:: Correctness of Non Overwriting Programs. {P}art {I}
:: by Yatsuka Nakamura
::
:: Copyright (c) 2003-2019 Association of Mizar Users

::----------------------
::----------------------
:: Non overwriting program is a program where each variable used in it
::is written only just one time, but the control variables used for
::for-statement are exceptional. Contrarily, variables are allowed
:: There are other restriction for non overwriting program. For statements,
::only the followings are allowed: substituting-statement, if-else-statement,
::for-statement(with break and without break), function(correct one)-call
::-statement and return-statement.
:: Grammars of non overwriting program is like one of C-language.
:: For type of variables, 'int','real","char" and "float" can be used, and
::and array of them can also be used. For operation, "+", "-" and "*"
::are used for a type int, "+","-","*" and "/" are used for a type float.
:: User can also define structures like in C.
:: Non overwriting program can be translated to (predicative) logic
::formula in definition part to define functions. If a new function
::is correctly defined, a corresponding program is correct, if it does not use
::arrays. If it uses arrays, area check is necessary in the following
::theorem.
:: Semantic correctness is shown by some theorems following the definition.
::These theorems must tie up the result of the program and mathematical concepts
::introduced before.
:: Correctness is proven function-wise. We must use only
::correctness-proven functions to define a new function(to write a new
::program as a form of a function).
:: Here, we present two program of division function of two natural
::numbers and of two integers. An algorithm is checked for each case, by
::proving correctness of the definitions.
:: We also do an area check of index of arrays used in one of the programs.
::---------
:: type correspondence:
:: int .....> Integer
:: float .....> Real
:: char ......> Subset of A
::---------
:: statement correspondence:
:: We use tr(statement_i) for translated logic formula corresponding
:: to statement_i.
:: i=j+k-l; ....> i=j+k-l
:: i=j*k; ....> i=j*k
:: x=y*z/s; ....> x=y*z/s
:: statement_1;statement2;statement3;...
:: .......> tr(statement_1)& tr(statement_2)& tr(statement_3)&...
:: if (statement_1){statement_2;statement_3;...}
:: .......> tr(statement_1) implies tr(statement_2)& tr(statement_3)&..;
:: if (statement_1){statement_2;statement_3;...} else statement_4;
:: .......> (tr(statement_1) implies tr(statement_2)& tr(statement_3)&..)&
:: (not tr(statement_1) implies tr(statement_4));
:: for (i=1;i++;i<=n)statement_1;
:: .......> for i being Integer st 1<=i & i<=n holds tr(statement_1);
:: for (i=1;i++;i<=n){statement_1;statement_2;...;if (statement_3)break;}
:: .......> (ex j being Integer st 1<=j & j<=n &
:: (for i being Integer st 1<=i & i<j holds
:: tr(statement_1 for i)& tr(statement_2 for i)&...
:: & not (statement_3 for i))&
:: &tr(statement_1 for j)& tr(statement_2 for j)&...
:: &statement_3 for j)
:: or
:: (for i being Integer st 1<=i & i<=n holds
:: tr(statement_1 for i)& tr(statement_2 for i)&...
:: & not (statement_3 for i));
:: ***If "break" is expected in the above "for statement",
:: then "or" part can be deleted.
::------
:: arrays correspondence:
:: int a[n+1] .....> ex a being FinSequence of INT st len a=n & ...;
:: float x[n+1] .....> ex x being FinSequence of REAL st len x=n & ...;
:: Declaration of variables corresponds to existential statement.
::------
:: various correctness problem:
:: 1. mathematical algorithm .....> a function is well defined in Mizar
:: 2. semantic correctness .....> by theorems connecting it with
:: other mathematical or computational concepts in Mizar
:: 3. area check of variable of array .....> by a theorem checking
:: the area, in Mizar
:: 4. Is the translation to logic formula correct?
:: ......> by other methods outside of Mizar
:: 5. overflow problem .....> by other theorems, maybe in Mizar
:: or corresponding float to other types not Real
:: 7. translation of usual programs to non overwriting programs
:: ......> by other methods outside of Mizar
::------
:: A concept of non overwriting is important, not only
:: because of proving correctness, but because of debugging and
:: safety of data.
:: As memory is now cheap enough, it is wise to save all history of
:: variables in a program.
::-------------------------
::-------------------------
theorem Th1: :: PRGCOR_1:1
for n, m, k being Nat holds (n + k) -' (m + k) = n -' m
proof end;

theorem Th2: :: PRGCOR_1:2
for n, k being Element of NAT st k > 0 & n mod (2 * k) >= k holds
( (n mod (2 * k)) - k = n mod k & (n mod k) + k = n mod (2 * k) )
proof end;

theorem Th3: :: PRGCOR_1:3
for n, k being Element of NAT st k > 0 & n mod (2 * k) >= k holds
n div k = ((n div (2 * k)) * 2) + 1
proof end;

theorem Th4: :: PRGCOR_1:4
for n, k being Element of NAT st k > 0 & n mod (2 * k) < k holds
n mod (2 * k) = n mod k
proof end;

theorem Th5: :: PRGCOR_1:5
for n, k being Element of NAT st k > 0 & n mod (2 * k) < k holds
n div k = (n div (2 * k)) * 2
proof end;

theorem Th6: :: PRGCOR_1:6
for m, n being Element of NAT st m > 0 holds
ex i being Element of NAT st
( ( for k2 being Element of NAT st k2 < i holds
m * (2 |^ k2) <= n ) & m * (2 |^ i) > n )
proof end;

theorem Th7: :: PRGCOR_1:7
for i being Integer
for f being FinSequence st 1 <= i & i <= len f holds
i in dom f
proof end;

:: Overwrting program to divide n by m (n>=0&m>0),where division / used is
:: special, because it is achieved by shifting a word.
:: int idiv1_prg(int n,int m){
:: int sm,sn,pn,i,j;
:: if(n<m){return 0;}
:: sm=m;
:: for (i=1;i<=n;i++){sm=sm*2; if (sm >n)break;}
:: pn=0;sn=n;sm=sm/2;
:: for (j=1;j<=i;j++){
:: if(sn>=sm){sn=sn-sm;sm=sm/2;pn=pn*2+1;} else {sm=sm/2;pn=pn*2;}
:: }
:: return pn;
:: }
:: Non overwrting program same as above, assuming n>=0 & m>0
:: int idiv1_prg(int n,int m){
:: int sm[n+1+1],sn[n+1+1],pn[n+1+1],i,j;
:: if (n<m){return 0;}
:: sm=m;
:: for (i=1;i<=n;i++){
:: sm[i+1]=sm[i]*2;
:: if(sm[i+1]>n){break;}
:: }
:: pn[i+1]=0;sn[i+1]=n;
:: for (j=1;j<=i;j++){
:: if(sn[i+1-(j-1)]>=sm[i+1-j]){sn[i+1-j]=sn[i+1-(j-1)]-sm[i+1-j];
:: pn[i+1-j]=pn[i+1-(j-1)]*2+1;}
:: else {sn[i+1-j]=sn[i+1-(j-1)]; pn[i+1-j]=pn[i+1-(j-1)]*2;}
:: }
:: return pn;
:: }
definition
let n, m be Integer;
assume that
A1: n >= 0 and
A2: m > 0 ;
func idiv1_prg (n,m) -> Integer means :Def1: :: PRGCOR_1:def 1
ex sm, sn, pn being FinSequence of INT st
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies it = 0 ) & ( not n < m implies ( sm . 1 = m & ex i being Integer st
( 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & it = pn . 1 ) ) ) );
existence
ex b1 being Integer ex sm, sn, pn being FinSequence of INT st
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies b1 = 0 ) & ( not n < m implies ( sm . 1 = m & ex i being Integer st
( 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & b1 = pn . 1 ) ) ) )
proof end;
uniqueness
for b1, b2 being Integer st ex sm, sn, pn being FinSequence of INT st
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies b1 = 0 ) & ( not n < m implies ( sm . 1 = m & ex i being Integer st
( 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & b1 = pn . 1 ) ) ) ) & ex sm, sn, pn being FinSequence of INT st
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies b2 = 0 ) & ( not n < m implies ( sm . 1 = m & ex i being Integer st
( 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & b2 = pn . 1 ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines idiv1_prg PRGCOR_1:def 1 :
for n, m being Integer st n >= 0 & m > 0 holds
for b3 being Integer holds
( b3 = idiv1_prg (n,m) iff ex sm, sn, pn being FinSequence of INT st
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies b3 = 0 ) & ( not n < m implies ( sm . 1 = m & ex i being Integer st
( 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & b3 = pn . 1 ) ) ) ) );

:: The following theorem is about array index area checking.
:: Each index of an array appeared in the program is checked
:: at the place just in front of the place the array is used,
:: if it remains in the defined area of array.
theorem :: PRGCOR_1:8
for n, m being Integer st n >= 0 holds
for sm, sn, pn being FinSequence of INT
for i being Integer st len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( not n < m implies ( sm . 1 = m & 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & pn . (i + 1) = 0 & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & idiv1_prg (n,m) = pn . 1 ) ) holds
( len sm = n + 1 & len sn = n + 1 & len pn = n + 1 & ( n < m implies idiv1_prg (n,m) = 0 ) & ( not n < m implies ( 1 in dom sm & sm . 1 = m & 1 <= i & i <= n & ( for k being Integer st 1 <= k & k < i holds
( k + 1 in dom sm & k in dom sm & sm . (k + 1) = (sm . k) * 2 & not sm . (k + 1) > n ) ) & i + 1 in dom sm & i in dom sm & sm . (i + 1) = (sm . i) * 2 & sm . (i + 1) > n & i + 1 in dom pn & pn . (i + 1) = 0 & i + 1 in dom sn & sn . (i + 1) = n & ( for j being Integer st 1 <= j & j <= i holds
( (i + 1) - (j - 1) in dom sn & (i + 1) - j in dom sm & ( sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( (i + 1) - j in dom sn & (i + 1) - j in dom sm & sn . ((i + 1) - j) = (sn . ((i + 1) - (j - 1))) - (sm . ((i + 1) - j)) & (i + 1) - j in dom pn & (i + 1) - (j - 1) in dom pn & pn . ((i + 1) - j) = ((pn . ((i + 1) - (j - 1))) * 2) + 1 ) ) & ( not sn . ((i + 1) - (j - 1)) >= sm . ((i + 1) - j) implies ( (i + 1) - j in dom sn & (i + 1) - (j - 1) in dom sn & sn . ((i + 1) - j) = sn . ((i + 1) - (j - 1)) & (i + 1) - j in dom pn & (i + 1) - (j - 1) in dom pn & pn . ((i + 1) - j) = (pn . ((i + 1) - (j - 1))) * 2 ) ) ) ) & 1 in dom pn & idiv1_prg (n,m) = pn . 1 ) ) )
proof end;

theorem Th9: :: PRGCOR_1:9
for n, m being Element of NAT st m > 0 holds
idiv1_prg (n,m) = n div m
proof end;

theorem Th10: :: PRGCOR_1:10
for n, m being Integer st n >= 0 & m > 0 holds
idiv1_prg (n,m) = n div m
proof end;

theorem Th11: :: PRGCOR_1:11
for n, m being Integer
for n2, m2 being Element of NAT holds
( ( m = 0 & n2 = n & m2 = m implies ( n div m = 0 & n2 div m2 = 0 ) ) & ( n >= 0 & m > 0 & n2 = n & m2 = m implies n div m = n2 div m2 ) & ( n >= 0 & m < 0 & n2 = n & m2 = - m implies ( ( m2 * (n2 div m2) = n2 implies n div m = - (n2 div m2) ) & ( m2 * (n2 div m2) <> n2 implies n div m = (- (n2 div m2)) - 1 ) ) ) & ( n < 0 & m > 0 & n2 = - n & m2 = m implies ( ( m2 * (n2 div m2) = n2 implies n div m = - (n2 div m2) ) & ( m2 * (n2 div m2) <> n2 implies n div m = (- (n2 div m2)) - 1 ) ) ) & ( n < 0 & m < 0 & n2 = - n & m2 = - m implies n div m = n2 div m2 ) )
proof end;

::
:: int idiv_prg(int n,int m){
:: int i;
:: if (m==0){return 0;}
:: if (n>=0 && m>0){return idiv1_prg(n,m);}
:: if (n>=0 && m<0){
:: i= idiv1_prg(n,-m);
:: if((-m)*i==n){return -i;} else{return -i-1;}
:: }
:: if (n<0 && m>0){
:: i= idiv1_prg(-n,m);
:: if(m*i== -n){return -i;} else{return -i-1;}
:: }
:: return idiv1_prg(-n,-m);
:: }
::
:: One time writing program
:: Same as above.
definition
let n, m be Integer;
func idiv_prg (n,m) -> Integer means :Def2: :: PRGCOR_1:def 2
ex i being Integer st
( ( m = 0 implies it = 0 ) & ( not m = 0 implies ( ( n >= 0 & m > 0 implies it = idiv1_prg (n,m) ) & ( ( not n >= 0 or not m > 0 ) implies ( ( n >= 0 & m < 0 implies ( i = idiv1_prg (n,(- m)) & ( (- m) * i = n implies it = - i ) & ( (- m) * i <> n implies it = (- i) - 1 ) ) ) & ( ( not n >= 0 or not m < 0 ) implies ( ( n < 0 & m > 0 implies ( i = idiv1_prg ((- n),m) & ( m * i = - n implies it = - i ) & ( m * i <> - n implies it = (- i) - 1 ) ) ) & ( ( not n < 0 or not m > 0 ) implies it = idiv1_prg ((- n),(- m)) ) ) ) ) ) ) ) );
existence
ex b1, i being Integer st
( ( m = 0 implies b1 = 0 ) & ( not m = 0 implies ( ( n >= 0 & m > 0 implies b1 = idiv1_prg (n,m) ) & ( ( not n >= 0 or not m > 0 ) implies ( ( n >= 0 & m < 0 implies ( i = idiv1_prg (n,(- m)) & ( (- m) * i = n implies b1 = - i ) & ( (- m) * i <> n implies b1 = (- i) - 1 ) ) ) & ( ( not n >= 0 or not m < 0 ) implies ( ( n < 0 & m > 0 implies ( i = idiv1_prg ((- n),m) & ( m * i = - n implies b1 = - i ) & ( m * i <> - n implies b1 = (- i) - 1 ) ) ) & ( ( not n < 0 or not m > 0 ) implies b1 = idiv1_prg ((- n),(- m)) ) ) ) ) ) ) ) )
proof end;
uniqueness
for b1, b2 being Integer st ex i being Integer st
( ( m = 0 implies b1 = 0 ) & ( not m = 0 implies ( ( n >= 0 & m > 0 implies b1 = idiv1_prg (n,m) ) & ( ( not n >= 0 or not m > 0 ) implies ( ( n >= 0 & m < 0 implies ( i = idiv1_prg (n,(- m)) & ( (- m) * i = n implies b1 = - i ) & ( (- m) * i <> n implies b1 = (- i) - 1 ) ) ) & ( ( not n >= 0 or not m < 0 ) implies ( ( n < 0 & m > 0 implies ( i = idiv1_prg ((- n),m) & ( m * i = - n implies b1 = - i ) & ( m * i <> - n implies b1 = (- i) - 1 ) ) ) & ( ( not n < 0 or not m > 0 ) implies b1 = idiv1_prg ((- n),(- m)) ) ) ) ) ) ) ) ) & ex i being Integer st
( ( m = 0 implies b2 = 0 ) & ( not m = 0 implies ( ( n >= 0 & m > 0 implies b2 = idiv1_prg (n,m) ) & ( ( not n >= 0 or not m > 0 ) implies ( ( n >= 0 & m < 0 implies ( i = idiv1_prg (n,(- m)) & ( (- m) * i = n implies b2 = - i ) & ( (- m) * i <> n implies b2 = (- i) - 1 ) ) ) & ( ( not n >= 0 or not m < 0 ) implies ( ( n < 0 & m > 0 implies ( i = idiv1_prg ((- n),m) & ( m * i = - n implies b2 = - i ) & ( m * i <> - n implies b2 = (- i) - 1 ) ) ) & ( ( not n < 0 or not m > 0 ) implies b2 = idiv1_prg ((- n),(- m)) ) ) ) ) ) ) ) ) holds
b1 = b2
;
end;

:: deftheorem Def2 defines idiv_prg PRGCOR_1:def 2 :
for n, m, b3 being Integer holds
( b3 = idiv_prg (n,m) iff ex i being Integer st
( ( m = 0 implies b3 = 0 ) & ( not m = 0 implies ( ( n >= 0 & m > 0 implies b3 = idiv1_prg (n,m) ) & ( ( not n >= 0 or not m > 0 ) implies ( ( n >= 0 & m < 0 implies ( i = idiv1_prg (n,(- m)) & ( (- m) * i = n implies b3 = - i ) & ( (- m) * i <> n implies b3 = (- i) - 1 ) ) ) & ( ( not n >= 0 or not m < 0 ) implies ( ( n < 0 & m > 0 implies ( i = idiv1_prg ((- n),m) & ( m * i = - n implies b3 = - i ) & ( m * i <> - n implies b3 = (- i) - 1 ) ) ) & ( ( not n < 0 or not m > 0 ) implies b3 = idiv1_prg ((- n),(- m)) ) ) ) ) ) ) ) ) );

theorem :: PRGCOR_1:12
for n, m being Integer holds idiv_prg (n,m) = n div m
proof end;