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Specification cube
enrich
   nat
with
   sorts cube('a);
   functions 
       mkcube : nat × nat × nat × 'a → cube('a);
       #x . : cube('a) → nat;
       #y . : cube('a) → nat;
       #z . : cube('a) → nat;
   
   variables 
       cu : cube('a);
       nx : nat;
       ny : nat;
       nz : nat;
       a : 'a;
       b : 'a;
   
   partial functions 
       . [ . ] : cube('a) × nat × nat × nat → 'a with λ cu, nx, ny, nz. nx < #x cu ∧ ny < #y cu ∧ nz < #z cu;
       . [ . := . ] : cube('a) × nat × nat × nat × 'a → cube('a) with λ cu, nx, ny, nz, a. nx < #x cu ∧ ny < #y cu ∧ nz < #z cu;
   
   
   induction 
      cube('a) generated by mkcube, [ := ];
   
   axioms 
      extension: 
        cu1 = cu2
      ↔   #x cu1 = #x cu2 ∧ #y cu1 = #y cu2 ∧ #z cu1 = #z cu2
        ∧ (∀ nx, ny, nz. nx < #x cu1 ∧ ny < #y cu1 ∧ nz < #z cu1 → cu1[nx, ny, nz] = cu2[nx, ny, nz]);
      used for: ls;
      
      width-mkcube: #x mkcube(nx, ny, nz, a) = (ny = 0 ∨ nz = 0 ⊃ 0;nx);
      used for: s, ls;
      
      depth-mkcube: #y mkcube(nx, ny, nz, a) = (nx = 0 ∨ nz = 0 ⊃ 0;ny);
      used for: s, ls;
      
      height-mkcube: #z mkcube(nx, ny, nz, a) = (nx = 0 ∨ ny = 0 ⊃ 0;nz);
      used for: s, ls;
      
      get-mkcube: nx < nx0, ny < ny0, nz < nz0 ⊦ mkcube(nx0, ny0, nz0, a)[nx, ny, nz] = a;
      used for: s, ls;
      
      width-put: #x cu[nx, ny, nz := a] = #x cu;
      used for: s, ls;
      
      depth-put: #y cu[nx, ny, nz := a] = #y cu;
      used for: s, ls;
      
      height-put: #z cu[nx, ny, nz := a] = #z cu;
      used for: s, ls;
      
      put-same: nx < #x cu, ny < #y cu, nz < #z cu ⊦ (cu[nx, ny, nz := a])[nx, ny, nz] = a;
      used for: s, ls;
      
      put-otherwidth: nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];
      used for: s, ls, lc;
      
      put-otherdepth: ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];
      used for: s, ls, lc;
      
      put-otherheight: nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];
      used for: s, ls, lc;
   
   
end enrich