holds f.(g.y) = y)
  holds f is one-to-one & f is onto & g is one-to-one & g is onto &
  g = f" & f = g"
proof
  let X,Y be non empty set, f be Function of X,Y, g be Function of Y,X;
  assume
A1: for x be Element of X holds g.(f.x) = x;
  assume
A2: for y be Element of Y holds f.(g.y) = y;
  now let x be Element of X;
    thus (g*f).x = g.(f.x) by FUNCT_2:15
      .= x by A1;
  end;
  then
P2: g*f = id X by FUNCT_2:124;
  now let y be Element of Y;
    thus (f*g).y = f.(g.y) by FUNCT_2:15
      .= y by A2;
  end;
  then
P4: f*g = id Y by FUNCT_2:124;
  thus
P5: f is one-to-one & f is onto by P2,P4,FUNCT_2:23;
  thus
P6: g is one-to-one & g is onto by P2,P4,FUNCT_2:23;
  rng f = Y by P5,FUNCT_2:def 3;
  hence g = f" by FUNCT_2:30,P2,FUNCT_2:23;
  rng g = X by P6,FUNCT_2:def 3;
  hence f = g" by FUNCT_2:30,P4,FUNCT_2:23;
end;

begin :: State array

definition
  func AES-Statearray -> Function of 128-tuples_on BOOLEAN,
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefStatearray:
  for input be Element of 128-tuples_on BOOLEAN
  for i,j be Nat st i in Seg 4 & j in Seg 4
  holds ((it.input).i).j = mid (input,1+(i-'1)*8+(j-'1)*32,
  1+(i-'1)*8+(j-'1)*32+7);
  existence
  proof
    defpred P0[Element of 128-tuples_on BOOLEAN,set] means
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st $2 = z & for i,j be Nat st i in Seg 4 & j in Seg 4
    holds (z.i).j = mid ($1,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7);
A1: for x being Element of 128-tuples_on BOOLEAN
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[x,z]
    proof
      let x be Element of 128-tuples_on BOOLEAN;
      x in 128-tuples_on BOOLEAN;
      then
A01:  ex s be Element of (BOOLEAN)* st x = s & len s = 128;
      defpred P[Nat,set] means
      ex zi be Element of (4-tuples_on (8-tuples_on BOOLEAN))
      st $2 = zi & for j be Nat st j in Seg 4
      holds zi.j = mid (x,1+($1-'1)*8+(j-'1)*32,1+($1-'1)*8+(j-'1)*32+7);
Q1:   for k be Nat st k in Seg 4
      ex x being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,x]
      proof
        let k be Nat;
        assume k in Seg 4;
        then
Q110:   1 <= k & k <= 4 by FINSEQ_1:1;
        then 1-1 <= k-1 by XREAL_1:9;
        then k-'1 = k-1 by XREAL_0:def 2;
        then k-'1 <= 4-1 by Q110,XREAL_1:9;
        then
Q112:   (k-'1)*8 <= 3*8 by XREAL_1:64;
        defpred Pi[Nat,set] means
        $2 = mid (x,1+(k-'1)*8+($1-'1)*32,1+(k-'1)*8+($1-'1)*32+7);
Q12:    for j be Nat st j in Seg 4
        ex xi being Element of (8-tuples_on BOOLEAN) st Pi[j,xi]
        proof
          let j be Nat;
          assume j in Seg 4;
          then
Q130:     1 <= j & j <= 4 by FINSEQ_1:1;
          then 1-1 <= j-1 by XREAL_1:9;
          then j-'1 = j-1 by XREAL_0:def 2;
          then j-'1 <= 4-1 by Q130,XREAL_1:9;
          then
Q133:     (j-'1)*32 <= 3*32 by XREAL_1:64;
          (k-'1)*8+(j-'1)*32 <= 24+96 by Q133,Q112,XREAL_1:7;
          then
Q134:     1+((k-'1)*8+(j-'1)*32) <= 1+120 by XREAL_1:7;
Q136:     (1+(k-'1)*8+(j-'1)*32)+7 <= 121+7 by Q134,XREAL_1:7;
          1+0 <= 1+((k-'1)*8+(j-'1)*32) by XREAL_1:7;
          then
Q14:      1 <= 1+(k-'1)*8+(j-'1)*32 & 1+(k-'1)*8+(j-'1)*32 <= len x
          by Q134,XXREAL_0:2,A01;
Q150:     1+0 <= 1+(k-'1)*8+(j-'1)*32+7 by XREAL_1:7;
          reconsider mmd = mid (x,1+(k-'1)*8+(j-'1)*32,
          1+(k-'1)*8+(j-'1)*32+7) as Element of (BOOLEAN)* by FINSEQ_1:def 11;
          1+(k-'1)*8+(j-'1)*32+0 <= 1+(k-'1)*8+(j-'1)*32+7 by XREAL_1:6;
          then len mid (x,1+(k-'1)*8+(j-'1)*32,1+(k-'1)*8+(j-'1)*32+7) =
          (1+(k-'1)*8+(j-'1)*32+7)-'(1+(k-'1)*8+(j-'1)*32)+1
          by FINSEQ_6:118,Q14,Q136,A01,Q150
            .= 7+1 by NAT_D:34
            .= 8;
          then mmd in (8-tuples_on BOOLEAN);
          then reconsider xi = mid (x,1+(k-'1)*8+(j-'1)*32,
          1+(k-'1)*8+(j-'1)*32+7) as Element of (8-tuples_on BOOLEAN);
          xi = mid (x,1+(k-'1)*8+(j-'1)*32,1+(k-'1)*8+(j-'1)*32+7);
          hence thesis;
        end;
        consider zi be FinSequence of (8-tuples_on BOOLEAN) such that
Q13:    dom zi = Seg 4 & for i be Nat st i in Seg 4
        holds Pi[i,zi.i] from FINSEQ_1:sch 5(Q12);
Q14:    len zi = 4 by Q13,FINSEQ_1:def 3;
        reconsider zi as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zi in 4-tuples_on (8-tuples_on BOOLEAN) by Q14;
        then reconsider zi as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        for j be Nat st j in Seg 4
        holds zi.j = mid (x,1+(k-'1)*8+(j-'1)*32,1+(k-'1)*8+(j-'1)*32+7)
        by Q13;
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 &
      for i be Nat st i in Seg 4 holds P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))by Q3;
      then reconsider z as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      for i,j be Nat st i in Seg 4 & j in Seg 4
      holds (z.i).j = mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)
      proof
        let i,j be Nat;
        assume
P11:    i in Seg 4 & j in Seg 4;
        then consider zi be Element of (4-tuples_on (8-tuples_on BOOLEAN))
        such that
P12:    z.i = zi & for j be Nat st j in Seg 4
        holds zi.j = mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7) by Q2;
        thus (z.i).j = mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)
        by P11,P12;
      end;
      hence ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      st P0[x,z];
    end;
    consider I be Function of 128-tuples_on BOOLEAN,
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x being Element of 128-tuples_on BOOLEAN
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    now let input be Element of 128-tuples_on BOOLEAN;
      ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      st I.input = z & for i,j be Nat st i in Seg 4 & j in Seg 4
      holds (z.i).j = mid (input,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)
      by A2;
      hence for i,j be Nat st i in Seg 4 & j in Seg 4
      holds ((I.input).i).j = mid (input,1+(i-'1)*8+(j-'1)*32,
      1+(i-'1)*8+(j-'1)*32+7);
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let H1,H2 be Function of 128-tuples_on BOOLEAN,
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for input be Element of 128-tuples_on BOOLEAN
    for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ((H1.input).i).j = mid (input,1+(i-'1)*8+(j-'1)*32,
    1+(i-'1)*8+(j-'1)*32+7);
    assume
A2: for input be Element of 128-tuples_on BOOLEAN
    for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ((H2.input).i).j = mid (input,1+(i-'1)*8+(j-'1)*32,
    1+(i-'1)*8+(j-'1)*32+7);
    now let input be Element of 128-tuples_on BOOLEAN;
      (H1.input) in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P3:   ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st (H1.input) = s & len s = 4;
      (H2.input) in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P4:   ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st (H2.input) = s & len s = 4;
      now let i be Nat;
        assume 1 <=i & i <= len (H1.input);
        then
P6:     i in Seg 4 by P3;
        then i in dom(H1.input) by FINSEQ_1:def 3,P3;
        then (H1.input).i in rng (H1.input) by FUNCT_1:3;
        then (H1.input).i in 4-tuples_on (8-tuples_on BOOLEAN);
        then
P8:     ex s be Element of (8-tuples_on BOOLEAN)*
        st (H1.input).i = s & len s = 4;
        reconsider H1i = (H1.input).i as
        Element of (8-tuples_on BOOLEAN)* by P8;
        i in dom(H2.input) by FINSEQ_1:def 3,P4,P6;
        then (H2.input).i in rng (H2.input) by FUNCT_1:3;
        then (H2.input).i in 4-tuples_on (8-tuples_on BOOLEAN);
        then
P11:    ex s be Element of (8-tuples_on BOOLEAN)*
        st (H2.input).i = s & len s = 4;
        reconsider H2i = (H2.input).i as
        Element of (8-tuples_on BOOLEAN)* by P11;
        now let j be Nat;
          assume 1 <=j & j <= len H1i;
          then
P14:      j in Seg 4 by P8;
          then ((H1.input).i).j = mid (input,1+(i-'1)*8+(j-'1)*32,
          1+(i-'1)*8+(j-'1)*32+7) by A1,P6;
          hence H1i.j = H2i.j by A2,P6,P14;
        end;
        hence (H1.input).i = (H2.input).i by P8,P11,FINSEQ_1:def 17;
      end;
      hence H1.input = H2.input by P3,P4,FINSEQ_1:def 17;
    end;
    hence H1 = H2 by FUNCT_2:63;
  end;
end;

theorem LMStat0:
  for k be Nat st 1 <= k & k <= 128 ex i,j be Nat st i in Seg 4 & j in Seg 4 &
  1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7
proof
  let k be Nat;
  assume
A1: 1 <= k & k <= 128;
A3: k = 32*(k div 32)+(k mod 32) by NAT_D:2;
  reconsider m = k div 32 as Nat;
  reconsider n = k mod 32 as Nat;
  k div 32 <= (32*4) div 32 by A1,NAT_2:24;
  then
M1: m <= 4 by NAT_D:18;
  per cases;
  suppose
A4: n = 0;
A5: 1 <= m
    proof
      assume not 1 <= m;
      then m = 0 by NAT_1:14;
      hence contradiction by A1,A3,A4;
    end;
    set j = m;
A8: j in Seg 4 by M1,A5;
    set i = 4;
A10: i in Seg 4;
A11: j-'1 = j-1 by XREAL_1:233,A5;
A13: k = 32*(k div 32)+(k mod 32) by NAT_D:2
      .= 32*(j-'1)+8*((i-1)+1) by A4,A11
      .= 32*(j-'1)+8*((i-'1)+1) by XREAL_1:233
      .= 1+(i-'1)*8+(j-'1)*32+7;
    1+(i-'1)*8+(j-'1)*32+0 <= 1+(i-'1)*8+(j-'1)*32+7 by XREAL_1:7;
    hence ex i,j be Nat st i in Seg 4 & j in Seg 4 &
    1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7 by A8,A10,A13;
  end;
  suppose
A14: n <> 0;
    then
XX0: 1 <= n by NAT_1:14;
XX1: n <= 32 by NAT_D:1;
    m <> 4
    proof
      assume
U1:   m = 4;
U2:   k = 32*4+n by NAT_D:2,U1
        .= 128+n;
      128+1 <= 128+n by XX0,XREAL_1:7;
      hence contradiction by U2,XXREAL_0:2,A1;
    end;
    then m < 4 by XXREAL_0:1,M1;
    then
A15: m+1 <= 4 by NAT_1:13;
A16: 1 <= m+1 by NAT_1:11;
    set j = m+1;
A18: j in Seg 4 by A15,A16;
A19: j-'1 = j-1 by XREAL_1:233,NAT_1:11
      .=m;
A20: k = 32*(j-'1)+n by NAT_D:2,A19;
A22: n = 8*(n div 8)+(n mod 8) by NAT_D:2;
    reconsider s = n div 8 as Nat;
    reconsider t = n mod 8 as Nat;
    n div 8 <= (8*4) div 8 by XX1,NAT_2:24;
    then
M2: n div 8 <= 4 by NAT_D:18;
    now per cases;
    suppose
A23:  t = 0;
A24:  1 <= s
      proof
        assume not 1 <= s;
        then n = 8*0+0 by NAT_1:14,A22,A23;
        hence contradiction by A14;
      end;
      set i = s;
A28:  i in Seg 4 by M2,A24;
A29:  i-'1 = i-1 by XREAL_1:233,A24;
A30:  n = 8*s+0 by NAT_D:2,A23
        .= 8*(i-'1)+8*1 by A29;
      1+(i-'1)*8+(j-'1)*32+0 <= 1+(i-'1)*8+(j-'1)*32+7 by XREAL_1:7;
      hence ex i,j be Nat st i in Seg 4 & j in Seg 4 &
      1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7
      by A28,A18,A20,A30;
    end;
    suppose
      t <> 0;
      then
XX0:  1 <= t by NAT_1:14;
XXX1: t <= 8 by NAT_D:1;
      s <> 4
      proof
        assume
U1:     s = 4;
U2:     n = 8*4+t by NAT_D:2,U1
          .= 32+t;
        32+1 <= 32+t by XX0,XREAL_1:7;
        hence contradiction by U2,XXREAL_0:2,XX1;
      end;
      then s < 4 by XXREAL_0:1,M2;
      then
B15:  s+1 <= 4 by NAT_1:13;
B16:  1 <= s+1 by NAT_1:11;
      set i = s+1;
B18:  i in Seg 4 by B15,B16;
B19:  i-'1 = i-1 by XREAL_1:233,NAT_1:11
        .=s;
B20:  n = 8*(i-'1)+t by NAT_D:2,B19;
B220: 32*(j-'1)+8*(i-'1)+1 <= 32*(j-'1)+8*(i-'1)+t by XX0,XREAL_1:7;
      32*(j-'1)+8*(i-'1)+t <= 32*(j-'1)+8*(i-'1)+8 by XXX1,XREAL_1:7;
      then k <= 1+8*(i-'1)+32*(j-'1)+7 by A20,B20;
      hence ex i,j be Nat st i in Seg 4 & j in Seg 4 &
      1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7
      by B220,A20,B20,B18,A18;
    end;
  end;
  hence ex i,j be Nat st i in Seg 4 & j in Seg 4 & 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 1+(i-'1)*8+(j-'1)*32+7;
end;
end;

theorem LMStat2A:
  for i,j,i0,j0 be Nat st i in Seg 4 & j in Seg 4 & i0 in Seg 4 &
  j0 in Seg 4 & not (i = i0 & j = j0)
  holds {k where k is Nat : 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 8+(i-'1)*8+(j-'1)*32} /\ {k where k is Nat :
  1+(i0-'1)*8+(j0-'1)*32 <= k & k <= 8+(i0-'1)*8+(j0-'1)*32} = {}
proof
  let i,j,i0,j0 be Nat;
  assume
AS: i in Seg 4 & j in Seg 4 & i0 in Seg 4 &
  j0 in Seg 4 & not (i = i0 & j = j0);
  set A = {k where k is Nat : 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 8+(i-'1)*8+(j-'1)*32};
  set B = {k where k is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= k &
  k <= 8+(i0-'1)*8+(j0-'1)*32};
A1: 1 <= j & j <= 4 by AS,FINSEQ_1:1;
A2: 1 <= i & i <= 4 by AS,FINSEQ_1:1;
B1: 1 <= j0 & j0 <= 4 by AS,FINSEQ_1:1;
B2: 1 <= i0 & i0 <= 4 by AS,FINSEQ_1:1;
P1: (j-'1) = j-1 by XREAL_1:233,A1;
P2: (i-'1) = i-1 by XREAL_1:233,A2;
P3: (j0-'1) = j0-1 by XREAL_1:233,B1;
P4: (i0-'1) = i0-1 by XREAL_1:233,B2;
  i-1 <= 4-1 by A2,XREAL_1:9;
  then
R2: i-'1 <= 3 by XREAL_1:233,A2;
  i0-1 <= 4-1 by B2,XREAL_1:9;
  then
R4: i0-'1 <= 3 by XREAL_1:233,B2;
  per cases;
  suppose
A2: j <> j0;
    now per cases by A2,XXREAL_0:1;
    suppose
      j < j0;
      then (j-'1) < (j0-'1) by XREAL_1:14,P1,P3;
      then (j-'1)+1 <= (j0-'1) by NAT_1:13;
      then
A12:  ((j-'1)+1)*32 <= (j0-'1)*32 by XREAL_1:64;
      (i-'1)*8 <= 3*8 by R2,XREAL_1:64;
      then 8+(i-'1)*8 <= 8+24 by XREAL_1:6;
      then 8+(i-'1)*8+(j-'1)*32 <= 32+(j-'1)*32 by XREAL_1:6;
      then
A13:  8+(i-'1)*8+(j-'1)*32 <= (j0-'1)*32 by A12,XXREAL_0:2;
      0+(j0-'1)*32 <= (i0-'1)*8+(j0-'1)*32 by XREAL_1:6;
      then (j0-'1)*32+0 < (i0-'1)*8+(j0-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i-'1)*8+(j-'1)*32 < 1+(i0-'1)*8+(j0-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A17,A14,XXREAL_0:2,A16;
      end;
    end;
    suppose
      j0 < j;
      then (j0-'1) < (j-'1) by XREAL_1:14,P1,P3;
      then (j0-'1)+1 <= (j-'1) by NAT_1:13;
      then
A12:  ((j0-'1)+1)*32 <= (j-'1)*32 by XREAL_1:64;
      (i0-'1)*8 <= 3*8 by R4,XREAL_1:64;
      then 8+(i0-'1)*8 <= 8+24 by XREAL_1:6;
      then 8+(i0-'1)*8+(j0-'1)*32 <= 32+(j0-'1)*32 by XREAL_1:6;
      then
A13:  8+(i0-'1)*8+(j0-'1)*32 <= (j-'1)*32 by A12,XXREAL_0:2;
      0+(j-'1)*32 <= (i-'1)*8+(j-'1)*32 by XREAL_1:6;
      then (j-'1)*32+0 < (i-'1)*8+(j-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i0-'1)*8+(j0-'1)*32 < 1+(i-'1)*8+(j-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A14,XXREAL_0:2,A17;
      end;
    end;
  end;
  hence A /\ B = {};
  end;
  suppose
A2: j = j0;
    now per cases by A2,AS,XXREAL_0:1;
      suppose i < i0;
      then (i-'1) < (i0-'1) by XREAL_1:14,P2,P4;
      then (i-'1)+1 <= (i0-'1) by NAT_1:13;
      then ((i-'1)+1)*8 <= (i0-'1)*8 by XREAL_1:64;
      then
A13:  (i-'1)*8+8+(j-'1)*32 <= (i0-'1)*8+(j0-'1)*32 by A2,XREAL_1:6;
      (i0-'1)*8+(j0-'1)*32+0 < (i0-'1)*8+(j0-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i-'1)*8+(j-'1)*32 < 1+(i0-'1)*8+(j0-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A17,A14,XXREAL_0:2;
      end;
    end;
    suppose
      i0 < i;
      then (i0-'1) < (i-'1) by XREAL_1:14,P2,P4;
      then (i0-'1)+1 <= (i-'1) by NAT_1:13;
      then ((i0-'1)+1)*8 <= (i-'1)*8 by XREAL_1:64;
      then
A13:  (i0-'1)*8+8+(j0-'1)*32 <= (i-'1)*8+(j-'1)*32 by A2,XREAL_1:6;
      (i-'1)*8+(j-'1)*32+0 < (i-'1)*8+(j-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i0-'1)*8+(j0-'1)*32 < 1+(i-'1)*8+(j-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A14,XXREAL_0:2,A17;
      end;
    end;
  end;
  hence A /\ B = {};
end;
end;

theorem LMStat2:
  for k,i,j,i0,j0 be Nat st 1 <= k & k <= 128 & i in Seg 4 & j in Seg 4 &
  i0 in Seg 4 & j0 in Seg 4 & 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 1+(i-'1)*8+(j-'1)*32+7 & 1+(i0-'1)*8+(j0-'1)*32 <= k &
  k <= 1+(i0-'1)*8+(j0-'1)*32+7 holds i = i0 & j = j0
proof
  let k,i,j,i0,j0 be Nat;
  assume
AS: 1 <= k & k <= 128 & i in Seg 4 & j in Seg 4 & i0 in Seg 4 & j0 in Seg 4 &
  1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7 &
  1+(i0-'1)*8+(j0-'1)*32 <= k & k <= 1+(i0-'1)*8+(j0-'1)*32+7;
  assume not (i = i0 & j = j0);
  then
A2: {n where n is Nat : 1+(i-'1)*8+(j-'1)*32 <= n & n <= 8+(i-'1)*8+(j-'1)*32}
  /\ {n where n is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= n &
  n <= 8+(i0-'1)*8+(j0-'1)*32} = {} by LMStat2A,AS;
A3: k in {n where n is Nat : 1+(i-'1)*8+(j-'1)*32 <= n &
  n <= 8+(i-'1)*8+(j-'1)*32} by AS;
  k in {n where n is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= n &
  n <= 8+(i0-'1)*8+(j0-'1)*32} by AS;
  hence contradiction by A3,XBOOLE_0:def 4,A2;
end;

theorem LMStat1:
  AES-Statearray is one-to-one
proof
  for x1,x2 be object st x1 in 128-tuples_on BOOLEAN &
  x2 in 128-tuples_on BOOLEAN &
  (AES-Statearray).x1 = (AES-Statearray).x2 holds x1 = x2
  proof
    let x1,x2 be object;
    assume
A1: x1 in 128-tuples_on BOOLEAN & x2 in 128-tuples_on BOOLEAN &
    (AES-Statearray).x1 = (AES-Statearray).x2;
    then reconsider xx1 = x1,xx2 = x2 as Element of 128-tuples_on BOOLEAN;
P1: ex s be Element of (BOOLEAN)* st xx1 = s & len s = 128 by A1;
P2: ex s be Element of (BOOLEAN)* st xx2 = s & len s = 128 by A1;
    now let k be Nat;
      assume
P5:   1 <= k & k <= len xx1;
      consider i,j be Nat such that
A4:   i in Seg 4 & j in Seg 4 & 1+(i-'1)*8+(j-'1)*32 <= k &
      k <= 1+(i-'1)*8+(j-'1)*32+7 by LMStat0,P5,P1;
      mid (xx1,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7) is
      Element of (BOOLEAN)* by FINSEQ_1:def 11;
      then reconsider A1ij = (((AES-Statearray).xx1).i).j as
      FinSequence of (BOOLEAN) by DefStatearray,A4;
      mid (xx2,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7) is
      Element of (BOOLEAN)* by FINSEQ_1:def 11;
      then reconsider A2ij = (((AES-Statearray).xx2).i).j as
      FinSequence of (BOOLEAN) by DefStatearray,A4;
A50:  1+(i-'1)*8+(j-'1)*32-((i-'1)*8+(j-'1)*32) <= k-((i-'1)*8+(j-'1)*32)
      by A4,XREAL_1:9;
      then reconsider n = k-((i-'1)*8+(j-'1)*32) as Element of NAT by INT_1:3;
F41:  k-((i-'1)*8+(j-'1)*32) <= 1+(i-'1)*8+(j-'1)*32+7-((i-'1)*8+(j-'1)*32)
      by A4,XREAL_1:9;
F1:   1 <= 1+((i-'1)*8+(j-'1)*32) by NAT_1:11;
F2:   1+(i-'1)*8+(j-'1)*32 <= 1+(i-'1)*8+(j-'1)*32+7 by NAT_1:11;
Q110: 1 <= i & i <= 4 by A4,FINSEQ_1:1;
      then 1-1 <= i-1 by XREAL_1:9;
      then i-'1 = i-1 by XREAL_0:def 2;
      then i-'1 <= 4-1 by Q110,XREAL_1:9;
      then
Q112: (i-'1)*8 <= 3*8 by XREAL_1:64;
Q130: 1 <= j & j <= 4 by A4,FINSEQ_1:1;
      then 1-1 <= j-1 by XREAL_1:9;
      then j-'1 = j-1 by XREAL_0:def 2;
      then j-'1 <= 4-1 by Q130,XREAL_1:9;
      then
Q133: (j-'1)*32 <= 3*32 by XREAL_1:64;
      (i-'1)*8+(j-'1)*32 <= 24+96 by Q133,Q112,XREAL_1:7;
      then 1+((i-'1)*8+(j-'1)*32) <= 1+120 by XREAL_1:7;
      then
Q135: 1+(i-'1)*8+(j-'1)*32+7 <= 121+7 by XREAL_1:6;
F5:   n <= (1+(i-'1)*8+(j-'1)*32+7)-(1+(i-'1)*8+(j-'1)*32)+1 by F41;
A6:   k = n-1+(1+(i-'1)*8+(j-'1)*32);
      thus xx1.k = (mid (xx1,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)).n
      by F1,F2,Q135,P1,A50,F5,A6,FINSEQ_6:122
        .= A2ij.n by DefStatearray,A4,A1
        .= (mid (xx2,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)).n
        by DefStatearray,A4
        .= xx2.k by F1,F2,P2,Q135,A50,F5,A6,FINSEQ_6:122;
    end;
  hence thesis by P1,P2,FINSEQ_1:def 17;
end;
hence thesis by FUNCT_2:19;
end;

theorem LMStat3:
  AES-Statearray is onto
proof
  for y be object st y in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  ex x be object st x in 128-tuples_on BOOLEAN & y = AES-Statearray.x
  proof
    let y be object;
    assume y in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
B10:ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
    st y = s & len s = 4;
    then reconsider z = y as Element of (4-tuples_on (8-tuples_on BOOLEAN))*;
    defpred PK[Nat,set] means
    ex i,j,n be Nat, zij be Element of (8-tuples_on BOOLEAN)
    st i in Seg 4 & j in Seg 4 & n in Seg 8 & 1+(i-'1)*8+(j-'1)*32 <= $1 &
    $1 <= 1+(i-'1)*8+(j-'1)*32+7 & n = $1-((i-'1)*8+(j-'1)*32) & zij = (z.i).j
    & $2 = zij.n;
Q12: for k be Nat st k in Seg 128 ex z being Element of BOOLEAN st PK[k,z]
    proof
      let k be Nat;
      assume k in Seg 128;
      then 1 <= k & k <= 128 by FINSEQ_1:1;
      then consider i,j be Nat such that
A4:   i in Seg 4 & j in Seg 4 & 1+(i-'1)*8+(j-'1)*32 <= k &
      k <= 1+(i-'1)*8+(j-'1)*32+7 by LMStat0;
      i in dom z by FINSEQ_1:def 3,A4,B10;
      then z.i in rng z by FUNCT_1:3;
      then z.i in 4-tuples_on (8-tuples_on BOOLEAN);
      then
B10:  ex s be Element of (8-tuples_on BOOLEAN)* st z.i = s & len s = 4;
      then reconsider zi = z.i as Element of (8-tuples_on BOOLEAN)*;
      j in dom zi by B10,FINSEQ_1:def 3,A4;
      then (zi).j in rng zi by FUNCT_1:3;
      then reconsider zij = (z.i).j as Element of 8-tuples_on BOOLEAN;
A50:  1+(i-'1)*8+(j-'1)*32-((i-'1)*8+(j-'1)*32) <= k-((i-'1)*8+(j-'1)*32)
      by A4,XREAL_1:9;
      then reconsider n = k-((i-'1)*8+(j-'1)*32) as Element of NAT by INT_1:3;
      k-((i-'1)*8+(j-'1)*32) <= 1+(i-'1)*8+(j-'1)*32+7-((i-'1)*8+(j-'1)*32)
      by A4,XREAL_1:9;
      then
G4:   n in Seg 8 by A50;
      reconsider z = zij.n as Element of BOOLEAN;
      take z;
      thus thesis by A4,G4;
    end;
    consider x be FinSequence of BOOLEAN
    such that
Q13: dom x = Seg 128 & for i be Nat st i in Seg 128
    holds PK[i,x.i] from FINSEQ_1:sch 5(Q12);
Q14: len x = 128 by Q13,FINSEQ_1:def 3;
    reconsider x as Element of (BOOLEAN)* by FINSEQ_1:def 11;
    x in 128-tuples_on BOOLEAN by Q14;
    then reconsider x as Element of 128-tuples_on BOOLEAN;
P2: for i,j be Nat st i in Seg 4 & j in Seg 4
    holds (z.i).j = mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)
    proof
      let i,j be Nat;
      assume
P21:  i in Seg 4 & j in Seg 4;
      then i in dom z by FINSEQ_1:def 3,B10;
      then z.i in rng z by FUNCT_1:3;
      then z.i in 4-tuples_on (8-tuples_on BOOLEAN);
      then
P8:   ex s be Element of (8-tuples_on BOOLEAN)* st z.i = s & len s = 4;
      reconsider zi = z.i as Element of (8-tuples_on BOOLEAN)* by P8;
      j in dom zi by P8,FINSEQ_1:def 3,P21;
      then zi.j in rng zi by FUNCT_1:3;
      then zi.j in (8-tuples_on BOOLEAN);
      then
P11:  ex s be Element of (BOOLEAN)* st zi.j = s & len s = 8;
      reconsider zij = zi.j as Element of (BOOLEAN)* by P11;
Q110: 1 <= i & i <= 4 by P21,FINSEQ_1:1;
      then 1-1 <= i-1 by XREAL_1:9;
      then i-'1 = i-1 by XREAL_0:def 2;
      then i-'1 <= 4-1 by Q110,XREAL_1:9;
      then
Q112: (i-'1)*8 <= 3*8 by XREAL_1:64;
Q130: 1 <= j & j <= 4 by P21,FINSEQ_1:1;
      then 1-1 <= j-1 by XREAL_1:9;
      then j-'1 = j-1 by XREAL_0:def 2;
      then j-'1 <= 4-1 by Q130,XREAL_1:9;
      then
Q133: (j-'1)*32 <= 3*32 by XREAL_1:64;
      (i-'1)*8+(j-'1)*32 <= 24+96 by Q133,Q112,XREAL_1:7;
      then
Q134: 1+((i-'1)*8+(j-'1)*32) <= 1+120 by XREAL_1:7;
      then
G1:   1+((i-'1)*8+(j-'1)*32) <= len x by XXREAL_0:2,Q14;
G0:   1 <= 1+((i-'1)*8+(j-'1)*32) by NAT_1:11;
G2:   1 <= 1+((i-'1)*8+(j-'1)*32+7) by NAT_1:11;
G3:   1+((i-'1)*8+(j-'1)*32)+0 <= 1+(i-'1)*8+(j-'1)*32+7 by XREAL_1:7;
Q135: 1+(i-'1)*8+(j-'1)*32+7 <= 121+7 by XREAL_1:6,Q134;
      then
F3:   1+(i-'1)*8+(j-'1)*32+7 <= len x by Q13,FINSEQ_1:def 3;
P13:  len (mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)) =
      (1+(i-'1)*8+(j-'1)*32+7)-'(1+(i-'1)*8+(j-'1)*32)+1
      by G1,G2,G3,G0,F3,FINSEQ_6:118
        .= (1+(i-'1)*8+(j-'1)*32+7)-(1+(i-'1)*8+(j-'1)*32)+1 by G3,XREAL_1:233
        .= 8;
      now let n be Nat;
        assume
F40:    1 <= n & n <= len zij;
F1:     1 <= 1+((i-'1)*8+(j-'1)*32) by NAT_1:11;
F2:     1+(i-'1)*8+(j-'1)*32 <= 1+(i-'1)*8+(j-'1)*32+7 by NAT_1:11;
F5:     n <= (1+(i-'1)*8+(j-'1)*32+7)-(1+(i-'1)*8+(j-'1)*32)+1 by F40,P11;
        reconsider k = n+((i-'1)*8+(j-'1)*32) as Nat;
A6:     k = n-1+(1+(i-'1)*8+(j-'1)*32);
        n <= n+((i-'1)*8+(j-'1)*32) by NAT_1:11;
        then
H1:     1 <= k by F40,XXREAL_0:2;
        reconsider k = n+((i-'1)*8+(j-'1)*32) as Nat;
H3:     k <= 8+((i-'1)*8+(j-'1)*32) by F40,P11,XREAL_1:7;
        then
H2:     k <= 128 by Q135,XXREAL_0:2;
        then k in Seg 128 by H1;
        then consider i0,j0,n0 be Nat,
        zi0j0 be Element of (8-tuples_on BOOLEAN) such that
AA1:    i0 in Seg 4 & j0 in Seg 4 & n0 in Seg 8 & 1+(i0-'1)*8+(j0-'1)*32 <= k &
        k <= 1+(i0-'1)*8+(j0-'1)*32+7 & n0 = k-((i0-'1)*8+(j0-'1)*32) &
        zi0j0 = (z.i0).j0 & x.k = zi0j0.n0 by Q13;
        1+((i-'1)*8+(j-'1)*32) <= n+((i-'1)*8+(j-'1)*32) by F40,XREAL_1:7;
        then 1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7 by H3;
        then i = i0 & j = j0 by LMStat2,AA1,P21,H1,H2;
        hence zij.n = (mid (x,1+(i-'1)*8+(j-'1)*32,1+(i-'1)*8+(j-'1)*32+7)).n
        by AA1,F1,F2,F3,F40,F5,A6,FINSEQ_6:122;
      end;
      hence thesis by FINSEQ_1:def 17,P11,P13;
    end;
    (AES-Statearray.x) in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
P3: ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
    st (AES-Statearray.x) = s & len s = 4;
    now let i be Nat;
      assume 1 <= i & i <= len (AES-Statearray.x);
      then
P6:   i in Seg 4 by P3;
      then i in dom(AES-Statearray.x) by FINSEQ_1:def 3,P3;
      then (AES-Statearray.x).i in rng (AES-Statearray.x) by FUNCT_1:3;
      then (AES-Statearray.x).i in 4-tuples_on (8-tuples_on BOOLEAN);
      then
P8:   ex s be Element of (8-tuples_on BOOLEAN)*
      st (AES-Statearray.x).i = s & len s = 4;
      reconsider H1i = (AES-Statearray.x).i as
      Element of (8-tuples_on BOOLEAN)* by P8;
      i in dom z by FINSEQ_1:def 3,B10,P6;
      then z.i in rng z by FUNCT_1:3;
      then z.i in 4-tuples_on (8-tuples_on BOOLEAN);
      then
P11:  ex s be Element of (8-tuples_on BOOLEAN)*
      st z.i = s & len s = 4;
      reconsider H2i = z.i as Element of (8-tuples_on BOOLEAN)* by P11;
      now let j be Nat;
        assume 1 <= j & j <= len H1i;
        then
P14:    j in Seg 4 by P8;
        then ((AES-Statearray.x).i).j = mid (x,1+(i-'1)*8+(j-'1)*32,
        1+(i-'1)*8+(j-'1)*32+7) by DefStatearray,P6;
        hence H1i.j = H2i.j by P2,P6,P14;
      end;
      hence (AES-Statearray.x).i = z.i by P8,P11,FINSEQ_1:def 17;
    end;
    hence thesis by P3,B10,FINSEQ_1:def 17;
  end;
  then 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) =
  rng AES-Statearray by FUNCT_2:10;
  hence thesis by FUNCT_2:def 3;
end;

registration
  cluster AES-Statearray -> bijective;
  correctness by LMStat1,LMStat3;
end;

theorem LMINV1:
  for cipher be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds AES-Statearray.((AES-Statearray)".(cipher)) = cipher
proof
  let cipher be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  set f = AES-Statearray;
L0: rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  by FUNCT_2:def 3;
  then reconsider g = f" as Function of
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),128-tuples_on BOOLEAN
  by FUNCT_2:25;
L2: (f")*f = id (128-tuples_on BOOLEAN) & f*(f") =
  id (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) by FUNCT_2:29,L0;
  then g is one-to-one & rng g = 128-tuples_on BOOLEAN by FUNCT_2:18;
  then f = g" by FUNCT_2:30,L2;
  hence thesis by FUNCT_2:26;
end;

begin :: SubBytes

reserve SBT for Permutation of (8-tuples_on BOOLEAN);

definition
  let SBT;
  func SubBytes(SBT) ->
  Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefSubBytes:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds for i,j be Nat st i in Seg 4 & j in Seg 4
  holds ex inputij be Element of 8-tuples_on BOOLEAN
  st inputij = (input.i).j & ((it.input).i).j = SBT.(inputij);
  existence
  proof
    defpred P0[Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))] means
    for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ex inputij be Element of 8-tuples_on BOOLEAN
    st inputij = ($1.i).j & ($2.i).j = SBT.(inputij);
A1: for text being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[text,z]
    proof
      let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      text in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q01:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st text = s & len s = 4;
      defpred P[Nat,set] means
      ex zk be Element of (4-tuples_on (8-tuples_on BOOLEAN))
      st $2 = zk & for j be Nat st j in Seg 4
      holds ex textij be Element of 8-tuples_on BOOLEAN
      st textij = (text.$1).j & zk.j = SBT.(textij);
Q1:   for k be Nat st k in Seg 4 ex zk being
      Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,zk]
      proof
        let k be Nat;
        assume k in Seg 4;
        then k in dom text by Q01,FINSEQ_1:def 3;
        then text.k in rng text by FUNCT_1:3;
        then text.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q13:    ex s be Element of (8-tuples_on BOOLEAN)* st text.k = s & len s = 4;
        then reconsider textk = text.k as Element of (8-tuples_on BOOLEAN)*;
        defpred Pi[Nat,set] means
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (textk).$1 &
        $2 = SBT.(textij);
Q18:    for j be Nat st j in Seg 4 ex xi being Element of (8-tuples_on BOOLEAN)
        st Pi[j,xi]
        proof
          let j be Nat;
          assume j in Seg 4;
          then j in dom(textk) by Q13,FINSEQ_1:def 3;
          then (textk).j in rng (textk) by FUNCT_1:3;
          then reconsider textkj = (textk).j as Element of 8-tuples_on BOOLEAN;
          SBT.(textkj) = SBT.(textkj);
          hence thesis;
        end;
        consider zk be FinSequence of (8-tuples_on BOOLEAN) such that
Q22:    dom zk = Seg 4 & for j be Nat st j in Seg 4
        holds Pi[j,zk.j] from FINSEQ_1:sch 5(Q18);
Q23:    len zk = 4 by Q22,FINSEQ_1:def 3;
        reconsider zk as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zk in 4-tuples_on (8-tuples_on BOOLEAN) by Q23;
        then reconsider zk as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        for j be Nat st j in Seg 4 holds
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (textk).j &
        zk.j = SBT.(textij) by Q22;
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 & for i be Nat st i in Seg 4
      holds P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by Q3;
      then reconsider z as Element of
      4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      take z;
      for i,j be Nat st i in Seg 4 & j in Seg 4 holds
      ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
      (z.i).j = SBT.(textij)
      proof
        let i,j be Nat;
        assume
Q4:     i in Seg 4 & j in Seg 4;
        then ex zi be Element of (4-tuples_on (8-tuples_on BOOLEAN))
        st z.i = zi & for j be Nat st j in Seg 4 holds
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        zi.j = SBT.(textij) by Q2;
        hence ex textij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & (z.i).j = SBT.(textij) by Q4;
      end;
      hence thesis;
    end;
    consider I be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let F1,F2 be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4 holds
    ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
    ((F1.text).i).j = SBT.(textij);
    assume
A2: for text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4 holds
    ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
    ((F2.text).i).j = SBT.(textij);
    now let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      now let i,j be Nat;
        assume
A3:     i in Seg 4 & j in Seg 4;
        then
A4:     ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        ((F1.text).i).j = SBT.(textij) by A1;
A5:     ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        ((F2.text).i).j = SBT.(textij) by A3,A2;
        thus ((F1.text).i).j = ((F2.text).i).j by A4,A5;
      end;
      hence F1.text = F2.text by LM01;
    end;
    hence F1 = F2 by FUNCT_2:63;
  end;
end;

definition
  let SBT;
  func InvSubBytes(SBT) ->
  Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefInvSubBytes:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds for i,j be Nat st i in Seg 4 & j in Seg 4 holds
  ex inputij be Element of 8-tuples_on BOOLEAN st inputij = (input.i).j &
  ((it.input).i).j = (SBT").(inputij);
  existence
  proof
    defpred P0[Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))] means
    for i,j be Nat st i in Seg 4 & j in Seg 4 holds
    ex inputij be Element of 8-tuples_on BOOLEAN st inputij = ($1.i).j &
    ($2.i).j = (SBT").(inputij);
A1: for text being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[text,z]
    proof
      let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      text in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q01:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st text = s &
      len s = 4;
      defpred P[Nat,set] means
      ex zk be Element of (4-tuples_on (8-tuples_on BOOLEAN)) st $2 = zk &
      for j be Nat st j in Seg 4 holds
      ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.$1).j &
      zk.j = (SBT").(textij);
Q1:   for k be Nat st k in Seg 4
      ex zk being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,zk]
      proof
        let k be Nat;
        assume k in Seg 4;
        then k in dom text by Q01,FINSEQ_1:def 3;
        then text.k in rng text by FUNCT_1:3;
        then text.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q13:    ex s be Element of (8-tuples_on BOOLEAN)* st text.k = s & len s = 4;
        then reconsider textk = text.k as Element of (8-tuples_on BOOLEAN)*;
        defpred Pi[Nat,set] means
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (textk).$1 &
        $2 = (SBT").(textij);
Q18:    for j be Nat st j in Seg 4 ex xi being Element of (8-tuples_on BOOLEAN)
        st Pi[j,xi]
        proof
          let j be Nat;
          assume j in Seg 4;
          then j in dom(textk) by Q13,FINSEQ_1:def 3;
          then (textk).j in rng (textk) by FUNCT_1:3;
          then reconsider textkj = (textk).j as Element of 8-tuples_on BOOLEAN;
          (SBT").(textkj) = (SBT").(textkj);
          hence thesis;
        end;
        consider zk be FinSequence of (8-tuples_on BOOLEAN) such that
Q22:    dom zk = Seg 4 & for j be Nat st j in Seg 4
        holds Pi[j,zk.j ] from FINSEQ_1:sch 5(Q18);
Q23:    len zk = 4 by Q22,FINSEQ_1:def 3;
        reconsider zk as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zk in 4-tuples_on (8-tuples_on BOOLEAN) by Q23;
        then reconsider zk as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        for j be Nat st j in Seg 4 holds
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (textk).j &
        zk.j = (SBT").(textij) by Q22;
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 & for i be Nat st i in Seg 4
      holds P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by Q3;
      then reconsider z as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      take z;
      for i,j be Nat st i in Seg 4 & j in Seg 4 holds
      ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
      (z.i).j = (SBT").(textij)
      proof
        let i,j be Nat;
        assume
Q4:     i in Seg 4 & j in Seg 4;
        then ex zi be Element of (4-tuples_on (8-tuples_on BOOLEAN))
        st z.i = zi & for j be Nat st j in Seg 4 holds
        ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        zi.j = (SBT").(textij) by Q2;
        hence ex textij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & (z.i).j = (SBT").(textij) by Q4;
      end;
      hence thesis;
    end;
    consider I be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let F1,F2 be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4 holds
    ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
    ((F1.text).i).j = (SBT").(textij);
    assume
A2: for text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4 holds
    ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
    ((F2.text).i).j = (SBT").(textij);
    now let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      now let i,j be Nat;
        assume
A3:     i in Seg 4 & j in Seg 4;
        then
A4:     ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        ((F1.text).i).j = (SBT").(textij) by A1;
A5:     ex textij be Element of 8-tuples_on BOOLEAN st textij = (text.i).j &
        ((F2.text).i).j = (SBT").(textij) by A3,A2;
        thus ((F1.text).i).j = ((F2.text).i).j by A4,A5;
      end;
      hence F1.text = F2.text by LM01;
    end;
    hence F1 = F2 by FUNCT_2:63;
  end;
end;

INV07A:
  for input be Element of 8-tuples_on BOOLEAN
  holds (SBT").(SBT.(input)) = input
proof
  let input be Element of 8-tuples_on BOOLEAN;
  thus (SBT").(SBT.(input)) = ((SBT")*SBT).input by FUNCT_2:15
    .= (id (8-tuples_on BOOLEAN)).input by FUNCT_2:61
    .= input;
end;

INV08A:
  for input be Element of 8-tuples_on BOOLEAN
  holds SBT.((SBT").(input)) = input
proof
  let input be Element of 8-tuples_on BOOLEAN;
  thus SBT.((SBT").(input)) = (SBT*(SBT")).input by FUNCT_2:15
    .= (id (8-tuples_on BOOLEAN)).input by FUNCT_2:61
    .= input;
end;

theorem INV07:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (InvSubBytes(SBT)).((SubBytes(SBT)).input) = input
proof
  let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  now let i,j be Nat;
    assume
A3: i in Seg 4 & j in Seg 4;
    then consider outputij be Element of 8-tuples_on BOOLEAN such that
A4: outputij = (((SubBytes(SBT)).input).i).j &
    (((InvSubBytes(SBT)).((SubBytes(SBT)).input)).i).j = (SBT").(outputij)
    by DefInvSubBytes;
    consider inputij be Element of 8-tuples_on BOOLEAN such that
A5: inputij = (input.i).j & (((SubBytes(SBT)).input).i).j = SBT.(inputij)
    by DefSubBytes,A3;
    thus (((InvSubBytes(SBT)).((SubBytes(SBT)).input)).i).j = (input.i).j
    by A4,A5,INV07A;
  end;
  hence (InvSubBytes(SBT)).((SubBytes(SBT)).input) = input by LM01;
end;

theorem INV08:
  for output be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (SubBytes(SBT)).((InvSubBytes(SBT)).output) = output
proof
  let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  now let i,j be Nat;
    assume
A3: i in Seg 4 & j in Seg 4;
    then consider outputij be Element of 8-tuples_on BOOLEAN such that
A4: outputij = (((InvSubBytes(SBT)).input).i).j &
    (((SubBytes(SBT)).((InvSubBytes(SBT)).input)).i).j = SBT.(outputij)
    by DefSubBytes;
    consider inputij be Element of 8-tuples_on BOOLEAN such that
A5: inputij = (input.i).j &
    (((InvSubBytes(SBT)).input).i).j = (SBT").(inputij) by DefInvSubBytes,A3;
    thus (((SubBytes(SBT)).((InvSubBytes(SBT)).input)).i).j = (input.i).j
    by A4,A5,INV08A;
  end;
  hence (SubBytes(SBT)).((InvSubBytes(SBT)).input) = input by LM01;
end;

theorem
  SubBytes(SBT) is one-to-one & SubBytes(SBT) is onto &
  InvSubBytes(SBT) is one-to-one & InvSubBytes(SBT) is onto &
  InvSubBytes(SBT) = (SubBytes(SBT))" & SubBytes(SBT) = (InvSubBytes(SBT))"
proof
  set f = SubBytes(SBT);
  set g = InvSubBytes(SBT);
P1: for x be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds g.(f.x) = x by INV07;
P2: for y be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds f.(g.y) = y by INV08;
  thus f is one-to-one & f is onto & g is one-to-one & g is onto &
  g = f" & f = g" by INV00,P1,P2;
end;

begin :: ShiftRows

definition
  func ShiftRows ->
  Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefShiftRows:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (for i be Nat st i in Seg 4 holds
  ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
  (it.input).i = Op-Shift(xi,5-i));
  existence
  proof
    defpred P0[Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))] means
    for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = $1.i &
    $2.i = Op-Shift(xi,5-i);
A1: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[x,z]
    proof
      let x be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      x in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q01:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st x = s &
      len s = 4;
      defpred P[Nat,set] means
      ex xk be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xk = x.$1 &
      $2 = Op-Shift(xk,5-$1);
Q1:   for k be Nat st k in Seg 4
      ex zk being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,zk]
      proof
        let k be Nat;
        assume k in Seg 4;
        then k in dom x by Q01,FINSEQ_1:def 3;
        then
Q11:    x.k in rng x by FUNCT_1:3;
        then x.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q13:    ex s be Element of (8-tuples_on BOOLEAN)* st x.k = s & len s = 4;
        then reconsider xk = x.k as Element of (8-tuples_on BOOLEAN)*;
        reconsider xk1 = xk as Element of 4-tuples_on (8-tuples_on BOOLEAN)
        by Q11;
        reconsider zk = Op-Shift(xk,5-k) as
        FinSequence of (8-tuples_on BOOLEAN);
Q15:    len zk = 4 by Q13,DESCIP_1:def 3;
        reconsider zk as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zk in 4-tuples_on (8-tuples_on BOOLEAN) by Q15;
        then reconsider zk as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        ex xk be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xk = x.k &
        zk = Op-Shift(xk1,5-k);
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 & for i be Nat st i in Seg 4
      holds P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by Q3;
      hence thesis by Q2;
    end;
    consider I be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let H1,H2 be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
    (H1.input).i = Op-Shift(xi,5-i));
    assume
A2: for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
    (H2.input).i = Op-Shift(xi,5-i));
    now let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      H1.input in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P3:   ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))* st H1.input = s &
      len s = 4;
      H2.input in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P4:   ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))* st H2.input = s &
      len s = 4;
      now let i be Nat;
        assume 1 <= i & i <= len (H1.input);
        then
XX2:    i in Seg 4 by P3;
        then
XX3:    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN)
        st xi = input.i & (H1.input).i = Op-Shift(xi,5-i) by A1;
XX4:    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN)
        st xi = input.i & (H2.input).i = Op-Shift(xi,5-i) by A2,XX2;
        thus (H1.input).i = (H2.input).i by XX3,XX4;
      end;
      hence H1.input = H2.input by P3,P4,FINSEQ_1:14;
    end;
    hence H1 = H2 by FUNCT_2:63;
  end;
end;

definition
  func InvShiftRows ->
  Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefInvShiftRows:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (for i be Nat st i in Seg 4 holds
  ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
  (it.input).i = Op-Shift(xi,i-1));
  existence
  proof
    defpred P0[Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))] means
    for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = $1.i &
    $2.i = Op-Shift(xi,i-1);
A1: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[x,z]
    proof
      let x be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      x in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q01:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st x = s &
      len s = 4;
      defpred P[Nat,set] means
      ex xk be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xk = x.$1 &
      $2 = Op-Shift(xk,$1-1);
Q1:   for k be Nat st k in Seg 4
      ex zk being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,zk]
      proof
        let k be Nat;
        assume k in Seg 4;
        then k in dom x by Q01,FINSEQ_1:def 3;
        then
Q11:    x.k in rng x by FUNCT_1:3;
        then x.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q13:    ex s be Element of (8-tuples_on BOOLEAN)* st x.k = s & len s = 4;
        then reconsider xk = x.k as Element of (8-tuples_on BOOLEAN)*;
        reconsider xk1 = xk as Element of 4-tuples_on (8-tuples_on BOOLEAN)
        by Q11;
        reconsider zk = Op-Shift(xk,k-1) as
        FinSequence of (8-tuples_on BOOLEAN);
Q15:    len zk = 4 by Q13,DESCIP_1:def 3;
        reconsider zk as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zk in 4-tuples_on (8-tuples_on BOOLEAN) by Q15;
        then reconsider zk as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        ex xk be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xk = x.k &
        zk = Op-Shift(xk1,k-1);
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 & for i be Nat st i in Seg 4 holds
      P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by Q3;
      hence thesis by Q2;
    end;
    consider I be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let H1,H2 be Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
    (H1.input).i = Op-Shift(xi,i-1));
    assume
A2: for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Nat st i in Seg 4 holds
    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
    (H2.input).i = Op-Shift(xi,i-1));
    now let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      H1.input in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P3:   ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))* st H1.input = s &
      len s = 4;
      H2.input in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
P4:   ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))* st H2.input = s &
      len s = 4;
      now let i be Nat;
        assume 1 <= i & i <= len (H1.input);
        then
XX2:    i in Seg 4 by P3;
        then
XX3:    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
        (H1.input).i = Op-Shift(xi,i-1) by A1;
XX4:    ex xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) st xi = input.i &
        (H2.input).i = Op-Shift(xi,i-1) by A2,XX2;
        thus (H1.input).i = (H2.input).i by XX3,XX4;
      end;
      hence H1.input = H2.input by P3,P4,FINSEQ_1:14;
    end;
    hence H1 = H2 by FUNCT_2:63;
  end;
end;

theorem INV04:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds InvShiftRows.(ShiftRows.input) = input
proof
  let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  InvShiftRows.(ShiftRows.input) in
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  then
P3: ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))*
  st InvShiftRows.(ShiftRows.input) = s & len s = 4;
  input in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  then
P4: ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))*
  st input = s & len s = 4;
  now let i be Nat;
    assume 1 <= i & i <= len (InvShiftRows.(ShiftRows.input));
    then
XX2: i in Seg 4 by P3;
    then consider xi be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
XX3: xi = input.i & (ShiftRows.input).i = Op-Shift(xi,5-i) by DefShiftRows;
    xi in 4-tuples_on (8-tuples_on BOOLEAN);
    then
YY1: ex s be Element of (8-tuples_on BOOLEAN)* st xi = s & len s = 4;
    consider yi be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
XX4: yi = (ShiftRows.input).i &
    (InvShiftRows.(ShiftRows.input)).i = Op-Shift(yi,i-1)
    by DefInvShiftRows,XX2;
    thus (InvShiftRows.(ShiftRows.input)).i = Op-Shift(xi,5-i+(i-1))
    by XX3,XX4,DESCIP_1:10,YY1
      .= input.i by DESCIP_1:12,YY1,XX3;
  end;
  hence (InvShiftRows.(ShiftRows.input)) = input by P3,P4,FINSEQ_1:14;
end;

theorem INV05:
  for output be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds ShiftRows.(InvShiftRows.output) = output
proof
  let output be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  ShiftRows.(InvShiftRows.output) in
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  then
P3: ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))*
  st ShiftRows.(InvShiftRows.output) = s & len s = 4;
  output in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  then
P4: ex s be Element of (4-tuples_on(8-tuples_on BOOLEAN))*
  st output = s & len s = 4;
  now let i be Nat;
    assume 1 <= i & i <= len (ShiftRows.(InvShiftRows.output));
    then
XX2: i in Seg 4 by P3;
    then consider xi be Element of 4-tuples_on (8-tuples_on BOOLEAN)
    such that
XX3: xi = output.i &
    (InvShiftRows.output).i = Op-Shift(xi,i-1) by DefInvShiftRows;
    xi in 4-tuples_on (8-tuples_on BOOLEAN);
    then
YY1: ex s be Element of (8-tuples_on BOOLEAN)* st xi = s & len s = 4;
    consider yi be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
XX4: yi = (InvShiftRows.output).i &
    (ShiftRows.(InvShiftRows.output)).i = Op-Shift(yi,5-i) by DefShiftRows,XX2;
    thus (ShiftRows.(InvShiftRows.output)).i = Op-Shift(xi,i-1+(5-i))
    by XX3,XX4,DESCIP_1:10,YY1
      .= output.i by DESCIP_1:12,YY1,XX3;
  end;
  hence (ShiftRows.(InvShiftRows.output)) = output by P3,P4,FINSEQ_1:14;
end;

theorem
  ShiftRows is one-to-one & ShiftRows is onto &
  InvShiftRows is one-to-one & InvShiftRows is onto &
  InvShiftRows = (ShiftRows)" & ShiftRows = (InvShiftRows)"
  by INV00,INV04,INV05;

begin :: AddRoundKey

definition
  func AddRoundKey ->
  Function of [:4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)):],
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :DefAddRoundKey:
  for text, key be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds for i,j be Nat st i in Seg 4 & j in Seg 4
  holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
  st textij = (text.i).j & keyij = (key.i).j &
  ((it.(text,key)).i).j = Op-XOR(textij,keyij);
  existence
  proof
    defpred P0[Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))] means
    for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
    st textij = ($1.i).j & keyij = ($2.i).j & ($3.i).j = Op-XOR(textij,keyij);
A1: for text,key
    being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P0[text,key,z]
    proof
      let text, key
      be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      text in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q01:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st text = s & len s = 4;
      key in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      then
Q02:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st key = s & len s = 4;
      defpred P[Nat,set] means
      ex zk be Element of (4-tuples_on (8-tuples_on BOOLEAN))
      st $2 = zk & for j be Nat st j in Seg 4
      holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
      st textij = (text.$1).j & keyij = (key.$1).j &
      zk.j = Op-XOR(textij,keyij);
Q1:   for k be Nat st k in Seg 4 ex zk being
      Element of (4-tuples_on (8-tuples_on BOOLEAN)) st P[k,zk]
      proof
        let k be Nat;
        assume
Q11:    k in Seg 4;
        then k in dom text by Q01,FINSEQ_1:def 3;
        then text.k in rng text by FUNCT_1:3;
        then text.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q13:    ex s be Element of (8-tuples_on BOOLEAN)*
        st text.k = s & len s = 4;
        then reconsider textk = text.k as Element of (8-tuples_on BOOLEAN)*;
        k in dom(key) by Q02,FINSEQ_1:def 3,Q11;
        then key.k in rng key by FUNCT_1:3;
        then key.k in 4-tuples_on (8-tuples_on BOOLEAN);
        then
Q16:    ex s be Element of (8-tuples_on BOOLEAN)*
        st key.k = s & len s = 4;
        then reconsider keyk = key.k as Element of (8-tuples_on BOOLEAN)*;
        defpred Pi[Nat,set] means
        ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (textk).$1 & keyij = (keyk).$1 & $2 = Op-XOR(textij,keyij);
Q18:    for j be Nat st j in Seg 4 ex xi being Element of (8-tuples_on BOOLEAN)
        st Pi[j,xi]
        proof
          let j be Nat;
          assume
Q19:      j in Seg 4;
          then j in dom(textk) by Q13,FINSEQ_1:def 3;
          then (textk).j in rng (textk) by FUNCT_1:3;
          then reconsider textkj = (textk).j as Element of 8-tuples_on BOOLEAN;
          j in dom(keyk) by Q16,FINSEQ_1:def 3,Q19;
          then (keyk).j in rng (keyk) by FUNCT_1:3;
          then reconsider keykj = (key.k).j as Element of 8-tuples_on BOOLEAN;
          Op-XOR(textkj,keykj) = Op-XOR(textkj,keykj);
          hence thesis;
        end;
        consider zk be FinSequence of (8-tuples_on BOOLEAN) such that
Q22:    dom zk = Seg 4 & for j be Nat st j in Seg 4
        holds Pi[j,zk.j] from FINSEQ_1:sch 5(Q18);
Q23:    len zk = 4 by Q22,FINSEQ_1:def 3;
        reconsider zk as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
        zk in 4-tuples_on (8-tuples_on BOOLEAN) by Q23;
        then reconsider zk as Element of 4-tuples_on (8-tuples_on BOOLEAN);
        for j be Nat st j in Seg 4
        holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (textk).j & keyij = (keyk).j & zk.j = Op-XOR(textij,keyij)
        by Q22;
        hence thesis;
      end;
      consider z be FinSequence of 4-tuples_on (8-tuples_on BOOLEAN) such that
Q2:   dom z = Seg 4 & for i be Nat st i in Seg 4
      holds P[i,z.i] from FINSEQ_1:sch 5(Q1);
Q3:   len z = 4 by Q2,FINSEQ_1:def 3;
      reconsider z as Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      by FINSEQ_1:def 11;
      z in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by Q3;
      then reconsider z as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      take z;
      for i,j be Nat st i in Seg 4 & j in Seg 4 holds
      ex textij,keyij be Element of 8-tuples_on BOOLEAN st textij =
      (text.i).j & keyij = (key.i).j & (z.i).j = Op-XOR(textij,keyij)
      proof
        let i,j be Nat;
        assume
Q4:     i in Seg 4 & j in Seg 4;
        then ex zi be Element of (4-tuples_on (8-tuples_on BOOLEAN))
        st z.i = zi & for j be Nat st j in Seg 4
        holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & keyij = (key.i).j &
        zi.j = Op-XOR(textij,keyij) by Q2;
        hence ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & keyij = (key.i).j &
        (z.i).j = Op-XOR(textij,keyij) by Q4;
      end;
      hence thesis;
    end;
    consider I be Function of [:4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)):],
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2: for x,y being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,y,I.(x,y)] from BINOP_1:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let F1,F2 be Function of [:4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)):],
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: for text,key be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
    st textij = (text.i).j & keyij = (key.i).j &
    ((F1.(text,key)).i).j = Op-XOR(textij,keyij);
    assume
A2: for text,key be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds for i,j be Nat st i in Seg 4 & j in Seg 4
    holds ex textij,keyij be Element of 8-tuples_on BOOLEAN
    st textij = (text.i).j & keyij = (key.i).j &
    ((F2.(text,key)).i).j = Op-XOR(textij,keyij);
    now let text,key be
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      now let i,j be Nat;
        assume
A3:     i in Seg 4 & j in Seg 4;
        then
A4:     ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & keyij = (key.i).j &
        ((F1.(text,key)).i).j = Op-XOR(textij,keyij) by A1;
A5:     ex textij,keyij be Element of 8-tuples_on BOOLEAN
        st textij = (text.i).j & keyij = (key.i).j &
        ((F2.(text,key)).i).j = Op-XOR(textij,keyij) by A3,A2;
        thus ((F1.(text,key)).i).j = ((F2.(text,key)).i).j by A4,A5;
      end;
      hence F1.(text,key) = F2.(text,key) by LM01;
    end;
    hence F1 = F2 by BINOP_1:2;
  end;
end;

begin :: Key Expansion

definition
  let SBT;
  let x be Element of 4-tuples_on (8-tuples_on BOOLEAN);
  func SubWord(SBT,x) -> Element of 4-tuples_on (8-tuples_on BOOLEAN) means
  for i be Element of Seg 4 holds it.i = SBT.(x.i);
  existence
  proof
    defpred P[Nat,set] means ex xi be Element of 8-tuples_on BOOLEAN
    st xi = x.$1 & $2 = SBT.(xi);
P1: for k be Nat st k in Seg 4 ex z being Element of 8-tuples_on BOOLEAN
    st P[k,z]
    proof
      let k be Nat;
      assume
AS:   k in Seg 4;
      x in 4-tuples_on (8-tuples_on BOOLEAN);
      then ex s be Element of (8-tuples_on BOOLEAN)* st x = s & len s = 4;
      then k in dom x by FINSEQ_1:def 3,AS;
      then x.k in rng x by FUNCT_1:3;
      then reconsider xk = x.k as Element of 8-tuples_on BOOLEAN;
      SBT.(xk) is Element of 8-tuples_on BOOLEAN;
      hence thesis;
    end;
    consider p being FinSequence of 8-tuples_on BOOLEAN such that
P3: dom p = Seg 4 & for k be Nat st k in Seg 4 holds
    P[k,p.k] from FINSEQ_1:sch 5(P1);
    reconsider p as Element of (8-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
    len p = 4 by P3,FINSEQ_1:def 3;
    then p in 4-tuples_on (8-tuples_on BOOLEAN);
    then reconsider p as Element of 4-tuples_on (8-tuples_on BOOLEAN);
    take p;
    now let i be Element of Seg 4;
      ex xi be Element of 8-tuples_on BOOLEAN st xi = x.i & p.i = SBT.(xi)
      by P3;
      hence p.i = SBT.(x.i);
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let H1,H2 be Element of 4-tuples_on (8-tuples_on BOOLEAN);
    assume
A1: for i be Element of Seg 4 holds H1.i = SBT.(x.i);
    assume
A2: for i be Element of Seg 4 holds H2.i = SBT.(x.i);
    H1 in 4-tuples_on (8-tuples_on BOOLEAN);
    then
P1: ex s be Element of (8-tuples_on BOOLEAN)* st H1 = s & len s = 4;
    H2 in 4-tuples_on (8-tuples_on BOOLEAN);
    then
P2: ex s be Element of (8-tuples_on BOOLEAN)* st H2 = s & len s = 4;
    now let i be Nat;
      assume 1 <= i & i <= len H1;
      then i in Seg 4 by P1;
      then reconsider j = i as Element of Seg 4;
      thus H1.i = (SBT).(x.j) by A1
        .= H2.i by A2;
    end;
    hence H1 = H2 by P1,P2,FINSEQ_1:14;
  end;
end;

definition
  let x be Element of 4-tuples_on (8-tuples_on BOOLEAN);
  func RotWord(x) -> Element of 4-tuples_on (8-tuples_on BOOLEAN) equals
  Op-LeftShift(x);
  correctness by DESCIP_1:6;
end;

definition
  let n,m be non zero Element of NAT;
  let s,t be Element of m-tuples_on (n-tuples_on BOOLEAN);
  func Op-WXOR(s,t) -> Element of m-tuples_on (n-tuples_on BOOLEAN) means
  for i be Element of Seg m holds it.i = Op-XOR(s.i,t.i);
  existence
  proof
    defpred P[Nat,set] means
    ex si,ti be Element of (n-tuples_on BOOLEAN) st si = s.$1 & ti = t.$1 &
    $2 = Op-XOR(si,ti);
P1: for k be Nat st k in Seg m ex z being Element of n-tuples_on BOOLEAN
    st P[k,z]
    proof
      let k be Nat;
      assume
AS:   k in Seg m;
      s in m-tuples_on (n-tuples_on BOOLEAN);
      then ex v be Element of (n-tuples_on BOOLEAN)* st s = v & len v = m;
      then k in dom s by FINSEQ_1:def 3,AS;
      then s.k in rng s by FUNCT_1:3;
      then reconsider sk = s.k as Element of n-tuples_on BOOLEAN;
      t in m-tuples_on (n-tuples_on BOOLEAN);
      then ex v be Element of (n-tuples_on BOOLEAN)* st t = v & len v = m;
      then k in dom t by FINSEQ_1:def 3,AS;
      then t.k in rng t by FUNCT_1:3;
      then reconsider tk = t.k as Element of n-tuples_on BOOLEAN;
      Op-XOR(sk,tk) is Element of n-tuples_on BOOLEAN;
      hence thesis;
    end;
    consider p being FinSequence of n-tuples_on BOOLEAN such that
P3: dom p = Seg m & for k be Nat st k in Seg m holds P[k,p.k]
    from FINSEQ_1:sch 5(P1);
P4: len p = m by P3,FINSEQ_1:def 3;
    p in (n-tuples_on BOOLEAN)* by FINSEQ_1:def 11;
    then p in m-tuples_on (n-tuples_on BOOLEAN) by P4;
    then reconsider p as Element of m-tuples_on (n-tuples_on BOOLEAN);
    take p;
    now let i be Element of Seg m;
      ex si,ti be Element of n-tuples_on BOOLEAN st si = s.i & ti = t.i &
      p.i = Op-XOR(si,ti) by P3;
      hence p.i = Op-XOR(s.i,t.i);
    end;
    hence thesis;
  end;
  uniqueness
  proof
    let H1,H2 be Element of m-tuples_on (n-tuples_on BOOLEAN);
    assume
A1: for i be Element of Seg m holds H1.i = Op-XOR(s.i,t.i);
    assume
A2: for i be Element of Seg m holds H2.i = Op-XOR(s.i,t.i);
    H1 in m-tuples_on (n-tuples_on BOOLEAN);
    then
P1: ex v be Element of (n-tuples_on BOOLEAN)* st H1 = v & len v = m;
    H2 in m-tuples_on (n-tuples_on BOOLEAN);
    then
P2: ex v be Element of (n-tuples_on BOOLEAN)* st H2 = v & len v = m;
    now let i be Nat;
      assume 1 <= i & i <= len H1;
      then i in Seg m by P1;
      then reconsider j = i as Element of Seg m;
      thus H1.i = Op-XOR(s.j,t.j) by A1
        .= H2.i by A2;
    end;
    hence H1 = H2 by P1,P2,FINSEQ_1:14;
  end;
end;

definition
  func Rcon -> Element of 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
  means
  it.1 = <* <*0,0,0,0*>^<*0,0,0,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.2 = <* <*0,0,0,0*>^<*0,0,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.3 = <* <*0,0,0,0*>^<*0,1,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.4 = <* <*0,0,0,0*>^<*1,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.5 = <* <*0,0,0,1*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.6 = <* <*0,0,1,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.7 = <* <*0,1,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.8 = <* <*1,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.9 = <* <*0,0,0,1*>^<*1,0,1,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
  it.10 = <* <*0,0,1,1*>^<*0,1,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *>;
  existence
  proof
X0: 0 in BOOLEAN by TARSKI:def 2,MARGREL1:def 11;
X1: 1 in BOOLEAN by TARSKI:def 2,MARGREL1:def 11;
P1: <*0,0,0,0*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0;
P2: <*0,0,0,1*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
P3: <*0,0,1,0*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
P4: <*0,1,0,0*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
P5: <*1,0,0,0*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
R1: <*1,0,1,1*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
R2: <*0,0,1,1*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
R3: <*0,1,1,0*> is Element of (4-tuples_on BOOLEAN) by LMGSEQ4,X0,X1;
    reconsider PP6 = <* <*0,0,0,0*>^<*0,0,0,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P2,LMGSEQ16;
    reconsider PP7 = <* <*0,0,0,0*>^<*0,0,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P3,LMGSEQ16;
    reconsider PP8 = <* <*0,0,0,0*>^<*0,1,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P4,LMGSEQ16;
    reconsider PP9 = <* <*0,0,0,0*>^<*1,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P5,LMGSEQ16;
    reconsider PP10 = <* <*0,0,0,1*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P2,LMGSEQ16;
    reconsider PP11 = <* <*0,0,1,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P3,LMGSEQ16;
    reconsider PP12 = <* <*0,1,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P4,LMGSEQ16;
    reconsider PP13 = <* <*1,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P5,LMGSEQ16;
    reconsider PP14 = <* <*0,0,0,1*>^<*1,0,1,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,P2,R1,LMGSEQ16;
    reconsider PP15 = <* <*0,0,1,1*>^<*0,1,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> as
    Element of 4-tuples_on (8-tuples_on BOOLEAN) by P1,R2,R3,LMGSEQ16;
    reconsider Q0 = <*PP6,PP7,PP8,PP9,PP10*> as FinSequence;
    reconsider Q1 = <*PP11,PP12,PP13,PP14,PP15*> as FinSequence;
    reconsider IT = Q0^Q1 as
    Element of 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) by LMGSEQ10;
A1: len Q0 = 5 & Q0.1 = PP6 & Q0.2 = PP7 & Q0.3 = PP8 & Q0.4 = PP9 &
    Q0.5 = PP10 by FINSEQ_4:78;
A2: len Q1 = 5 & Q1.1 = PP11 & Q1.2 = PP12 & Q1.3 = PP13 & Q1.4 = PP14 &
    Q1.5 = PP15 by FINSEQ_4:78;
    1 in Seg 5;
    then 1 in dom Q0 by FINSEQ_1:def 3,A1;
    then
R1: IT.1 = PP6 by A1,FINSEQ_1:def 7;
    2 in Seg 5;
    then 2 in dom Q0 by FINSEQ_1:def 3,A1;
    then
R2: IT.2 = PP7 by A1,FINSEQ_1:def 7;
    3 in Seg 5;
    then 3 in dom Q0 by FINSEQ_1:def 3,A1;
    then
R3: IT.3 = PP8 by A1,FINSEQ_1:def 7;
    4 in Seg 5;
    then 4 in dom Q0 by FINSEQ_1:def 3,A1;
    then
R4: IT.4 = PP9 by A1,FINSEQ_1:def 7;
    5 in Seg 5;
    then 5 in dom Q0 by FINSEQ_1:def 3,A1;
    then
R5: IT.5 = PP10 by A1,FINSEQ_1:def 7;
    1 in Seg 5;
    then 1 in dom Q1 by FINSEQ_1:def 3,A2;
    then
R10: IT.(5+1) = Q1.1 by A1,FINSEQ_1:def 7
      .= PP11 by FINSEQ_4:78;
    2 in Seg 5;
    then 2 in dom Q1 by FINSEQ_1:def 3,A2;
    then
R20: IT.(5+2) = Q1.2 by A1,FINSEQ_1:def 7
      .= PP12 by FINSEQ_4:78;
    3 in Seg 5;
    then 3 in dom Q1 by FINSEQ_1:def 3,A2;
    then
R30: IT.(5+3) = Q1.3 by A1,FINSEQ_1:def 7
      .= PP13 by FINSEQ_4:78;
    4 in Seg 5;
    then 4 in dom Q1 by FINSEQ_1:def 3,A2;
    then
R40: IT.(5+4) = Q1.4 by A1,FINSEQ_1:def 7
      .= PP14 by FINSEQ_4:78;
    5 in Seg 5;
    then 5 in dom Q1 by FINSEQ_1:def 3,A2;
    then
R50: IT.(5+5) = Q1.5 by A1,FINSEQ_1:def 7
      .= PP15 by FINSEQ_4:78;
    thus thesis by R1,R2,R3,R4,R5,R10,R20,R30,R40,R50;
  end;
  uniqueness
  proof
    let R1,R2 be Element of 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
    assume
A1: R1.1 = <* <*0,0,0,0*>^<*0,0,0,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.2 = <* <*0,0,0,0*>^<*0,0,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.3 = <* <*0,0,0,0*>^<*0,1,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.4 = <* <*0,0,0,0*>^<*1,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.5 = <* <*0,0,0,1*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.6 = <* <*0,0,1,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.7 = <* <*0,1,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.8 = <* <*1,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.9 = <* <*0,0,0,1*>^<*1,0,1,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R1.10 = <* <*0,0,1,1*>^<*0,1,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *>;
    assume
A2: R2.1 = <* <*0,0,0,0*>^<*0,0,0,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.2 = <* <*0,0,0,0*>^<*0,0,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.3 = <* <*0,0,0,0*>^<*0,1,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.4 = <* <*0,0,0,0*>^<*1,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.5 = <* <*0,0,0,1*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.6 = <* <*0,0,1,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.7 = <* <*0,1,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.8 = <* <*1,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.9 = <* <*0,0,0,1*>^<*1,0,1,1*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *> &
    R2.10 = <* <*0,0,1,1*>^<*0,1,1,0*>,<*0,0,0,0*>^<*0,0,0,0*>,
    <*0,0,0,0*>^<*0,0,0,0*>,<*0,0,0,0*>^<*0,0,0,0*> *>;
    R1 in 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
    then
XP1: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st R1 = v &
    len v = 10;
    R2 in 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
    then
XP2: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st R2 = v &
    len v = 10;
    for i be Nat st 1 <= i & i <= len R1 holds R1.i = R2.i
    proof
      let i be Nat;
      assume 1 <= i & i <= len R1;
      then i = 1 or ... or i = 10 by XP1;
      hence thesis by A1,A2;
    end;
    hence R1 = R2 by XP1,XP2,FINSEQ_1:14;
  end;
end;

definition
  let SBT;
  let m,i be Nat, w be Element of (4-tuples_on (8-tuples_on BOOLEAN));
  assume AS: (m = 4 or m = 6 or m = 8) & i < 4*(7+m) & m <= i;
  func KeyExTemp(SBT,m,i,w) -> Element of (4-tuples_on (8-tuples_on BOOLEAN))
  means
  (ex T3 be Element of (4-tuples_on (8-tuples_on BOOLEAN)) st T3 = Rcon.(i/m) &
  it = Op-WXOR(SubWord(SBT,RotWord(w)),T3)) if ((i mod m) = 0),
  (it = SubWord(SBT,w)) if (m = 8 & (i mod 8) = 4) otherwise it = w;
  existence
  proof
    thus (i mod m) = 0 implies ex A
    being Element of (4-tuples_on (8-tuples_on BOOLEAN))
    st (ex T3 be Element of (4-tuples_on (8-tuples_on BOOLEAN))
    st T3 = Rcon.(i/m) & A = Op-WXOR(SubWord(SBT,RotWord(w)),T3))
    proof
      assume
A1:   (i mod m) = 0;
      m <> 0 & m divides i by A1,INT_1:62,AS;
      then
LTT0: i/m is Integer by WSIERP_1:17;
LTT1: (4*(7+m))/m = (28/m)+4 by AS;
LTT2: m/m <= i/m by AS,XREAL_1:72;
LTT4: i/m in NAT by INT_1:3,LTT0;
LTT5: i/m < 28/m+4 by AS,XREAL_1:74,LTT1;
      i/m <= 10
      proof
        now per cases by AS;
          case m = 4;
            then i/m <10+1 by AS,XREAL_1:74,LTT1;
            hence thesis by NAT_1:13,LTT4;
          end;
          case m = 6;
            hence thesis by LTT5,XXREAL_0:2;
          end;
          case m = 8;
            hence thesis by LTT5,XXREAL_0:2;
          end;
        end;
        hence thesis;
      end;
      then
Q0:   i/m in Seg 10 by AS,LTT2,LTT4;
      reconsider j = i/m as Nat by LTT4;
      Rcon in 10-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      then ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN)) *
      st Rcon = v & len v = 10;
      then dom Rcon = Seg 10 by FINSEQ_1:def 3;
      then Rcon.j in rng Rcon by Q0,FUNCT_1:3;
      then reconsider T3 = Rcon.j as
      Element of (4-tuples_on (8-tuples_on BOOLEAN));
      Op-WXOR(SubWord(SBT,RotWord(w)),T3) is
      Element of (4-tuples_on (8-tuples_on BOOLEAN));
      hence thesis;
    end;
    thus m = 8 & (i mod 8) = 4 implies ex A being
    Element of (4-tuples_on (8-tuples_on BOOLEAN)) st (A = SubWord(SBT,w));
    thus not ((i mod m) = 0) & not (m = 8 & (i mod 8) = 4) implies
    ex A being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st A = w;
  end;
  uniqueness;
  consistency;
end;

definition
  let SBT;
  let m be Nat;
  assume AS: (m = 4 or m = 6 or m = 8);
  func KeyExpansionX(SBT,m) ->
  Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) means
  for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (for i be Element of NAT st i < m holds (it.Key).(i+1) = Key.(i+1))
  & (for i be Element of NAT st m <= i & i < 4*(7+m) holds
  ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
  Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
  st P = (it.Key).((i-m)+1) & Q = (it.Key).i &
  (it.Key).(i+1) = Op-WXOR(P,KeyExTemp(SBT,m,i,Q)));
  existence
  proof
    defpred P0[Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))] means
    (for i be Element of NAT st i < m holds $2.(i+1) = $1.(i+1))&
    (for i be Element of NAT st m <= i & i < 4*(7+m) holds
    ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
    Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
    st P = $2.((i-m)+1) & Q = ($2).i &
    $2.(i+1) = Op-WXOR(P,KeyExTemp(SBT,m,i,Q)));
A1: for x being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
    st P0[x,z]
    proof
      let x be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      defpred PP[Nat,set,set] means
      ex r,t be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      st r = $2 & t = $3 &
      (ex P0,Q0 be Element of 4-tuples_on (8-tuples_on BOOLEAN)
      st P0 = r.1 & Q0 = r.m & t.1 = Op-WXOR(P0,KeyExTemp(SBT,m,m*$1,Q0))) &
      for i be Nat st 1 <= i & i < m holds
      ex P be Element of 4-tuples_on (8-tuples_on BOOLEAN),
      Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
      st P = r.(i+1) & Q = t.i &
      t.(i+1) = Op-WXOR(P,KeyExTemp(SBT,m,m*$1+i,Q));
      0+m <= 7+m by XREAL_1:6;
      then
LMMLT47M: 1*m <= 4*(7+m) by XREAL_1:66;
      reconsider N2 = (4*(7+m) div m )+1 as Nat;
YY1:  for k being Nat st 1 <= k & k < N2
      for s being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      ex y being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      st PP[k,s,y]
      proof
        let k be Nat;
        assume 1 <= k & k < N2;
        let s be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
        defpred PX[Nat,set,set] means
        ex P,Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
        st P = s.($1+1) & Q = $2 &
        $3 = Op-WXOR(P,KeyExTemp(SBT,m,m*k+$1,Q));
        s in m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
        then ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
        st s = v & len v = m;
        then
QQ3:    dom s = Seg m by FINSEQ_1:def 3;
XX1:    for i being Nat st 1 <= i & i < m
        for z being Element of (4-tuples_on (8-tuples_on BOOLEAN))
        ex w being Element of (4-tuples_on (8-tuples_on BOOLEAN))
        st PX[i,z,w]
        proof
          let i be Nat;
          assume
AA1:      1 <= i & i < m;
          let z be Element of (4-tuples_on (8-tuples_on BOOLEAN));
          1 <= i+1 & i+1 <= m by NAT_1:13,AA1;
          then i+1 in Seg m;
          then s.(i+1) in rng s by QQ3,FUNCT_1:3;
          then reconsider P = s.(i+1) as
          Element of 4-tuples_on (8-tuples_on BOOLEAN);
          reconsider Q = z as Element of 4-tuples_on (8-tuples_on BOOLEAN);
          Op-WXOR(P,KeyExTemp(SBT,m,m*k+i,Q)) is
          Element of (4-tuples_on (8-tuples_on BOOLEAN));
          hence thesis;
        end;
        1 in dom s by AS,QQ3;
        then s.1 in rng s by FUNCT_1:3;
        then reconsider P0 = s.1 as
        Element of 4-tuples_on (8-tuples_on BOOLEAN);
        m in dom s by AS,QQ3;
        then s.m in rng s by FUNCT_1:3;
        then reconsider Q0 = s.m as
        Element of 4-tuples_on (8-tuples_on BOOLEAN);
        reconsider A0 = Op-WXOR(P0,KeyExTemp(SBT,m,m*k,Q0)) as
        Element of 4-tuples_on (8-tuples_on BOOLEAN);
        consider y being FinSequence of (4-tuples_on (8-tuples_on BOOLEAN))
        such that
A2:     len y = m & (y.1 = A0 or m = 0) & for i be Nat st 1 <= i &
        i < m holds PX[i,y.i,y.(i+1)] from RECDEF_1:sch 4 (XX1);
        y in (4-tuples_on (8-tuples_on BOOLEAN))* by FINSEQ_1:def 11;
        then y in m-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) by A2;
        hence thesis by AS,A2;
      end;
      consider z being
      FinSequence of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
A2:   len z = N2 & (z.1 = x or N2 = 0) & for k be Nat st 1 <= k &
      k < N2 holds PP[k,z.k,z.(k+1)] from RECDEF_1:sch 4 (YY1);
      defpred Q0[Nat,set] means
      ex i,j be Element of NAT,
      zi be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
      st (($1 mod m) <> 0 implies i = ($1 div m)+1 & j = $1 mod m) &
      (($1 mod m) = 0 implies i = ($1 div m) & j = m) & zi = z.i & $2 = zi.j;
YY2:  for k be Nat st k in Seg (4*(7+m))
      ex w being Element of (4-tuples_on (8-tuples_on BOOLEAN)) st Q0[k,w]
      proof
        let k be Nat;
        assume
A1:     k in Seg (4*(7+m));
QQ1:    1 <= k & k <= 4*(7+m) by A1,FINSEQ_1:1;
        then
QQ2:    k div m <= (4*(7+m)) div m by NAT_2:24;
        per cases;
        suppose
C1:       (k mod m) <> 0;
          reconsider j = (k mod m) as Element of NAT;
          reconsider i = (k div m)+1 as Element of NAT;
          1 <= i & i <= N2 by QQ2,XREAL_1:6,NAT_1:11;
          then i in Seg N2;
          then i in dom z by A2,FINSEQ_1:def 3;
          then z.i in rng z by FUNCT_1:3;
          then reconsider zi = z.i as
          Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
          zi in m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
          then ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
          st zi = v & len v = m;
          then
Q0:       dom zi = Seg m by FINSEQ_1:def 3;
          1 <= j & j <= m by C1,INT_1:58,AS,NAT_1:14;
          then j in dom zi by Q0;
          then zi.j in rng zi by FUNCT_1:3;
          then reconsider w = zi.j as
          Element of (4-tuples_on (8-tuples_on BOOLEAN));
          ((k mod m) <> 0 implies i = (k div m)+1 & j = k mod m) &
          ((k mod m) = 0 implies i = (k div m) & j = m) & zi = z.i &
          w = zi.j by C1;
          hence thesis;
        end;
        suppose
C2:       (k mod m) = 0;
          reconsider j = m as Element of NAT by ORDINAL1:def 12;
          reconsider i = (k div m) as Element of NAT;
QQ3:      1 <= i by NAT_D:24,QQ1,C2,NAT_2:13,AS;
          (k div m)+0 <= ((4*(7+m)) div m)+1 by QQ2,XREAL_1:7;
          then i in Seg N2 by QQ3;
          then i in dom z by A2,FINSEQ_1:def 3;
          then z.i in rng z by FUNCT_1:3;
          then reconsider zi = z.i as
          Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
          zi in m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
          then ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
          st zi = v & len v = m;
          then
Q0:       dom zi = Seg m by FINSEQ_1:def 3;
          j in Seg m by AS;
          then zi.j in rng zi by Q0,FUNCT_1:3;
          then reconsider w = zi.j as
          Element of (4-tuples_on (8-tuples_on BOOLEAN));
          ((k mod m) <> 0 implies i = (k div m)+1 & j = k mod m) &
          ((k mod m) = 0 implies i = (k div m) & j = m) & zi = z.i &
          w = zi.j by C2;
          hence thesis;
        end;
      end;
      consider u being FinSequence of (4-tuples_on (8-tuples_on BOOLEAN))
      such that
YY3:  dom u = Seg (4*(7+m)) & for k be Nat st k in Seg (4*(7+m))
      holds Q0[k,u.k] from FINSEQ_1:sch 5(YY2);
      4*(7+m) is Element of NAT by ORDINAL1:def 12;
      then
YY4:  len u = 4*(7+m) by YY3,FINSEQ_1:def 3;
      u in ((4-tuples_on (8-tuples_on BOOLEAN)))* by FINSEQ_1:def 11;
      then u in (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
      by YY4;
      then reconsider u as
      Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      take u;
LX3:  for i be Element of NAT st i < m holds u.(i+1) = x.(i+1)
      proof
        let k be Element of NAT;
        assume k < m;
        then
LX31:   1 <= k+1 & k+1 <= m by NAT_1:11,NAT_1:13;
        then 1 <= k+1 & k+1 <= 4*(7+m) by LMMLT47M,XXREAL_0:2;
        then k+1 in Seg (4*(7+m));
        then consider i,j be Element of NAT,
        zi be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        such that
LX34:   (((k+1) mod m) <> 0 implies i = ((k+1) div m)+1 &
        j = (k+1) mod m) & (((k+1) mod m) = 0 implies
        i = ((k+1) div m) & j = m) & zi = z.i & u.(k+1) = zi.j by YY3;
        per cases;
        suppose
C1:       ((k+1) mod m) <> 0;
C11:      (k+1) < m
          proof
            assume not (k+1) < m;
            then (k+1) = m by XXREAL_0:1,LX31;
            hence contradiction by NAT_D:25,C1;
          end;
          then (k+1) div m = 0 by NAT_D:27;
          hence u.(k+1) = x.(k+1) by C11,NAT_D:24,LX34,A2;
        end;
        suppose
C2:       ((k+1) mod m) = 0;
          (k+1) = m
          proof
            assume not (k+1) = m;
            then k+1 < m by LX31,XXREAL_0:1;
            hence contradiction by NAT_D:24,C2;
          end;
          hence u.(k+1) = x.(k+1) by LX34,C2,INT_1:49,A2;
        end;
      end;
      for k be Element of NAT st m <= k & k < 4*(7+m) holds
      ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
      Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
      st P = u.((k-m)+1) & Q = u.k & u.(k+1) = Op-WXOR(P,KeyExTemp(SBT,m,k,Q))
      proof
        let k be Element of NAT;
        assume
AS1:    m <= k & k < 4*(7+m);
        then 1 <= k & k <= 4*(7+m) by XXREAL_0:2,AS;
        then k in Seg (4*(7+m));
        then consider i,j be Element of NAT,
        zi be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        such that
LX34:   ((k mod m) <> 0 implies i = (k div m)+1 & j = k mod m) &
        ((k mod m) = 0 implies i = (k div m) & j = m) & zi = z.i &
        u.k = zi.j by YY3;
NLX32:  1 <= k+1 & k+1 <= 4*(7+m) by AS1,NAT_1:11,NAT_1:13;
        then k+1 in Seg (4*(7+m));
        then consider i1,j1 be Element of NAT,
        zi1 be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        such that
NLX34:  (((k+1) mod m) <> 0 implies i1 = ((k+1) div m)+1 &
        j1 = (k+1) mod m) & (((k+1) mod m) = 0 implies i1 = ((k+1) div m) &
        j1 = m) & zi1 = z.i1 & u.(k+1) = zi1.j1 by YY3;
        reconsider km0 = k-m as Element of NAT by AS1,XREAL_1:48,INT_1:3;
        reconsider km1 = km0+1 as Element of NAT;
        k+1-m <= 4*(7+m)-0 by NLX32,XREAL_1:13;
        then 1 <= km1 & km1 <= 4*(7+m) by NAT_1:11;
        then km1 in Seg (4*(7+m));
        then consider i2,j2 be Element of NAT,
        zi2 be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        such that
LLX34:  ((km1 mod m) <> 0 implies i2 = (km1 div m)+1 & j2 = km1 mod m) &
        ((km1 mod m) = 0 implies i2 = (km1 div m) & j2 = m) &
        zi2 = z.i2 & u.km1 = zi2.j2 by YY3;
        per cases;
        suppose
C1:       (k mod m) <> 0;
          reconsider i0 = k div m as Element of NAT;
DD1:      ((4*(7+m)) div m)+0 < ((4*(7+m)) div m)+1 by XREAL_1:8;
          k div m <= (4*(7+m)) div m by AS1,NAT_2:24;
          then 1 <= i0 & i0 < N2 by DD1,XXREAL_0:2,AS,NAT_2:13,AS1;
          then consider r,t be
          Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
C16:      r = z.i0 & t = z.(i0+1) &
          (ex P0,Q0 be Element of 4-tuples_on (8-tuples_on BOOLEAN)
          st P0 = r.1 & Q0 = r.m & t.1 = Op-WXOR(P0,KeyExTemp(SBT,m,m*i0,Q0)))
          & for n be Nat st 1 <= n & n < m holds
          ex P be Element of 4-tuples_on (8-tuples_on BOOLEAN),
          Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
          st P = r.(n+1) & Q = t.n &
          t.(n+1) = Op-WXOR(P,KeyExTemp(SBT,m,i0*m+n,Q)) by A2;
          1 <= j & j < m by AS,INT_1:58,LX34,C1,NAT_1:14;
          then consider P be Element of 4-tuples_on (8-tuples_on BOOLEAN),
          Q be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
C18:      P = r.(j+1) & Q = t.j &
          t.(j+1) = Op-WXOR(P,KeyExTemp(SBT,m,i0*m+j,Q)) by C16;
          per cases;
          suppose
NC1:        ((k+1) mod m) <> 0;
NC16:       zi1 = zi by NLX34,NC1,AS,XLMOD01,LX34,C1;
C21:        u.(k+1) = t.(j+1) by NLX34,NC16,NC1,AS,XLMOD02,LX34,C1,C16;
C22X:       km1 = (k+1)-m;
LC12:       i2 = ((k+1) div m)-1+1 by NC1,XLMOD03,C22X,LLX34,AS,XLMOD04
              .= i0 by AS,XLMOD01,NC1;
LC13:       j2 = j1 by LLX34,C22X,XLMOD03,NLX34;
C19:        u.(k-m+1) = r.(j+1)
            by LLX34,LC13,LC12,C16,NLX34,NC1,AS,XLMOD02,LX34,C1;
C22:        k = i0*m+j by AS,INT_1:59,LX34,C1;
            thus ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
            Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
            st P = u.((k-m)+1) & Q = u.k &
            u.(k+1) = Op-WXOR(P,KeyExTemp(SBT,m,k,Q))
            by C18,C19,C16,LX34,C1,C21,C22;
          end;
          suppose
MC1:        ((k+1) mod m) = 0;
NC13:       j1 = m-1+1 by NLX34,MC1
              .= j+1 by AS,XLMOD02X,MC1,LX34;
C21:        u.(k+1) = t.(j+1) by NLX34,MC1,XLMOD01X,NC13,C16;
C22X:       km1 = (k+1)-m;
LC12:       i2 = ((k+1) div m)-1 by C22X,MC1,XLMOD03,LLX34,AS,XLMOD04
              .= (k div m)+1-1 by AS,XLMOD01X,MC1
              .= i0;
C19:        u.(k-m+1) = r.(j+1) by LLX34,C22X,XLMOD03,NLX34,LC12,C16,NC13;
C22:        k = i0*m+j by AS,INT_1:59,LX34,C1;
            thus ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
            Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
            st P = u.((k-m)+1) & Q = u.k &
            u.(k+1) = Op-WXOR(P,KeyExTemp(SBT,m,k,Q))
            by C18,C19,LX34,C16,C1,C21,C22;
          end;
        end;
        suppose
C2:       (k mod m) = 0;
DD1:      ((4*(7+m)) div m)+0 < ((4*(7+m)) div m)+1 by XREAL_1:8;
          k div m <= (4*(7+m)) div m by AS1,NAT_2:24;
          then 1 <= i & i < N2 by DD1,XXREAL_0:2,C2,LX34,AS,NAT_2:13,AS1;
          then consider r,t be
          Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
          such that
C16:      r = z.i & t = z.(i+1) &
          (ex P0,Q0 be Element of 4-tuples_on (8-tuples_on BOOLEAN)
          st P0 = r.1 & Q0 = r.m & t.1 = Op-WXOR(P0,KeyExTemp(SBT,m,m*i,Q0)))
          & for n be Nat st 1 <= n & n < m holds
          ex P be Element of 4-tuples_on (8-tuples_on BOOLEAN),
          Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
          st P = r.(n+1) & Q = t.n &
          t.(n+1) = Op-WXOR(P,KeyExTemp(SBT,m,i*m+n,Q)) by A2;
NC1X:     ((k+1) mod m) = ((0 qua Nat)+1) mod m by C2,NAT_D:23
            .= 1 by NAT_D:14,AS;
C21:      u.(k+1) = t.1 by NLX34,NC1X,AS,XLMOD01,C2,LX34,C16;
C22X:     km1 = (k+1)-m;
LC12:     i2 = ((k+1) div m)-1+1 by NC1X,XLMOD03,C22X,LLX34,AS,XLMOD04
            .= i by AS,XLMOD01,NC1X,C2,LX34;
C19:      u.(k-m+1) = r.1 by LLX34,XLMOD03,C22X,LC12,C16,NC1X;
C22:      k = (k div m)*m+(k mod m) by AS,INT_1:59
            .= i*m by C2,LX34;
          thus ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
          Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
          st P = u.((k-m)+1) & Q = u.k &
          u.(k+1) = Op-WXOR(P,KeyExTemp(SBT,m,k,Q))
          by C19,LX34,C16,C2,C21,C22;
        end;
      end;
      hence P0[x,u] by LX3;
    end;
    consider I be Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
    such that
A2: for x being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let H1,H2 be Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
    assume
AA1: for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Element of NAT st i < m holds (H1.Key).(i+1) = Key.(i+1))
    & (for i be Element of NAT st m <= i & i < 4*(7+m) holds
    ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
    Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
    st P = (H1.Key).((i-m)+1) & Q = (H1.Key).i &
    (H1.Key).(i+1) = Op-WXOR(P,KeyExTemp(SBT,m,i,Q)));
    assume
AA2: for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds (for i be Element of NAT st i < m holds (H2.Key).(i+1) = Key.(i+1))
    & (for i be Element of NAT st m <= i & i < 4*(7+m) holds
    ex P be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
    Q be Element of 4-tuples_on (8-tuples_on BOOLEAN)
    st P = (H2.Key).((i-m)+1) & Q = (H2.Key).i &
    (H2.Key).(i+1) = Op-WXOR(P,KeyExTemp(SBT,m,i,Q)));
    now let input be
    Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      (H1.input) in (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      then
XX1:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st (H1.input) = s & len s = (4*(7+m));
      reconsider H1i = (H1.input) as
      Element of (4-tuples_on (8-tuples_on BOOLEAN))* by XX1;
      (H2.input) in (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      then
XX2:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st (H2.input) = s & len s = (4*(7+m));
      reconsider H2i = (H2.input) as
      Element of (4-tuples_on (8-tuples_on BOOLEAN))* by XX2;
      defpred PN[Nat] means
      (m <= $1 & $1 <= 4*(7+m)) implies for k be Element of NAT
      st 1 <= k & k <= $1 holds (H1.input).k = (H2.input).k;
PN0:  PN[0];
PN1:  for i be Nat st PN[i] holds PN[i+1]
      proof
        let i be Nat;
        assume
A1:     PN[i];
        assume
A2:     m <= i+1 & i+1 <= 4*(7+m);
        per cases;
        suppose
C10:      m = i+1;
          thus for k be Element of NAT st 1 <= k & k <= i+1 holds
          (H1.input).k = (H2.input).k
          proof
            let k be Element of NAT;
            assume
B1:         1 <= k & k <= i+1;
            k-1 < k by XREAL_1:44;
            then
B2:         k-1 < m by C10,B1,XXREAL_0:2;
            reconsider k1 = k-1 as Element of NAT by XREAL_1:48,B1,INT_1:3;
            thus (H1.input).k = input.(k1+1) by B2,AA1
              .= (H2.input).k by B2,AA2;
          end;
        end;
        suppose m <> i+1;
          then
C10X:     m < i+1 by A2,XXREAL_0:1;
          i < i+1 by XREAL_1:29;
          then
C11Z:     i < 4*(7+m) by A2,XXREAL_0:2;
          thus for k be Element of NAT st 1 <= k & k <= i+1 holds
          (H1.input).k = (H2.input).k
          proof
            let k be Element of NAT;
            assume
C13:        1 <= k & k <= i+1;
            then reconsider k1 = k-1 as Element of NAT by XREAL_1:48,INT_1:3;
            per cases;
            suppose
C14:          k1 < m;
              thus (H1.input).k = input.(k1+1) by C14,AA1
                .= (H2.input).k by C14,AA2;
            end;
            suppose
C15:          m <= k1;
              k-1 <= i+1-1 by C13,XREAL_1:9;
              then
C16:          m <= k1 & k1 < 4*(7+m) by C11Z,XXREAL_0:2,C15;
              then consider PP1 be
              Element of (4-tuples_on (8-tuples_on BOOLEAN)),
              QQ1 be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
C17:          PP1 = (H1.input).((k1-m)+1) & QQ1 = (H1.input).k1 &
              (H1.input).(k1+1) = Op-WXOR(PP1,KeyExTemp(SBT,m,k1,QQ1)) by AA1;
              consider PP2 be Element of (4-tuples_on (8-tuples_on BOOLEAN)),
              QQ2 be Element of 4-tuples_on (8-tuples_on BOOLEAN) such that
C18:          PP2 = (H2.input).((k1-m)+1) & QQ2 = (H2.input).k1 &
              (H2.input).(k1+1) = Op-WXOR(PP2,KeyExTemp(SBT,m,k1,QQ2))
              by AA2,C16;
C190:         k-1 <= i+1-1 by XREAL_1:9,C13;
              then
C191:         1 <= k1 & k1 <= i by C15,AS,XXREAL_0:2;
C24X:         0 <= k1-m by C15,XREAL_1:48;
              then
C25X:         1+0 <= k1-m+1 by XREAL_1:6;
              k1-(m-1) <= k1 by AS,XREAL_1:43;
              then
C25:          1 <= (k1-m)+1 & (k1-m)+1 <= i by C190,XXREAL_0:2,C25X;
              reconsider k1m1 = (k1-m)+1 as Element of NAT by C24X,INT_1:3;
C21:          (H1.input).k1m1 = (H2.input).k1m1 by A2,C10X,NAT_1:13,A1,C25;
              thus (H1.input).k = (H2.input).k
              by C21,C17,C18,C191,A2,C10X,NAT_1:13,A1;
            end;
          end;
        end;
      end;
L10:  for i be Nat holds PN[i] from NAT_1:sch 2(PN0,PN1);
L1:   now let i be Element of NAT;
        assume
A1:     m <=i & i <= 4*(7+m);
        1 <= i & i <= i by AS,A1,XXREAL_0:2;
        hence (H1.input).i = (H2.input).i by L10,A1;
      end;
      now let i0 be Nat;
        assume
P13:    1 <= i0 & i0 <= len H1i;
        then reconsider i = i0-1 as Element of NAT by XREAL_1:48,INT_1:3;
        now per cases;
        suppose
C1:       i0 <= m;
          i < i0 by XREAL_1:44;
          then
C11:      i < m by C1,XXREAL_0:2;
          thus H1i.i0 = input.(i+1) by C11,AA1
            .= H2i.i0 by C11,AA2;
        end;
        suppose
C3:       m < i0;
          i+1 in Seg len H1i by P13;
          hence H1i.i0 = H2i.i0 by L1,C3,XX1,P13;
        end;
      end;
      hence H1i.i0 = H2i.i0;
    end;
    hence H1.input = H2.input by XX1,XX2,FINSEQ_1:def 17;
  end;
  hence H1 = H2 by FUNCT_2:63;
end;
end;

definition
  let SBT;
  let m be Nat;
  func KeyExpansion(SBT,m) ->
  Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) means
  for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) holds
  ex w be Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
  st w = (KeyExpansionX(SBT,m)).Key & for i be Nat st i < 7+m
  holds (it.Key).(i+1) = <*w.(4*i+1),w.(4*i+2),w.(4*i+3),w.(4*i+4)*>;
  existence
  proof
    defpred P0[Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    Element of (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))]
    means
    ex w be Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
    st w = (KeyExpansionX(SBT,m)).$1 & for i be Nat st i < 7+m holds
    $2.(i+1) = <*w.(4*i+1),w.(4*i+2),w.(4*i+3),w.(4*i+4)*>;
A1: for x being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    ex z be
    Element of (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
    st P0[x,z]
    proof
      let x be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      reconsider w = (KeyExpansionX(SBT,m)).x as
      Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      w in (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN));
      then
XX1:  ex s be Element of (4-tuples_on (8-tuples_on BOOLEAN))*
      st w = s & len s = (4*(7+m));
      reconsider w0 = w as
      Element of (4-tuples_on (8-tuples_on BOOLEAN))* by XX1;
      reconsider m7 = 7+m as Element of NAT by ORDINAL1:def 12;
      reconsider m47 = (4*(7+m)) as Element of NAT by ORDINAL1:def 12;
      defpred P[Nat,set] means
      ex n be Element of NAT st n = $1-1 &
      $2 = <*w.(4*n+1),w.(4*n+2),w.(4*n+3),w.(4*n+4)*>;
P1:   for k be Nat st k in Seg m7
      ex z being Element of 4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
      st P[k,z]
      proof
        let k be Nat;
        assume k in Seg m7;
        then
ZZ1:    1 <= k & k <= m7 by FINSEQ_1:1;
        then reconsider n = k-1 as Element of NAT by XREAL_1:48,INT_1:3;
ZZ3:    4*(n+1) <= 4*m7 by ZZ1,XREAL_1:64;
ZZ4:    0+1 <= 4*n+1 by XREAL_1:7;
ZZ7:    4*n+1 <= 4*n+4 by XREAL_1:7;
ZZ8:    4*n+2 <= 4*n+4 by XREAL_1:7;
ZZ9:    4*n+3 <= 4*n+4 by XREAL_1:7;
        4*n+1 <= 4*m7 by ZZ7,ZZ3,XXREAL_0:2;
        then
X1:     (4*n+1) in Seg m47 by ZZ4;
ZZ10:   1 <= 4*n+2 by ZZ4,XREAL_1:7;
        4*n+2 <= 4*m7 by ZZ8,ZZ3,XXREAL_0:2;
        then
X2:     (4*n+2) in Seg m47 by ZZ10;
ZZ11:   1 <= 4*n+3 by ZZ4,XREAL_1:7;
        4*n+3 <= 4*m7 by ZZ9,ZZ3,XXREAL_0:2;
        then
X3:     (4*n+3) in Seg m47 by ZZ11;
ZZ12:   1 <= 4*n+4 by ZZ4,XREAL_1:7;
X4:     (4*n+4) in Seg m47 by ZZ3,ZZ12;
X5:     dom w = Seg m47 by FINSEQ_1:def 3,XX1;
        w.(4*n+1) in rng w by X5,X1,FUNCT_1:3;
        then reconsider w1 = w.(4*n+1) as
        Element of (4-tuples_on (8-tuples_on BOOLEAN));
        w.(4*n+2) in rng w by X5,X2,FUNCT_1:3;
        then reconsider w2 = w.(4*n+2) as
        Element of (4-tuples_on (8-tuples_on BOOLEAN));
        w.(4*n+3) in rng w by X5,X3,FUNCT_1:3;
        then reconsider w3 = w.(4*n+3) as
        Element of (4-tuples_on (8-tuples_on BOOLEAN));
        w.(4*n+4) in rng w by X5,X4,FUNCT_1:3;
        then reconsider w4 = w.(4*n+4) as
        Element of (4-tuples_on (8-tuples_on BOOLEAN));
        reconsider z = <*w1,w2,w3,w4*> as
        Element of 4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) by LMGSEQ4;
        z = <*w.(4*n+1),w.(4*n+2),w.(4*n+3),w.(4*n+4)*>;
        hence thesis;
      end;
      consider p being
      FinSequence of 4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) such that
P3:   dom p = Seg m7 & for k be Nat st k in Seg m7 holds P[k,p.k]
      from FINSEQ_1:sch 5(P1);
P4:   len p = m7 by P3,FINSEQ_1:def 3;
      p in (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
      by FINSEQ_1:def 11;
      then p in m7-tuples_on (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))
      by P4;
      then reconsider p as Element of
      (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
      take p;
      now let i be Nat;
        assume i < 7+m;
        then
AA2:    i+1 <= 7+m by NAT_1:13;
        1 <= i+1 by NAT_1:11;
        then i+1 in Seg m7 by AA2;
        then ex n be Element of NAT st n = (i+1)-1 &
        p.(i+1) = <*w.(4*n+1),w.(4*n+2),w.(4*n+3),w.(4*n+4)*> by P3;
        hence p.(i+1) = <*w.(4*i+1),w.(4*i+2),w.(4*i+3),w.(4*i+4)*>;
      end;
      hence thesis;
    end;
    consider I be Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
A2: for x being Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds P0[x,I.x] from FUNCT_2:sch 3(A1);
    take I;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let H1,H2 be Function of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
    assume
A1: for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) holds
    ex w be Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
    st w = (KeyExpansionX(SBT,m)).Key & for i be Nat st i < 7+m holds
    (H1.Key).(i+1) = <*w.(4*i+1),w.(4*i+2),w.(4*i+3),w.(4*i+4)*>;
    assume
A2: for Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) holds
    ex w be Element of (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN))
    st w = (KeyExpansionX(SBT,m)).Key & for i be Nat st i <7+m holds
    (H2.Key).(i+1) = <*w.(4*i+1),w.(4*i+2),w.(4*i+3),w.(4*i+4)*>;
    now let input be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      consider w1 be Element of
      (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) such that
P1:   w1 = (KeyExpansionX(SBT,m)).input &
      for i be Nat st i < 7+m holds
      (H1.input).(i+1) = <*w1.(4*i+1),w1.(4*i+2),w1.(4*i+3),w1.(4*i+4)*> by A1;
      consider w2 be Element of
      (4*(7+m))-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)) such that
P2:   w2 = (KeyExpansionX(SBT,m)).input &
      for i be Nat st i < 7+m holds
      (H2.input).(i+1) = <*w2.(4*i+1),w2.(4*i+2),w2.(4*i+3),w2.(4*i+4)*> by A2;
      (H1.input) in
      (7+m)-tuples_on(4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)));
      then
P3:   ex s be Element of (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
      st (H1.input) = s & len s = (7+m);
      (H2.input) in
      (7+m)-tuples_on (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
      then
P4:   ex s be Element of (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
      st (H2.input) = s & len s = (7+m);
      now let i be Nat;
        assume
P5:     1 <= i & i <= len (H1.input);
        then i-1 in NAT by XREAL_1:48,INT_1:3;
        then reconsider i0 = i-1 as Nat;
        i < (7+m)+1 by P3,P5,NAT_1:13;
        then
P6:     i-1 < (7+m)+1-1 by XREAL_1:14;
        thus (H1.input).i = (H1.input).(i0+1)
          .= <*w2.(4*i0+1),w2.(4*i0+2),w2.(4*i0+3),w2.(4*i0+4)*> by P6,P1,P2
          .= (H2.input).(i0+1) by P6,P2
          .= (H2.input).i;
      end;
      hence
      (H1.input) = (H2.input) by P3,P4,FINSEQ_1:def 17;
    end;
    hence H1 = H2 by FUNCT_2:63;
  end;
end;

begin :: Encryption and Decryption

reserve MCFunc for Permutation of
4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
reserve MixColumns for
Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));

definition
  let SBT;
  let MCFunc;
  let m be Nat;
  let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  let Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  func AES-ENC(SBT,MCFunc,text,Key) ->
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :defENC:
  ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
  st len seq = 7+m-1 & (ex Keyi1 be Element of
  4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
  seq.1 = AddRoundKey.(text,Keyi1)) &
  (for i be Nat st 1 <= i & i < 7+m-1 holds
  ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) &
  seq.(i+1) = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq.i),Keyi)) &
  ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
  it = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq.(7+m-1)),KeyNr);
  existence
  proof
    1+0 < 7+m by XREAL_1:8;
    then
N1: 0 < 7+m-1 by XREAL_1:50;
    then 7+m-1 in NAT by INT_1:3;
    then reconsider Nr = 7+m-1 as Nat;
ZZ1: (KeyExpansion(SBT,m)).(Key) in
    (Nr+1)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
    reconsider kky = (KeyExpansion(SBT,m)).(Key) as Element of
    (Nr+1)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
XX12: ex s be Element of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))*
    st kky = s & len s = (Nr+1) by ZZ1;
    defpred P[Nat,set,set] means
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = ((KeyExpansion(SBT,m)).(Key)).($1+1) &
    $3 = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).$2,Keyi);
    1+0 <= 7+m by XREAL_1:7;
    then 1 in Seg (Nr+1);
    then 1 in dom (kky) by FINSEQ_1:def 3,XX12;
    then ((KeyExpansion(SBT,m)).(Key)).1 in rng kky by FUNCT_1:3;
    then reconsider Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    reconsider I0 = AddRoundKey.(text,Keyi1) as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
X1: for n being Nat st 1 <= n & n < Nr
    for z being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds ex y being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P[n,z,y]
    proof
      let n be Nat;
      assume
X11:  1 <= n & n < Nr;
      let z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
X111: n+1 <= Nr+1 by XREAL_1:7,X11;
      0+1 <= n+1 by XREAL_1:7;
      then n+1 in Seg (Nr+1) by X111;
      then n+1 in dom (kky) by FINSEQ_1:def 3,XX12;
      then ((KeyExpansion(SBT,m)).(Key)).(n+1) in rng kky by FUNCT_1:3;
      then reconsider Keyi = ((KeyExpansion(SBT,m)).(Key)).(n+1) as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      reconsider y = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).z,Keyi) as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      take y;
      thus P[n,z,y];
    end;
    consider seq be
    FinSequence of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
X2: len seq = Nr & (seq.1 = I0 or Nr = 0) &
    for i be Nat st 1 <= i & i < Nr holds P[i,seq.i,seq.(i+1)]
    from RECDEF_1:sch 4(X1);
    Nr in Seg Nr by FINSEQ_1:3,N1;
    then
    Nr in dom seq by FINSEQ_1:def 3,X2;
    then seq.Nr in rng seq by FUNCT_1:3;
    then reconsider seq10 = seq.Nr as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    Nr+1 in Seg (Nr+1) by FINSEQ_1:3;
    then Nr+1 in dom (kky) by FINSEQ_1:def 3,XX12;
    then ((KeyExpansion(SBT,m)).(Key)).(Nr+1) in rng kky by FUNCT_1:3;
    then reconsider KeyNr = ((KeyExpansion(SBT,m)).(Key)).(Nr+1) as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    reconsider w = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq10),KeyNr) as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    w = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq.Nr),KeyNr);
    hence thesis by XREAL_1:8,X2;
  end;
  uniqueness
  proof
    let s1,s2 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    1+0 < 7+m by XREAL_1:8;
    then 0 < 7+m-1 by XREAL_1:50;
    then 7+m-1 in NAT by INT_1:3;
    then reconsider Nr = 7+m-1 as Nat;
    assume
A1: ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
    st len seq = 7+m-1 &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
    seq.1 = AddRoundKey.(text,Keyi1)) & (for i be Nat st 1 <= i & i < 7+m-1
    holds ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) &
    seq.(i+1) = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq.i),Keyi)) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
    s1 = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq.(7+m-1)),KeyNr);
    assume
A2: ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
    st len seq = 7+m-1 &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
    seq.1 = AddRoundKey.(text,Keyi1)) & (for i be Nat st 1 <= i & i < 7+m-1
    holds ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) &
    seq.(i+1) = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq.i),Keyi)) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
    s2 = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq.(7+m-1)),KeyNr);
    consider seq1 be
    FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P1: len seq1 = Nr &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
    seq1.1 = AddRoundKey.(text,Keyi1)) & (for i be Nat st 1 <= i & i < Nr
    holds ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) &
    seq1.(i+1) = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq1.i),Keyi))
    & ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(Nr+1) &
    s1 = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq1.(Nr)),KeyNr) by A1;
    consider seq2 be
    FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P2: len seq2 = Nr &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
    seq2.1 = AddRoundKey.(text,Keyi1)) & (for i be Nat st 1 <= i & i < Nr
    holds ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) &
    seq2.(i+1) = AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq2.i),Keyi))
    & ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(Nr+1) &
    s2 = AddRoundKey.((ShiftRows*SubBytes(SBT)).(seq2.(Nr)),KeyNr) by A2;
    defpred EQ[Nat] means
    1 <= $1 & $1 <= len seq1 implies seq1.$1 = seq2.$1;
Q50: EQ[0];
Q51: for i be Nat st EQ[i] holds EQ[i+1]
    proof
      let i be Nat;
      assume
Q52:  EQ[i];
      assume 1 <= i+1 & i+1 <= len seq1;
      then
Q54:  1-1 <= i+1-1 & i+1-1 <= len seq1-1 by XREAL_1:9;
Q550: (len seq1)-1 <= (len seq1)-0 by XREAL_1:13;
      per cases;
      suppose
C1:     i = 0;
        thus seq1.(i+1) = seq2.(i+1) by C1,P1,P2;
      end;
      suppose
Q560:   i <> 0;
        Nr-1 < Nr-0 by XREAL_1:15;
        then
XX1:    1 <= i & i < Nr by Q560,NAT_1:14,P1,Q54,XXREAL_0:2;
        then
Q60:    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) & seq1.(i+1) =
        AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq1.i),Keyi) by P1;
        ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) & seq2.(i+1) =
        AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(seq2.i),Keyi) by P2,XX1;
        hence seq1.(i+1) = seq2.(i+1)
        by Q560,NAT_1:14,Q550,Q54,XXREAL_0:2,Q52,Q60;
      end;
    end;
    for i be Nat holds EQ[i] from NAT_1:sch 2(Q50,Q51);
    hence s1 = s2 by P1,P2,FINSEQ_1:14;
  end;
end;

definition
  let SBT;
  let MCFunc;
  let m be Nat;
  let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  let Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  func AES-DEC(SBT,MCFunc,text,Key) ->
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  :defDEC:
  ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
  st len seq = 7+m-1 &
  (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  seq.1 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))) &
  (for i be Nat st 1 <= i & i < 7+m-1 holds
  ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
  seq.(i+1) =
  (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq.i,Keyi))) &
  ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
  it = AddRoundKey.(seq.(7+m-1),KeyNr);
  existence
  proof
    1+0 < 7+m by XREAL_1:8;
    then
N1: 0 < 7+m-1 by XREAL_1:50;
    then 7+m-1 in NAT by INT_1:3;
    then reconsider Nr = 7+m-1 as Nat;
ZZ1: Rev((KeyExpansion(SBT,m)).(Key)) in
    (Nr+1)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
    reconsider kky = Rev((KeyExpansion(SBT,m)).(Key)) as Element of
    (Nr+1)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
XX12: ex s be Element of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))*
    st kky = s & len s = Nr+1 by ZZ1;
    defpred P[Nat,set,set] means
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).($1+1) &
    $3 = (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.($2,Keyi));
    1+0 <= 7+m by XREAL_1:7;
    then 1 in Seg (Nr+1);
    then 1 in dom (kky) by FINSEQ_1:def 3,XX12;
    then (Rev((KeyExpansion(SBT,m)).(Key))).1 in rng kky by FUNCT_1:3;
    then reconsider Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    reconsider I0 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))
    as Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
X1: for n being Nat st 1 <= n & n < Nr
    for z being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    holds ex y being Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st P[n,z,y]
    proof
      let n be Nat;
      assume
X11:  1 <= n & n < Nr;
      let z be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
X111: n+1 <= Nr+1 by XREAL_1:7,X11;
      0+1 <= n+1 by XREAL_1:7;
      then n+1 in Seg (Nr+1) by X111;
      then n+1 in dom (kky) by FINSEQ_1:def 3,XX12;
      then (Rev((KeyExpansion(SBT,m)).(Key))).(n+1) in rng kky by FUNCT_1:3;
      then reconsider Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(n+1) as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      reconsider y =
      (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(z,Keyi)) as
      Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
      take y;
      thus P[n,z,y];
    end;
    consider seq be
    FinSequence of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) such that
X2: len seq = Nr & (seq.1 = I0 or Nr = 0) &
    for i be Nat st 1 <= i & i < Nr holds P[i,seq.i,seq.(i+1)]
    from RECDEF_1:sch 4(X1);
    Nr in Seg Nr by FINSEQ_1:3,N1;
    then Nr in dom seq by FINSEQ_1:def 3,X2;
    then seq.Nr in rng seq by FUNCT_1:3;
    then reconsider seq10 = seq.Nr as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    Nr+1 in Seg (Nr+1) by FINSEQ_1:3;
    then Nr+1 in dom (kky) by FINSEQ_1:def 3,XX12;
    then (Rev((KeyExpansion(SBT,m)).(Key))).(Nr+1) in rng kky by FUNCT_1:3;
    then reconsider KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(Nr+1) as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    reconsider w = AddRoundKey.((seq10),KeyNr) as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    w = AddRoundKey.(seq.Nr,KeyNr);
    hence thesis by X2,XREAL_1:8;
  end;
  uniqueness
  proof
    let s1,s2 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    1+0 < 7+m by XREAL_1:8;
    then 0 < 7+m-1 by XREAL_1:50;
    then 7+m-1 in NAT by INT_1:3;
    then reconsider Nr = 7+m-1 as Nat;
    assume
A1: ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
    st len seq = 7+m-1 &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
    seq.1 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))) &
    (for i be Nat st 1 <= i & i < 7+m-1 holds
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
    seq.(i+1) =
    (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq.i,Keyi))) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
    s1 = AddRoundKey.(seq.(7+m-1),KeyNr);
    assume
A2: ex seq be FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))
    st len seq = 7+m-1 &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
    seq.1 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))) &
    (for i be Nat st 1 <= i & i < 7+m-1 holds
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
    seq.(i+1) =
    (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq.i,Keyi))) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
    s2 = AddRoundKey.(seq.(7+m-1),KeyNr);
    consider seq1 be
    FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P1: len seq1 = Nr &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
    seq1.1 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))) &
    (for i be Nat st 1 <= i & i < Nr holds
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
    seq1.(i+1) =
    (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq1.i,Keyi))) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
    s1 = AddRoundKey.(seq1.(7+m-1),KeyNr) by A1;
    consider seq2 be
    FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P2: len seq2 = Nr &
    (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
    seq2.1 = (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(text,Keyi1))) &
    (for i be Nat st 1 <= i & i < Nr holds
    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
    seq2.(i+1) =
    (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq2.i,Keyi))) &
    ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
    s2 = AddRoundKey.(seq2.(7+m-1),KeyNr) by A2;
    defpred EQ[Nat] means
    1 <= $1 & $1 <= len seq1 implies seq1.$1 = seq2.$1;
Q50: EQ[0];
Q51: for i be Nat st EQ[i] holds EQ[i+1]
    proof
      let i be Nat;
      assume
Q52:  EQ[i];
      assume 1 <= i+1 & i+1 <= len seq1;
      then
Q54:  1-1 <= i+1-1 & i+1-1 <= (len seq1)-1 by XREAL_1:9;
Q550: (len seq1)-1 <= (len seq1)-0 by XREAL_1:13;
      per cases;
      suppose
C1:     i = 0;
        thus seq1.(i+1) = seq2.(i+1) by C1,P1,P2;
      end;
      suppose
Q560:   i <> 0;
        Nr-1 < Nr-0 by XREAL_1:15;
        then
XX1:    1 <= i & i < Nr by Q560,NAT_1:14,P1,Q54,XXREAL_0:2;
        then
Q60:    ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
        seq1.(i+1) =
        (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq1.i,Keyi))
        by P1;
        ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
        st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
        seq2.(i+1) =
        (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(seq2.i,Keyi))
        by P2,XX1;
        hence seq1.(i+1) = seq2.(i+1)
        by Q560,NAT_1:14,Q550,Q54,XXREAL_0:2,Q52,Q60;
      end;
    end;
    for i be Nat holds EQ[i] from NAT_1:sch 2(Q50,Q51);
    hence s1 = s2 by FINSEQ_1:14,P1,P2;
  end;
end;

theorem INV01:
  for input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds (MCFunc").(MCFunc.input) = input
proof
  let input be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  thus (MCFunc").(MCFunc.(input)) = ((MCFunc")*MCFunc).input by FUNCT_2:15
    .= (id (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))).input
    by FUNCT_2:61
    .= input;
end;

theorem
  for output be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  holds MCFunc.((MCFunc").output) = output
proof
  let output be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  thus (MCFunc).((MCFunc").(output)) = (MCFunc*(MCFunc")).output by FUNCT_2:15
    .= (id (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)))).output
    by FUNCT_2:61
    .= output;
end;

theorem LAST01:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) holds
  (InvSubBytes(SBT)*InvShiftRows).((ShiftRows*SubBytes(SBT)).text) = text
proof
  let m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  thus (InvSubBytes(SBT)*InvShiftRows).((ShiftRows*SubBytes(SBT)).text) =
  (InvSubBytes(SBT)*InvShiftRows).(ShiftRows.((SubBytes(SBT)).text))
  by FUNCT_2:15
    .= (InvSubBytes(SBT)).(InvShiftRows.(ShiftRows.((SubBytes(SBT)).text)))
    by FUNCT_2:15
    .= (InvSubBytes(SBT)). ((SubBytes(SBT)).text) by INV04
    .= text by INV07;
end;

theorem LAST02:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) holds
  (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).
  ((MCFunc*ShiftRows*SubBytes(SBT)).text) = text
proof
  let m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  thus (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).
  ((MCFunc*ShiftRows*SubBytes(SBT)).text) =
  (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).
  ((MCFunc*ShiftRows).((SubBytes(SBT)).text)) by FUNCT_2:15
    .= (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).
    (MCFunc.(ShiftRows.((SubBytes(SBT)).text))) by FUNCT_2:15
    .= (InvSubBytes(SBT)*InvShiftRows).((MCFunc").
    (MCFunc.(ShiftRows.((SubBytes(SBT)).text)))) by FUNCT_2:15
    .= (InvSubBytes(SBT)*InvShiftRows).(ShiftRows.((SubBytes(SBT)).text))
    by INV01
    .= (InvSubBytes(SBT)).(InvShiftRows.(ShiftRows.((SubBytes(SBT)).text)))
    by FUNCT_2:15
    .= (InvSubBytes(SBT)).((SubBytes(SBT)).text) by INV04
    .= text by INV07;
end;

theorem LAST03:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  dkeyi,ekeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & dkeyi = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  ekeyi = ((KeyExpansion(SBT,m)).(Key)).(7+m)
  holds AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi) = text
proof
  let m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  dkeyi,ekeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & dkeyi = (Rev((KeyExpansion(SBT,m)).(key))).1 &
  ekeyi = ((KeyExpansion(SBT,m)).(key)).(7+m);
  set p = (KeyExpansion(SBT,m)).(key);
  (KeyExpansion(SBT,m)).(key) in
  (7+m)-tuples_on(4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)));
  then
B0: ex s be Element of (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
  st (KeyExpansion(SBT,m)).(key) = s & len s = (7+m);
  1+0 < 7+m by XREAL_1:8;
  then 1 in Seg (7+m);
  then
B1: 1 in dom p by FINSEQ_1:def 3,B0;
A0: dkeyi = p.((len p)-1+1) by AS,FINSEQ_5:58,B1
    .= ekeyi by B0,AS;
  now let i,j be Nat;
    assume
A3: i in Seg 4 & j in Seg 4;
    then consider etextij,ekeyij be Element of 8-tuples_on BOOLEAN such that
A4: etextij = (text.i).j & ekeyij = (ekeyi.i).j &
    ((AddRoundKey.(text,ekeyi)).i).j = Op-XOR(etextij,ekeyij)
    by DefAddRoundKey;
    consider dtextij,dkeyij be Element of 8-tuples_on BOOLEAN such that
A5: dtextij = ((AddRoundKey.(text,ekeyi)).i).j & dkeyij = (dkeyi.i).j &
    ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j =
    Op-XOR(dtextij,dkeyij) by DefAddRoundKey,A3;
    thus ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j = (text.i).j
    by A4,A5,A0,DESCIP_1:17;
  end;
  hence AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi) = text by LM01;
end;

LAST04:
  for m be Nat,
  text,otext be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Keyi1,KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
  otext = AddRoundKey.((ShiftRows*SubBytes(SBT)).text,KeyNr)
  holds (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(otext,Keyi1)) = text
proof
  let m be Nat,
  text,otext be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Keyi1,KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
  otext = AddRoundKey.((ShiftRows*SubBytes(SBT)).text,KeyNr);
  (AddRoundKey.(otext,Keyi1)) = (ShiftRows*SubBytes(SBT)).text by AS,LAST03;
  hence (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey. (otext,Keyi1))
  = text by LAST01;
end;

theorem LAST05:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  dkeyi,ekeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & dkeyi = ((KeyExpansion(SBT,m)).(key)).1 &
  ekeyi = (Rev((KeyExpansion(SBT,m)).(key))).(7+m)
  holds AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi) = text
proof
  let m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  dkeyi,ekeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & dkeyi = ((KeyExpansion(SBT,m)).(key)).1 &
  ekeyi = (Rev((KeyExpansion(SBT,m)).(key))).(7+m);
  set p = (KeyExpansion(SBT,m)).(key);
  (KeyExpansion(SBT,m)).(key) in
  (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
  then
B0: ex s be Element of (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
  st (KeyExpansion(SBT,m)).(key) = s & len s = (7+m);
  1+0 < 7+m by XREAL_1:8;
  then 7+m in Seg (7+m);
  then
B1: 7+m in dom p by FINSEQ_1:def 3,B0;
A0: ekeyi = p.((len p)-(7+m)+1) by AS,FINSEQ_5:58,B1
    .= dkeyi by B0,AS;
  now let i,j be Nat;
    assume
A3: i in Seg 4 & j in Seg 4;
    then consider etextij,ekeyij be Element of 8-tuples_on BOOLEAN such that
A4: etextij = (text.i).j & ekeyij = (ekeyi.i).j &
    ((AddRoundKey.(text,ekeyi)).i).j = Op-XOR(etextij,ekeyij)
    by DefAddRoundKey;
    consider dtextij,dkeyij be Element of 8-tuples_on BOOLEAN such that
A5: dtextij = ((AddRoundKey.(text,ekeyi)).i).j & dkeyij = (dkeyi.i).j &
    ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j =
    Op-XOR(dtextij,dkeyij) by DefAddRoundKey,A3;
    thus ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j =
    (text.i).j by A4,A5,A0,DESCIP_1:17;
  end;
  hence AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi) = text by LM01;
end;

theorem
  for m be Nat,
  text,otext be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Keyi1,KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
  KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
  otext = AddRoundKey.((ShiftRows*SubBytes(SBT)).text,KeyNr)
  holds (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(otext,Keyi1)) = text
proof
  let m be Nat,
  text,otext be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Keyi1,KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
  KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
  otext = AddRoundKey.((ShiftRows*SubBytes(SBT)).text,KeyNr);
  (AddRoundKey.(otext,Keyi1)) = (ShiftRows*SubBytes(SBT)).text by AS,LAST05;
  hence (InvSubBytes(SBT)*InvShiftRows).(AddRoundKey.(otext,Keyi1)) = text
  by LAST01;
end;

theorem LAST08:
  for m,i be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  eKeyi,dKeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & i <= 7+m-1 &
  eKeyi = ((KeyExpansion(SBT,m)).(Key)).(7+m-i) &
  dKeyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1)
  holds AddRoundKey.(AddRoundKey.(text,eKeyi),dKeyi) = text
proof
  let m,i be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  ekeyi,dkeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & i <= 7+m-1 &
  ekeyi = ((KeyExpansion(SBT,m)).(key)).(7+m-i) &
  dkeyi = (Rev((KeyExpansion(SBT,m)).(key))).(i+1);
  set p = (KeyExpansion(SBT,m)).(key);
  (KeyExpansion(SBT,m)).(key) in
  (7+m)-tuples_on(4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)));
  then
B0: ex s be Element of (4-tuples_on (4-tuples_on (8-tuples_on BOOLEAN)))*
  st (KeyExpansion(SBT,m)).(key) = s & len s = (7+m);
  i+1 <= 7+m-1+1 by AS,XREAL_1:7;
  then 1 <= i+1 & i+1 <= 7+m by NAT_1:11;
  then i+1 in Seg (7+m);
  then
B1: i+1 in dom p by FINSEQ_1:def 3,B0;
A0: dkeyi = p.((len p)-(i+1)+1) by AS,FINSEQ_5:58,B1
    .= ekeyi by B0,AS;
  now let i,j be Nat;
    assume
A3: i in Seg 4 & j in Seg 4;
    then consider etextij,ekeyij be Element of 8-tuples_on BOOLEAN such that
A4: etextij = (text.i).j & ekeyij = (ekeyi.i).j &
    ((AddRoundKey.(text,ekeyi)).i).j = Op-XOR(etextij,ekeyij)
    by DefAddRoundKey;
    consider dtextij,dkeyij be Element of 8-tuples_on BOOLEAN such that
A5: dtextij = ((AddRoundKey.(text,ekeyi)).i).j & dkeyij = (dkeyi.i).j &
    ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j =
    Op-XOR(dtextij,dkeyij) by DefAddRoundKey,A3;
    thus ((AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi)).i).j =
    (text.i).j by A4,A5,A0,DESCIP_1:17;
  end;
  hence AddRoundKey.(AddRoundKey.(text,ekeyi),dkeyi) = text by LM01;
end;

LAST07:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  eKeyi,dKeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8) & eKeyi = ((KeyExpansion(SBT,m)).(Key)).1 &
  dKeyi = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m)
  holds AddRoundKey.(AddRoundKey.(text,eKeyi),dKeyi) = text
proof
  let m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  eKeyi,dKeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  assume
AS: (m = 4 or m = 6 or m = 8) & eKeyi = ((KeyExpansion(SBT,m)).(Key)).1 &
  dKeyi = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m);
  1+0 < 7+m by XREAL_1:8;
  then 0 < 7+m-1 by XREAL_1:50;
  then 7+m-1 in NAT by INT_1:3;
  then reconsider i = 7+m-1 as Nat;
P2: eKeyi = ((KeyExpansion(SBT,m)).(Key)).(7+m-i) by AS;
  dKeyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) by AS;
  hence thesis by AS,P2,LAST08;
end;

theorem LASTXX:
  for m be Nat,
  text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st (m = 4 or m = 6 or m = 8)
  holds AES-DEC(SBT,MCFunc,AES-ENC(SBT,MCFunc,text,Key),Key) = text
proof
  let m be Nat;
  let text be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  let Key be Element of m-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  1+0 < 7+m by XREAL_1:8;
  then
N1: 0 < 7+m-1 by XREAL_1:50;
  then 7+m-1 in NAT by INT_1:3;
  then reconsider Nr = 7+m-1 as Nat;
A0: 1 <= Nr by NAT_1:14,N1;
  assume
AS: (m = 4 or m = 6 or m = 8);
  consider eseq be
  FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P1: len eseq = Nr &
  (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
  eseq.1 = AddRoundKey.(text,Keyi1)) & (for i be Nat st 1 <= i & i < Nr holds
  ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi = ((KeyExpansion(SBT,m)).(Key)).(i+1) & eseq.(i+1) =
  AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(eseq.i),Keyi)) &
  ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st KeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
  AES-ENC(SBT,MCFunc,text,Key) =
  AddRoundKey.((ShiftRows*SubBytes(SBT)).(eseq.Nr),KeyNr) by defENC;
  consider dseq be
  FinSequence of (4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))) such that
P2: len dseq = Nr &
  (ex Keyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  dseq.1 = (InvSubBytes(SBT)*InvShiftRows).
  (AddRoundKey.(AES-ENC(SBT,MCFunc,text,Key),Keyi1))) &
  (for i be Nat st 1 <= i & i < Nr holds
  ex Keyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st Keyi = (Rev((KeyExpansion(SBT,m)).(Key))).(i+1) &
  dseq.(i+1) =
  (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(dseq.i,Keyi))) &
  ex KeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  st KeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
  AES-DEC(SBT,MCFunc,AES-ENC(SBT,MCFunc,text,Key),Key) =
  AddRoundKey.(dseq.Nr,KeyNr) by defDEC;
  consider eKeyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  such that
P11: eKeyi1 = ((KeyExpansion(SBT,m)).(Key)).1 &
  eseq.1 = AddRoundKey.(text,eKeyi1) by P1;
  consider eKeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  such that
P12: eKeyNr = ((KeyExpansion(SBT,m)).(Key)).(7+m) &
  AES-ENC(SBT,MCFunc,text,Key) =
  AddRoundKey.((ShiftRows*SubBytes(SBT)).(eseq.Nr),eKeyNr) by P1;
  consider dKeyi1 be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  such that
P21: dKeyi1 = (Rev((KeyExpansion(SBT,m)).(Key))).1 &
  dseq.1 = (InvSubBytes(SBT)*InvShiftRows).
  (AddRoundKey.(AES-ENC(SBT,MCFunc,text,Key),dKeyi1)) by P2;
  consider dKeyNr be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
  such that
P22: dKeyNr = (Rev((KeyExpansion(SBT,m)).(Key))).(7+m) &
  AES-DEC(SBT,MCFunc,AES-ENC(SBT,MCFunc,text,Key),Key) =
  AddRoundKey.(dseq.Nr,dKeyNr) by P2;
  defpred PQ[Nat] means
  $1 < Nr implies dseq.($1+1) = eseq.(Nr-$1);
  Nr in Seg Nr by A0;
  then Nr in dom eseq by P1,FINSEQ_1:def 3;
  then eseq.Nr in rng eseq by FUNCT_1:3;
  then reconsider esqm = eseq.Nr as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  dseq.(1+0) = esqm by P12,P21,AS,LAST04
    .= eseq.(Nr-0);
  then
PN1: PQ[0];
PN2: for i be Nat st PQ[i] holds PQ[i+1]
  proof
    let i be Nat;
    assume
A1: PQ[i];
    assume
A2: i+1 < Nr;
A4: i <= i+1 by NAT_1:11;
    consider dKeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    such that
A6: dKeyi = (Rev((KeyExpansion(SBT,m)).(Key))).((i+1)+1) &
    dseq.((i+1)+1) =
    (InvSubBytes(SBT)*InvShiftRows*(MCFunc")).(AddRoundKey.(dseq.(i+1),dKeyi))
    by P2,A2,NAT_1:11;
X11: 0 < Nr-(i+1) by A2,XREAL_1:50;
    then Nr-(i+1) in NAT by INT_1:3;
    then reconsider m7i1 = Nr-(i+1) as Nat;
    1 <= m7i1 by NAT_1:14,X11;
    then
A9: 1 <= Nr-(i+1) & Nr-(i+1) < Nr by XREAL_1:44;
    consider eKeyi be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    such that
A10: eKeyi = ((KeyExpansion(SBT,m)).(Key)).(m7i1+1) &
    eseq.(m7i1+1) =
    AddRoundKey.((MCFunc*ShiftRows*SubBytes(SBT)).(eseq.(m7i1)),eKeyi)
    by P1,A9;
    m7i1 in Seg Nr by A9;
    then m7i1 in dom eseq by P1,FINSEQ_1:def 3;
    then eseq.m7i1 in rng eseq by FUNCT_1:3;
    then reconsider esq7mi1 = eseq.m7i1 as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    reconsider MSSesq7mi1 = (MCFunc*ShiftRows*SubBytes(SBT)).esq7mi1 as
    Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
XXX: eKeyi = ((KeyExpansion(SBT,m)).(Key)).(7+m-(i+1)) by A10;
A12: AddRoundKey.(eseq.(Nr-i),dKeyi) = MSSesq7mi1 by A10,A2,AS,A6,XXX,LAST08;
    thus dseq.((i+1)+1) = eseq.(Nr-(i+1)) by A6,A4,A2,XXREAL_0:2,A1,A12,LAST02;
  end;
P30: for k be Nat holds PQ[k] from NAT_1:sch 2(PN1,PN2);
  5+m < 6+m by XREAL_1:8;
  then
P31: dseq.(5+m+1) = eseq.(Nr-(5+m)) by P30;
  1 <= 1 & 1 <= 1+(5+m) by NAT_1:11;
  then 1 in Seg Nr;
  then 1 in dom eseq by P1,FINSEQ_1:def 3;
  then eseq.1 in rng eseq by FUNCT_1:3;
  then reconsider esq1 = eseq.1 as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  thus AES-DEC(SBT,MCFunc,AES-ENC(SBT,MCFunc,text,Key),Key) = text
  by P22,P31,P11,AS,LAST07;
end;

theorem LR8D1:
  for D be non empty set, n,m be non zero Element of NAT,
  r be Element of n-tuples_on D st m <= n & 8 <= n-m
  holds Op-Left(Op-Right(r,m),8) is Element of 8-tuples_on D
proof
  let D be non empty set,
  n,m be non zero Element of NAT,
  r be Element of n-tuples_on D;
  assume
A1: m <= n & 8 <= n-m;
  r in { s where s is Element of D*: len s = n};
  then consider s be Element of D* such that
A2: r = s & len s = n;
  len Op-Right(r,m) = n - m by A1,A2,RFINSEQ:def 1;
  then len(Op-Left(Op-Right(r,m),8)) = 8 by A1,FINSEQ_1:59;
  hence thesis by FINSEQ_2:92;
end;

Lm1:
  for D be non empty set, n be non zero Element of NAT,
  r be Element of n-tuples_on D st 8 <= n & 8 <= n-8 & 16 <= n &
  8 <= n-16 & 24 <= n & 8 <= n-24
  holds <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),Op-Left(Op-Right(r,16),8),
  Op-Left(Op-Right(r,24),8) *> is Element of 4-tuples_on (8-tuples_on D)
proof
  let D be non empty set, n be non zero Element of NAT,
  r be Element of n-tuples_on D;
  assume 8 <= n & 8 <= n-8 & 16 <= n & 8 <= n-16 & 24 <= n & 8 <= n-24;
  then Op-Left(r,8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,8),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,16),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,24),8) is Element of 8-tuples_on D by DESCIP_1:1,LR8D1;
  hence thesis by LMGSEQ4;
end;

Lm2:
  for D be non empty set, n,m,l,p,q be non zero Element of NAT,
  r be Element of n-tuples_on D st m <= n & 8 <= n-m & l = m+8 & l <= n &
  8 <= n-l & p = m+16 & p <= n & 8 <= n-p & q = m+24 & q <= n & 8 <= n-q
  holds <* Op-Left(Op-Right(r,m),8),Op-Left(Op-Right(r,l),8),
  Op-Left(Op-Right(r,p),8),Op-Left(Op-Right(r,q),8) *> is
  Element of 4-tuples_on (8-tuples_on D)
proof
  let D be non empty set, n,m,l,p,q be non zero Element of NAT,
  r be Element of n-tuples_on D;
  assume m <= n & 8 <= n-m & l = m+8 & l <= n & 8 <= n-l &
  p = m+16 & p <= n & 8 <= n-p & q = m+24 & q <= n & 8 <= n-q;
  then Op-Left(Op-Right(r,m),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,l),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,p),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,q),8) is Element of 8-tuples_on D by LR8D1;
  hence thesis by LMGSEQ4;
end;

Lm3:
  for D be non empty set, n,m,l,p,q be non zero Element of NAT,
  r be Element of n-tuples_on D st m <= n & 8 <= n-m & l = m+8 & l <= n &
  8 <= n-l & p = m+16 & p <= n & 8 <= n-p & q = m+24 & q <= n & 8 = n-q
  holds <* Op-Left(Op-Right(r,m),8),Op-Left(Op-Right(r,l),8),
  Op-Left(Op-Right(r,p),8),Op-Right(r,q) *> is
  Element of 4-tuples_on (8-tuples_on D)
proof
  let D be non empty set, n,m,l,p,q be non zero Element of NAT,
  r be Element of n-tuples_on D;
  assume m <= n & 8 <= n-m & l = m+8 & l <= n & 8 <= n-l &
  p = m+16 & p <= n & 8 <= n-p & q = m+24 & q <= n & 8 = n-q;
  then Op-Left(Op-Right(r,m),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,l),8) is Element of 8-tuples_on D &
  Op-Left(Op-Right(r,p),8) is Element of 8-tuples_on D &
  Op-Right(r,q) is Element of 8-tuples_on D by DESCIP_1:2,LR8D1;
  hence thesis by LMGSEQ4;
end;

definition
  let r be Element of 128-tuples_on BOOLEAN;
  func AES-KeyInitState128(r) ->
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  it.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
    Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
  it.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
    Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
  it.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
    Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
  it.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
    Op-Left(Op-Right(r,112),8),Op-Right(r,120) *>;
  existence
  proof
    set R1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *>;
    set R2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *>;
    set R3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *>;
    set R4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Right(r,120) *>;
    8 <= 128-8 & 8 <= 128-16 & 8 <= 128-24;
    then reconsider R1 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm1;
    8 <= 128-32 & 8 <= 128-40 & 8 <= 128-48 & 8 <= 128-56 &
    40 = 32+8 & 48 = 32+16 & 56 = 32+24;
    then reconsider R2 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 128-64 & 8 <= 128-72 & 8 <= 128-80 & 8 <= 128-88 &
    72 = 64+8 & 80 = 64+16 & 88 = 64+24;
    then reconsider R3 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 128-96 & 8 <= 128-104 & 8 <= 128-112 & 8 = 128-120 &
    104 = 96+8 & 112 = 96+16 & 120 = 96+24 & 8 = 128-120;
    then reconsider R4 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm3;
    set T1 = <*R1,R2*>;
    set T2 = <*R3,R4*>;
    set T = T1^T2;
A4: T.1 = T1.1 & ... & T.2 = T1.2 by FINSEQ_3:154;
A5: T.(2+1) = T2.1 & ... & T.(2+2) = T2.2 by FINSEQ_3:155;
    len T = 4 & T is FinSequence of 4-tuples_on (8-tuples_on BOOLEAN)
    by CARD_1:def 7;
    then reconsider T as Element of
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by FINSEQ_2:92;
    take T;
    thus thesis by A4,A5,FINSEQ_1:44;
  end;
  uniqueness
  proof
    let p,q be Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A6: p.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    p.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    p.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    p.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Right(r,120) *>;
    assume
A7: q.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    q.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    q.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    q.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Right(r,120) *>;
    p in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A8: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st p = v &
    len v = 4;
    q in 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A9: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st q = v &
    len v = 4;
    for i be Nat st 1 <= i & i <= len p holds p.i = q.i
    proof
      let i be Nat;
      assume 1 <= i & i <= len p;
      then i = 1 or ... or i = 4 by A8;
      hence thesis by A6,A7;
    end;
    hence p = q by A8,A9,FINSEQ_1:14;
  end;
end;

definition
  let r be Element of 192-tuples_on BOOLEAN;
  func AES-KeyInitState192(r) ->
  Element of 6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  it.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
    Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
  it.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
    Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
  it.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
    Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
  it.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
    Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
  it.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
    Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
  it.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
    Op-Left(Op-Right(r,176),8),Op-Right(r,184) *>;
  existence
  proof
    set R1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *>;
    set R2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *>;
    set R3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *>;
    set R4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *>;
    set R5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *>;
    set R6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Right(r,184) *>;
    8 <= 192-8 & 8 <= 192-16 & 8 <= 192-24;
    then reconsider R1 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm1;
    8 <= 192-32 & 8 <= 192-40 & 8 <= 192-48 & 8 <= 192-56 &
    40 = 32+8 & 48 = 32+16 & 56 = 32+24;
    then reconsider R2 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 192-64 & 8 <= 192-72 & 8 <= 192-80 & 8 <= 192-88 &
    72 = 64+8 & 80 = 64+16 & 88 = 64+24;
    then reconsider R3 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 192-96 & 8 <= 192-104 & 8 <= 192-112 & 8 <= 192-120 &
    104 = 96+8 & 112 = 96+16 & 120 = 96+24;
    then reconsider R4 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 192-128 & 8 <= 192-136 & 8 <= 192-144 & 8 <= 192-152 &
    136 = 128+8 & 144 = 128+16 & 152 = 128+24;
    then reconsider R5 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 192-160 & 8 <= 192-168 & 8 <= 192-176 & 8 = 192-184 &
    168 = 160+8 & 176 = 160+16 & 184 = 160+24 & 8 = 192-184;
    then reconsider R6 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm3;
    set T1 = <*R1,R2,R3*>;
    set T2 = <*R4,R5,R6*>;
    set T = T1^T2;
A4: T.1 = T1.1 & ... & T.3 = T1.3 by FINSEQ_3:154;
A5: T.(3+1) = T2.1 & ... & T.(3+3) = T2.3 by FINSEQ_3:155;
    len T = 6 & T is FinSequence of 4-tuples_on (8-tuples_on BOOLEAN)
    by CARD_1:def 7;
    then reconsider T as Element of
    6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by FINSEQ_2:92;
    take T;
    thus thesis by A4,A5,FINSEQ_1:45;
  end;
  uniqueness
  proof
    let p,q be Element of 6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A6: p.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    p.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    p.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    p.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
    p.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
    p.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Right(r,184) *>;
    assume
A7: q.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    q.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    q.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    q.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
    q.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
    q.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Right(r,184) *>;
    p in 6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A8: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st p = v &
    len v = 6;
    q in 6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A9: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st q = v &
    len v = 6;
    for i be Nat st 1 <= i & i <= len p holds p.i = q.i
    proof
      let i be Nat;
      assume 1 <= i & i <= len p;
      then i = 1 or ... or i = 6 by A8;
      hence thesis by A6,A7;
    end;
    hence p = q by A8,A9,FINSEQ_1:14;
  end;
end;

definition
  let r be Element of 256-tuples_on BOOLEAN;
  func AES-KeyInitState256(r) ->
  Element of 8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) means
  it.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
    Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
  it.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
    Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
  it.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
    Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
  it.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
    Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
  it.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
    Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
  it.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
    Op-Left(Op-Right(r,176),8),Op-Left(Op-Right(r,184),8) *> &
  it.7 = <* Op-Left(Op-Right(r,192),8),Op-Left(Op-Right(r,200),8),
    Op-Left(Op-Right(r,208),8),Op-Left(Op-Right(r,216),8) *> &
  it.8 = <* Op-Left(Op-Right(r,224),8),Op-Left(Op-Right(r,232),8),
    Op-Left(Op-Right(r,240),8),Op-Right(r,248) *>;
  existence
  proof
    set R1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *>;
    set R2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *>;
    set R3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *>;
    set R4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *>;
    set R5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *>;
    set R6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Left(Op-Right(r,184),8) *>;
    set R7 = <* Op-Left(Op-Right(r,192),8),Op-Left(Op-Right(r,200),8),
      Op-Left(Op-Right(r,208),8),Op-Left(Op-Right(r,216),8) *>;
    set R8 = <* Op-Left(Op-Right(r,224),8),Op-Left(Op-Right(r,232),8),
      Op-Left(Op-Right(r,240),8),Op-Right(r,248) *>;
    8 <= 256-8 & 8 <= 256-16 & 8 <= 256-24;
    then reconsider R1 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm1;
    8 <= 256-32 & 8 <= 256-40 & 8 <= 256-48 & 8 <= 256-56 &
    40 = 32+8 & 48 = 32+16 & 56 = 32+24;
    then reconsider R2 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-64 & 8 <= 256-72 & 8 <= 256-80 & 8 <= 256-88 &
    72 = 64+8 & 80 = 64+16 & 88 = 64+24;
    then reconsider R3 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-96 & 8 <= 256-104 & 8 <= 256-112 & 8 <= 256-120 &
    104 = 96+8 & 112 = 96+16 & 120 = 96+24;
    then reconsider R4 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-128 & 8 <= 256-136 & 8 <= 256-144 & 8 <= 256-152 &
    136 = 128+8 & 144 = 128+16 & 152 = 128+24;
    then reconsider R5 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-160 & 8 <= 256-168 & 8 <= 256-176 & 8 <= 256-184 &
    168 = 160+8 & 176 = 160+16 & 184 = 160+24;
    then reconsider R6 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-192 & 8 <= 256-200 & 8 <= 256-208 & 8 <= 256-216 &
    200 = 192+8 & 208 = 192+16 & 216 = 192+24;
    then reconsider R7 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm2;
    8 <= 256-224 & 8 <= 256-232 & 8 <= 256-240 & 8 = 256-248 &
    232 = 224+8 & 240 = 224+16 & 248 = 224+24 & 8 = 256-248;
    then reconsider R8 as Element of 4-tuples_on (8-tuples_on BOOLEAN) by Lm3;
    set T1 = <*R1,R2,R3,R4*>;
    set T2 = <*R5,R6,R7,R8*>;
    set T = T1^T2;
A4: T.1 = T1.1 & ... & T.4 = T1.4 by FINSEQ_3:154;
A5: T.(4+1) = T2.1 & ... & T.(4+4) = T2.4 by FINSEQ_3:155;
    len T = 8 & T is FinSequence of 4-tuples_on (8-tuples_on BOOLEAN)
    by CARD_1:def 7;
    then reconsider T as Element of
    8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)) by FINSEQ_2:92;
    take T;
    thus thesis by A4,A5,FINSEQ_4:76;
  end;
  uniqueness
  proof
    let p,q be Element of 8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    assume
A6: p.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    p.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    p.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    p.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
    p.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
    p.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Left(Op-Right(r,184),8) *> &
    p.7 = <* Op-Left(Op-Right(r,192),8),Op-Left(Op-Right(r,200),8),
      Op-Left(Op-Right(r,208),8),Op-Left(Op-Right(r,216),8) *> &
    p.8 = <* Op-Left(Op-Right(r,224),8),Op-Left(Op-Right(r,232),8),
      Op-Left(Op-Right(r,240),8),Op-Right(r,248) *>;
    assume
A7: q.1 = <* Op-Left(r,8),Op-Left(Op-Right(r,8),8),
      Op-Left(Op-Right(r,16),8),Op-Left(Op-Right(r,24),8) *> &
    q.2 = <* Op-Left(Op-Right(r,32),8),Op-Left(Op-Right(r,40),8),
      Op-Left(Op-Right(r,48),8),Op-Left(Op-Right(r,56),8) *> &
    q.3 = <* Op-Left(Op-Right(r,64),8),Op-Left(Op-Right(r,72),8),
      Op-Left(Op-Right(r,80),8),Op-Left(Op-Right(r,88),8) *> &
    q.4 = <* Op-Left(Op-Right(r,96),8),Op-Left(Op-Right(r,104),8),
      Op-Left(Op-Right(r,112),8),Op-Left(Op-Right(r,120),8) *> &
    q.5 = <* Op-Left(Op-Right(r,128),8),Op-Left(Op-Right(r,136),8),
      Op-Left(Op-Right(r,144),8),Op-Left(Op-Right(r,152),8) *> &
    q.6 = <* Op-Left(Op-Right(r,160),8),Op-Left(Op-Right(r,168),8),
      Op-Left(Op-Right(r,176),8),Op-Left(Op-Right(r,184),8) *> &
    q.7 = <* Op-Left(Op-Right(r,192),8),Op-Left(Op-Right(r,200),8),
      Op-Left(Op-Right(r,208),8),Op-Left(Op-Right(r,216),8) *> &
    q.8 = <* Op-Left(Op-Right(r,224),8),Op-Left(Op-Right(r,232),8),
      Op-Left(Op-Right(r,240),8),Op-Right(r,248) *>;
    p in 8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A8: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st p = v &
    len v = 8;
    q in 8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
    then
A9: ex v be Element of (4-tuples_on (8-tuples_on BOOLEAN))* st q = v &
    len v = 8;
    for i be Nat st 1 <= i & i <= len p holds p.i = q.i
    proof
      let i be Nat;
      assume 1 <= i & i <= len p;
      then i = 1 or ... or i = 8 by A8;
      hence thesis by A6,A7;
    end;
    hence p = q by A8,A9,FINSEQ_1:14;
  end;
end;

definition
  let SBT,MixColumns;
  let message be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 128-tuples_on BOOLEAN;
  func AES128-ENC(SBT,MixColumns,message,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-ENC(SBT,MixColumns,AES-Statearray.message,
  AES-KeyInitState128(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is Function of
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),128-tuples_on BOOLEAN
    by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

definition
  let SBT,MixColumns;
  let cipher be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 128-tuples_on BOOLEAN;
  func AES128-DEC(SBT,MixColumns,cipher,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,AES-Statearray.cipher,
  AES-KeyInitState128(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is
    Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    128-tuples_on BOOLEAN by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

theorem
  for SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message,Key be Element of 128-tuples_on BOOLEAN holds
  AES128-DEC(SBT,MixColumns,AES128-ENC(SBT,MixColumns,message,Key),Key) =
  message
proof
  let SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message,Key be Element of 128-tuples_on BOOLEAN;
  reconsider text = AES-Statearray.message as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider sKey = AES-KeyInitState128(Key) as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider cipher = AES-ENC(SBT,MixColumns,text,sKey) as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider CBLOCK = AES128-ENC(SBT,MixColumns,message,Key) as
  Element of 128-tuples_on BOOLEAN;
  AES128-DEC(SBT,MixColumns,CBLOCK,Key) =
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,cipher,sKey)) by LMINV1
    .=(AES-Statearray)".text by LASTXX;
  hence thesis by FUNCT_2:26;
end;

definition
  let SBT,MixColumns;
  let message be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 192-tuples_on BOOLEAN;
  func AES192-ENC(SBT,MixColumns,message,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-ENC(SBT,MixColumns,AES-Statearray.message,
  AES-KeyInitState192(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is Function of
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),128-tuples_on BOOLEAN
    by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

definition
  let SBT,MixColumns;
  let cipher be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 192-tuples_on BOOLEAN;
  func AES192-DEC(SBT,MixColumns,cipher,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,AES-Statearray.cipher,
  AES-KeyInitState192(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is
    Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    128-tuples_on BOOLEAN by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

theorem
  for SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message be Element of 128-tuples_on BOOLEAN,
  Key be Element of 192-tuples_on BOOLEAN holds
  AES192-DEC(SBT,MixColumns,AES192-ENC(SBT,MixColumns,message,Key),Key) =
  message
proof
  let SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message be Element of 128-tuples_on BOOLEAN,
  Key be Element of 192-tuples_on BOOLEAN;
  reconsider text = AES-Statearray.message as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider sKey = AES-KeyInitState192(Key) as
  Element of 6-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider cipher = AES-ENC(SBT,MixColumns,text,sKey) as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider CBLOCK = AES192-ENC(SBT,MixColumns,message,Key) as
  Element of 128-tuples_on BOOLEAN;
  AES192-DEC(SBT,MixColumns,CBLOCK,Key) =
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,cipher,sKey)) by LMINV1
    .=(AES-Statearray)".text by LASTXX;
  hence thesis by FUNCT_2:26;
end;

definition
  let SBT,MixColumns;
  let message be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 256-tuples_on BOOLEAN;
  func AES256-ENC(SBT,MixColumns,message,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-ENC(SBT,MixColumns,AES-Statearray.message,
  AES-KeyInitState256(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is Function of
    4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),128-tuples_on BOOLEAN
    by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

definition
  let SBT,MixColumns;
  let cipher be Element of 128-tuples_on BOOLEAN;
  let Key be Element of 256-tuples_on BOOLEAN;
  func AES256-DEC(SBT,MixColumns,cipher,Key) ->
  Element of 128-tuples_on BOOLEAN equals
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,AES-Statearray.cipher,
  AES-KeyInitState256(Key)));
  correctness
  proof
    rng AES-Statearray = 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN))
    by FUNCT_2:def 3;
    then (AES-Statearray)" is
    Function of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
    128-tuples_on BOOLEAN by FUNCT_2:25;
    hence thesis by FUNCT_2:5;
  end;
end;

theorem
  for SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message be Element of 128-tuples_on BOOLEAN,
  Key be Element of 256-tuples_on BOOLEAN holds
  AES256-DEC(SBT,MixColumns,AES256-ENC(SBT,MixColumns,message,Key),Key) =
  message
proof
  let SBT be Permutation of (8-tuples_on BOOLEAN),
  MixColumns be Permutation of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN)),
  message be Element of 128-tuples_on BOOLEAN,
  Key be Element of 256-tuples_on BOOLEAN;
  reconsider text = AES-Statearray.message as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider sKey = AES-KeyInitState256(Key) as
  Element of 8-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider cipher = AES-ENC(SBT,MixColumns,text,sKey) as
  Element of 4-tuples_on(4-tuples_on (8-tuples_on BOOLEAN));
  reconsider CBLOCK = AES256-ENC(SBT,MixColumns,message,Key) as
  Element of 128-tuples_on BOOLEAN;
  AES256-DEC(SBT,MixColumns,CBLOCK,Key) =
  (AES-Statearray)".(AES-DEC(SBT,MixColumns,cipher,sKey)) by LMINV1
    .=(AES-Statearray)".text by LASTXX;
  hence thesis by FUNCT_2:26;
end;