[Imprint][Data Privacy Statement][E-Mail]



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Proof step 1 from proof eq-write-2-c (conclusion) (back to tree)

⊦ c ≠ c0 → ( (mx[r, c := a0])[r0, c0 := a1] = (mx[r0, c0 := b1])[r, c := b0] ↔ (r < #r mx ∧ c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof eq-write-2-c (previous step) (back to tree)

¬ ( (mx[r, c := a0])[r0, c0 := a1] = (mx[r0, c0 := b1])[r, c := b0] ↔ (r < #r mx ∧ c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)), c ≠ c0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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]); rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx;

Proof step 3 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0 ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (r < #r mx ∧ c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r < #r mx

Additional information

Used formulas: r < #r mx

Proof step 4 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (r < #r mx ∧ c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 5 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: c < #c mx

Additional information

Used formulas: c < #c mx

Proof step 6 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 7 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a0 = b0 ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: a0 = b0

Additional information

Used formulas: a0 = b0

Proof step 8 from proof eq-write-2-c (previous step) (back to tree)

a0 = b0, c ≠ c0, c < #c mx, r < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a0 = b0 ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 < #r mx

Additional information

Used formulas: r0 < #r mx

Proof step 10 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c < #c mx, r < #r mx, r0 < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < #r mx, r < #r mx, c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: c0 < #c mx

Additional information

Used formulas: c0 < #c mx

Proof step 12 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 13 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: a1 = b1

Additional information

Used formulas: a1 = b1

Proof step 14 from proof eq-write-2-c (previous step) (back to tree)

a1 = b1, c ≠ c0, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0];

Proof step 15 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := b0])[r0, c0 := b1])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 = r1

Additional information

Used formulas: r0 = r1

Proof step 16 from proof eq-write-2-c (previous step) (back to tree)

r0 = r1, ((mx[r, c := b0])[r0, c0 := b1])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := b0])[r1, c0 := b1])[r1, c1] ≠ ((mx[r1, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c0 < #c mx, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: c0 = c1

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; Used formulas: c0 = c1

Proof step 18 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := b0])[r1, c0 := b1])[r1, c1] ≠ ((mx[r1, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c0 ≠ c1, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, c0 < #c mx ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0];

Proof step 19 from proof eq-write-2-c (previous step) (back to tree)

(mx[r, c := b0])[r1, c1] ≠ ((mx[r1, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c0 ≠ c1, c0 < #c mx, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r = r1

Additional information

Used simplifier rules: put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: r = r1

Proof step 20 from proof eq-write-2-c (previous step) (back to tree)

r = r1, (mx[r, c := b0])[r1, c1] ≠ ((mx[r1, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c0 ≠ c1, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, c0 < #c mx ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 21 from proof eq-write-2-c (previous step) (back to tree)

(mx[r1, c := b0])[r1, c1] ≠ ((mx[r1, c0 := b1])[r1, c := b0])[r1, c1], c ≠ c0, c0 ≠ c1, c0 < #c mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c = c1

Additional information

Used simplifier rules: put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: c = c1

Proof step 22 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := b0])[r0, c0 := b1])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r0 ≠ r1, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0];

Proof step 23 from proof eq-write-2-c (previous step) (back to tree)

(mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r0 ≠ r1, c0 < #c mx, r0 < #r mx, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r = r1

Additional information

Used simplifier rules: put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: r = r1

Proof step 24 from proof eq-write-2-c (previous step) (back to tree)

r = r1, (mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r0 ≠ r1, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 25 from proof eq-write-2-c (previous step) (back to tree)

(mx[r1, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r1, c := b0])[r1, c1], c ≠ c0, r0 ≠ r1, c0 < #c mx, r0 < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c = c1

Additional information

Used simplifier rules: put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: c = c1

Proof step 26 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, c < #c mx, r < #r mx, r0 < #r mx, c0 < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c < #c mx, ∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: r0, c0

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; Used terms: r0, c0

Proof step 28 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c < #c mx, r < #r mx, r0 < #r mx, ¬ c0 < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 29 from proof eq-write-2-c (previous step) (back to tree)

(mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r0 < #r mx, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx, c < c0, c1 < c0, #c mx ≤ c0 ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r = r1

Additional information

Used simplifier rules: put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; ngp-07-c: ⊦ c < c0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; Used formulas: r = r1

Proof step 30 from proof eq-write-2-c (previous step) (back to tree)

r = r1, (mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, c1 < c0, c < c0, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, r0 < #r mx, #c mx ≤ c0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 31 from proof eq-write-2-c (previous step) (back to tree)

(mx[r1, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r1, c := b0])[r1, c1], c ≠ c0, r0 < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx, c < c0, c1 < c0, #c mx ≤ c0 ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c = c1

Additional information

Used simplifier rules: put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; ngp-07-c: ⊦ c < c0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; Used formulas: c = c1

Proof step 32 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c < #c mx, r < #r mx, ¬ r0 < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := b0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-r: r ≠ r0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 33 from proof eq-write-2-c (previous step) (back to tree)

(mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r < #r mx, c < #c mx, r1 < #r mx, c1 < #c mx, r < r0, r1 < r0, #r mx ≤ r0 ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r = r1

Additional information

Used simplifier rules: put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; ngp-07-r: ⊦ r < r0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; Used formulas: r = r1

Proof step 34 from proof eq-write-2-c (previous step) (back to tree)

r = r1, (mx[r, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r, c := b0])[r1, c1], c ≠ c0, r1 < r0, r < r0, c1 < #c mx, r1 < #r mx, c < #c mx, r < #r mx, #r mx ≤ r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 35 from proof eq-write-2-c (previous step) (back to tree)

(mx[r1, c := b0])[r1, c1] ≠ ((mx[r0, c0 := b1])[r1, c := b0])[r1, c1], c ≠ c0, c < #c mx, r1 < #r mx, c1 < #c mx, r1 < r0, #r mx ≤ r0 ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c = c1

Additional information

Used simplifier rules: put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; ngp-07-r: ⊦ r < r0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; Used formulas: c = c1

Proof step 36 from proof eq-write-2-c (previous step) (back to tree)

a0 ≠ b0, c ≠ c0, c < #c mx, r < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ a0 = b0 ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 37 from proof eq-write-2-c (previous step) (back to tree)

a0 ≠ b0, c ≠ c0, r < #r mx, c < #c mx, ∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: r, c

Additional information

Used simplifier rules: put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; Used terms: r, c

Proof step 38 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, ¬ c < #c mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 39 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 < #r mx

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; cngetput-r: r ≠ r0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) Used formulas: r0 < #r mx

Proof step 40 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, r0 < #r mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 41 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < #r mx, r < #r mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: c0 < #c mx

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) Used formulas: c0 < #c mx

Proof step 42 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r < #r mx, r0 < #r mx, c0 < #c mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat

Proof step 43 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c0 < c, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: a1 = b1

Additional information

Used formulas: a1 = b1

Proof step 44 from proof eq-write-2-c (previous step) (back to tree)

a1 = b1, c ≠ c0, c0 < c, r < #r mx, r0 < #r mx, c0 < #c mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat

Proof step 45 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := a0])[r0, c0 := b1])[r1, c1] ≠ (mx[r0, c0 := b1])[r1, c1], c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c0 < c, r1 < #r mx, c1 < #c mx, c1 < c, #c mx ≤ c ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 = r1

Additional information

Used simplifier rules: ngp-07-c: ⊦ c < c0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: r0 = r1

Proof step 46 from proof eq-write-2-c (previous step) (back to tree)

r0 = r1, ((mx[r, c := a0])[r0, c0 := b1])[r1, c1] ≠ (mx[r0, c0 := b1])[r1, c1], c ≠ c0, c1 < c, c1 < #c mx, r1 < #r mx, c0 < c, r < #r mx, r0 < #r mx, c0 < #c mx, #c mx ≤ c ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 47 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := a0])[r1, c0 := b1])[r1, c1] ≠ (mx[r1, c0 := b1])[r1, c1], c ≠ c0, c0 < #c mx, r < #r mx, c0 < c, r1 < #r mx, c1 < #c mx, c1 < c, #c mx ≤ c ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c0 = c1

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; ngp-07-c: ⊦ c < c0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: c0 = c1

Proof step 48 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, c0 < c, r < #r mx, r0 < #r mx, c0 < #c mx, #c mx ≤ c ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 49 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, c0 < #c mx, r0 < #r mx, r < #r mx, c0 < c, #c mx ≤ c, ∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: r0, c0

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; Used terms: r0, c0

Proof step 50 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, ¬ r < #r mx ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = ((mx[r0, c0 := b1])[r, c := b0])[r1, c1]) ↔ (r < #r mx ∧ c < #c mx → a0 = b0) ∧ (r0 < #r mx ∧ c0 < #c mx → a1 = b1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-r: r ≠ r0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 51 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 < #r mx

Additional information

Used simplifier rules: cngetput-r: r ≠ r0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) Used formulas: r0 < #r mx

Proof step 52 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < #r mx, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ r0 < #r mx ∧ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat

Proof step 53 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < #r mx, r0 < r, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: c0 < #c mx

Additional information

Used simplifier rules: cngetput-r: r ≠ r0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) Used formulas: c0 < #c mx

Proof step 54 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < r, r0 < #r mx, c0 < #c mx, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ c0 < #c mx → a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 55 from proof eq-write-2-c (previous step) (back to tree)

c ≠ c0, r0 < r, r0 < #r mx, c0 < #c mx, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: a1 = b1

Additional information

Used formulas: a1 = b1

Proof step 56 from proof eq-write-2-c (previous step) (back to tree)

a1 = b1, c ≠ c0, c0 < #c mx, r0 < #r mx, r0 < r, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: cngetput-c: c ≠ c0 ⊦ (mx[r, c := a])[r0, c0] = mx[r0, c0]; fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat

Proof step 57 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := a0])[r0, c0 := b1])[r1, c1] ≠ (mx[r0, c0 := b1])[r1, c1], c ≠ c0, r0 < r, r0 < #r mx, c0 < #c mx, r1 < #r mx, c1 < #c mx, r1 < r, #r mx ≤ r ⊦

Heuristic cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 = r1

Additional information

Used simplifier rules: ngp-07-r: ⊦ r < r0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; put-otherrow: ⊦ r ≠ r0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: r0 = r1

Proof step 58 from proof eq-write-2-c (previous step) (back to tree)

r0 = r1, ((mx[r, c := a0])[r0, c0 := b1])[r1, c1] ≠ (mx[r0, c0 := b1])[r1, c1], c ≠ c0, r1 < r, c1 < #c mx, r1 < #r mx, c0 < #c mx, r0 < #r mx, r0 < r, #r mx ≤ r ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 59 from proof eq-write-2-c (previous step) (back to tree)

((mx[r, c := a0])[r1, c0 := b1])[r1, c1] ≠ (mx[r1, c0 := b1])[r1, c1], c ≠ c0, c0 < #c mx, r1 < #r mx, c1 < #c mx, r1 < r, #r mx ≤ r ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: c0 = c1

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; ngp-07-r: ⊦ r < r0 → (mx[r0, c0 := a])[r, c] = mx[r, c]; put-othercol: ⊦ c ≠ c0 → (mx[r, c := a])[r0, c0] = mx[r0, c0]; Used formulas: c0 = c1

Proof step 60 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, c0 < #c mx, r0 < #r mx, r0 < r, #r mx ≤ r ⊦ (∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1]) ↔ a1 = b1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 61 from proof eq-write-2-c (previous step) (back to tree)

a1 ≠ b1, c ≠ c0, r0 < r, r0 < #r mx, c0 < #c mx, #r mx ≤ r, ∀ r1, c1. r1 < #r mx ∧ c1 < #c mx → ((mx[r, c := a0])[r0, c0 := a1])[r1, c1] = (mx[r0, c0 := b1])[r1, c1] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: r0, c0

Additional information

Used simplifier rules: rowno-put: ⊦ #r mx[r, c := a] = #r mx; colno-put: ⊦ #c mx[r, c := a] = #c mx; cgetput: r < #r mx, c < #c mx ⊦ (mx[r, c := a])[r, c] = a; put-same: ⊦ r < #r mx ∧ c < #c mx → (mx[r, c := a])[r, c] = a; Used terms: r0, c0

Proof Tree for eq-write-2-c (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree