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Specification matrix
generic specification
   parameter elem
   using nat
target
   sorts matrix;
   functions 
       mkmatrix : nat × nat × elem → matrix;
       #r . : matrix → nat;
       #c . : matrix → nat;
   
   variables 
       mx : matrix;
       r : nat;
       c : nat;
       a : elem;
   
   partial functions 
       . [ . ] : matrix × nat × nat → elem with λ mx, r, c. r < #r mx ∧ c < #c mx;
       . [ . ] : matrix × nat × nat × elem → matrix with λ mx, r, c, a. r < #r mx ∧ c < #c mx;
   
   
   induction 
      matrix generated by mkmatrix, [ ];
   
   axioms 
      extension: 
      mx1 = mx2 ↔ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 ∧ (∀ r, c. r < #r mx1 ∧ c < #c mx1 → mx1[r, c] = mx2[r, c]);
      used for: ls;
      
      rowno-mkmatrix: #r mkmatrix(r, c, a) = r;
      used for: s, ls;
      
      colno-mkmatrix: #c mkmatrix(r, c, a) = c;
      used for: s, ls;
      
      get-mkmatrix: r < r0 ∧ c < c0 → mkmatrix(r0, c0, a)[r, c] = a;
      used for: s, ls;
      
      rowno-put: #r mx[r, c, a] = #r mx;
      used for: s, ls;
      
      colno-put: #c mx[r, c, a] = #c mx;
      used for: s, ls;
      
      put-same: r < #r mx ∧ c < #c mx → mx[r, c, a][r, c] = a;
      used for: s, ls, lc;
      
      put-otherrow: r ≠ r0 → mx[r, c, a][r0, c0] = mx[r0, c0];
      used for: s, ls, lc;
      
      put-othercol: c ≠ c0 → mx[r, c, a][r0, c0] = mx[r0, c0];
      used for: s, ls, lc;
   
   
end generic specification