Proof step 1 from proof write3 (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a] ≠ (cu[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a] ⊦

Rule argument

Additional information

Proof step 2 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  (((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz], nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Used simplifier rules: 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]); width-put: ⊦ #x cu[nx, ny, nz := a] = #x cu; depth-put: ⊦ #y cu[nx, ny, nz := a] = #y cu; height-put: ⊦ #z cu[nx, ny, nz := a] = #z cu; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 3 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 4:

  ny0 = ny, (((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz], nx < #x cu, ny < #y cu, nz < #z cu ⊦

• Proof step 20:

  (((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny0, nz0 := a])[nx, ny, nz], ny0 ≠ ny, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used formulas: ny0 = ny

Proof step 4 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  (((cu[nx0, ny, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz], nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 5 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 6:

  nz0 = nz, (((cu[nx0, ny, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz], nx < #x cu, ny < #y cu, nz < #z cu ⊦

• Proof step 14:

  (((cu[nx0, ny, nz0 := a0])[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny, nz0 := a])[nx, ny, nz], nz0 ≠ nz, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used formulas: nz0 = nz

Proof step 6 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  (((cu[nx0, ny, nz := a0])[nx1, ny1, nz1 := b])[nx0, ny, nz := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny, nz := a])[nx, ny, nz], nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 7 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 8:

  (((cu[nx0, ny, nz := a0])[nx1, ny1, nz1 := b])[nx0, ny, nz := a])[nx, ny, nz] ≠ ((cu[nx1, ny1, nz1 := b])[nx0, ny, nz := a])[nx, ny, nz], nx0 ≠ nx, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: width-put: ⊦ #x cu[nx, ny, nz := a] = #x cu; depth-put: ⊦ #y cu[nx, ny, nz := a] = #y cu; height-put: ⊦ #z cu[nx, ny, nz := a] = #z cu; put-same: nx < #x cu, ny < #y cu, nz < #z cu ⊦ (cu[nx, ny, nz := a])[nx, ny, nz] = a; Used formulas: nx0 = nx

Proof step 8 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  ((cu[nx0, ny, nz := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], nx0 ≠ nx, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];

Proof step 9 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 10:

  ny1 = ny, ((cu[nx0, ny, nz := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], nx0 ≠ nx, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: ny1 = ny

Proof step 10 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  ((cu[nx0, ny, nz := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], nx0 ≠ nx, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 11 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 12:

  nz1 = nz, ((cu[nx0, ny, nz := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], nx0 ≠ nx, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nz1 = nz

Proof step 12 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  ((cu[nx0, ny, nz := a0])[nx1, ny, nz := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz := b])[nx, ny, nz], nx0 ≠ nx, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 13 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And closed the goal!

Rule argument

Additional information

Used simplifier rules: width-put: ⊦ #x cu[nx, ny, nz := a] = #x cu; depth-put: ⊦ #y cu[nx, ny, nz := a] = #y cu; height-put: ⊦ #z cu[nx, ny, nz := a] = #z cu; put-same: nx < #x cu, ny < #y cu, nz < #z cu ⊦ (cu[nx, ny, nz := a])[nx, ny, nz] = a; put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nx1 = nx

Proof step 14 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  ((cu[nx0, ny, nz0 := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], nz0 ≠ nz, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];

Proof step 15 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 16:

  ny1 = ny, ((cu[nx0, ny, nz0 := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], nz0 ≠ nz, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: ny1 = ny

Proof step 16 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  ((cu[nx0, ny, nz0 := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], nz0 ≠ nz, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 17 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 18:

  nz1 = nz, ((cu[nx0, ny, nz0 := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], nz0 ≠ nz, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nz1 = nz

Proof step 18 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  ((cu[nx0, ny, nz0 := a0])[nx1, ny, nz := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz := b])[nx, ny, nz], nz0 ≠ nz, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 19 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And closed the goal!

Rule argument

Additional information

Used simplifier rules: width-put: ⊦ #x cu[nx, ny, nz := a] = #x cu; depth-put: ⊦ #y cu[nx, ny, nz := a] = #y cu; height-put: ⊦ #z cu[nx, ny, nz := a] = #z cu; put-same: nx < #x cu, ny < #y cu, nz < #z cu ⊦ (cu[nx, ny, nz := a])[nx, ny, nz] = a; put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nx1 = nx

Proof step 20 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  ((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], ny0 ≠ ny, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0];

Proof step 21 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 22:

  ny1 = ny, ((cu[nx0, ny0, nz0 := a0])[nx1, ny1, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny1, nz1 := b])[nx, ny, nz], ny0 ≠ ny, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: ny1 = ny

Proof step 22 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

  ((cu[nx0, ny0, nz0 := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], ny0 ≠ ny, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 23 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 24:

  nz1 = nz, ((cu[nx0, ny0, nz0 := a0])[nx1, ny, nz1 := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz1 := b])[nx, ny, nz], ny0 ≠ ny, nx < #x cu, ny < #y cu, nz < #z cu ⊦

Rule argument

Additional information

Used simplifier rules: put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherheight: ⊦ nz ≠ nz0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nz1 = nz

Proof step 24 from proof write3 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

  ((cu[nx0, ny0, nz0 := a0])[nx1, ny, nz := b])[nx, ny, nz] ≠ (cu[nx1, ny, nz := b])[nx, ny, nz], ny0 ≠ ny, nz < #z cu, ny < #y cu, nx < #x cu ⊦

Rule argument

Additional information

Proof step 25 from proof write3 (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And closed the goal!

Rule argument

Additional information

Used simplifier rules: width-put: ⊦ #x cu[nx, ny, nz := a] = #x cu; depth-put: ⊦ #y cu[nx, ny, nz := a] = #y cu; height-put: ⊦ #z cu[nx, ny, nz := a] = #z cu; put-same: nx < #x cu, ny < #y cu, nz < #z cu ⊦ (cu[nx, ny, nz := a])[nx, ny, nz] = a; put-otherdepth: ⊦ ny ≠ ny0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; put-otherwidth: ⊦ nx ≠ nx0 → (cu[nx, ny, nz := a])[nx0, ny0, nz0] = cu[nx0, ny0, nz0]; Used formulas: nx1 = nx