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Proof step 1 from proof pot-square-est-02 (conclusion) (back to tree)

⊦ 1 < i → ¬ 4 ^ i ≤ (i + 1) * (i + 1)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pot-square-est-02 (previous step) (back to tree)

1 < i, 4 ^ i ≤ (i + 1) * (i + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1

Proof step 3 from proof pot-square-est-02 (previous step) (back to tree)

0 < i, 1 < i, 4 ^ i ≤ (i + 1) * (i + 1) ⊦

User applied rule induction

And obtained the following new goals

Rule argument

indvar: i→n(i) wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: i→n(i)

Proof step 4 from proof pot-square-est-02 (previous step) (back to tree)

0 < i, 1 < i, 4 ^ i ≤ (i + 1) * (i + 1), ∀ i0. i→n(i0) < i→n(i) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof pot-square-est-02 (previous step) (back to tree)

1 < i, 0 < i, 4 ^ i ≤ (i + 1) * (i + 1), ∀ i0. i→n(i0) < i→n(i) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [i0] → [i - 1]

Additional information

Used simplifier rules: ⊦ 1 < i → i→n(i - 1) = i→n(i) -1; (s) ⊦ n -1 = n - 1; (s) ⊦ i ≤ j ↔ ¬ j < i; (ss) ⊦ i < j → ¬ j < i; (s) 0 ≤ i ⊦ (* m) ≤ i→n(i) ↔ n→i(m) ≤ i; (s) ⊦ 1 ≤ i ↔ 0 < i; (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j); (s) ⊦ 1 < i → abs(i) = i; (s) ⊦ ¬ i < i; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: i - 1

Proof step 6 from proof pot-square-est-02 (previous step) (back to tree)

0 < i, 1 < i, 4 ^ i ≤ (i + 1) * (i + 1), ∀ i0. i→n(i0) < i→n(i) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦ 0 < i - 1 ∧ 1 < i - 1 ∧ 4 ^ (i - 1) ≤ ((i - 1) + 1) * ((i - 1) + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2 add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 ⊦ i + (j - i) = j; (s) ⊦ i < j - i0 ↔ i + i0 < j; (s)

Proof step 7 from proof pot-square-est-02 (previous step) (back to tree)

1 < i, 0 < i, 4 ^ i ≤ (i + 1) * (i + 1), ∀ i0. i→n(i0) < i→n(i) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦ 2 < i ∧ 4 ^ (i - 1) ≤ i * i

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: int-abs, name:elim-sub, seq: ⊦ i0 = i - j ↔ i = i0 + j, subst:[i, j, i0] → [i, 1, i1]

Additional information

Used simplifier rules: ⊦ i0 = i - j ↔ i = i0 + j; (elim) Used terms: i, 1, i1

Proof step 8 from proof pot-square-est-02 (previous step) (back to tree)

i = i1 + 1, 0 < i, 1 < i, 4 ^ i ≤ (i + 1) * (i + 1), ∀ i0. i→n(i0) < i→n(i) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦ 2 < i ∧ 4 ^ i1 ≤ i * i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: succ: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j; forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs add-c: ⊦ i + j = j + i; from specification int-basic2 add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s) ⊦ ~1 < i ↔ 0 ≤ i; (s)

Proof step 9 from proof pot-square-est-02 (previous step) (back to tree)

0 < i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦ 1 < i1 ∧ 4 ^ i1 ≤ (i1 + 1) * (i1 + 1)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: 1 < i1

Additional information

Used simplifier rules: ⊦ i < j ↔ ¬ j ≤ i; (ss) ⊦ 0 < i → (i ≤ 1 ↔ i = 1); (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: 1 < i1

Proof step 10 from proof pot-square-est-02 (previous step) (back to tree)

0 < i1, 1 < i1, 0 ≤ i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦ 1 < i1 ∧ 4 ^ i1 ≤ (i1 + 1) * (i1 + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1 ⊦ i ≤ j ↔ ¬ j < i; (ss)

Proof step 11 from proof pot-square-est-02 (previous step) (back to tree)

(i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4

Additional information

Used simplifier rules: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s) ⊦ i < j ↔ ¬ j ≤ i; (ss) ⊦ i < j → (j ≤ i ↔ false); (as) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4

Proof step 12 from proof pot-square-est-02 (previous step) (back to tree)

0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (i1 + 1) * (i1 + 1) * 4 < (2 + i1) * (2 + i1)

Additional information

Used simplifier rules: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s) ⊦ i < j ↔ ¬ j ≤ i; (ss) forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); (s) ⊦ i < j → (j ≤ i ↔ false); (as) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: (i1 + 1) * (i1 + 1) * 4 < (2 + i1) * (2 + i1)

Proof step 13 from proof pot-square-est-02 (previous step) (back to tree)

(i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, (i1 + 1) * (i1 + 1) * 4 < (2 + i1) * (2 + i1), 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (2 + i1) * (2 + i1) = i1 * i1 + 4 * i1 + 4

Additional information

Used formulas: (2 + i1) * (2 + i1) = i1 * i1 + 4 * i1 + 4

Proof step 14 from proof pot-square-est-02 (previous step) (back to tree)

(2 + i1) * (2 + i1) = i1 * i1 + 4 * i1 + 4, (i1 + 1) * (i1 + 1) * 4 < (2 + i1) * (2 + i1), 0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 15 from proof pot-square-est-02 (previous step) (back to tree)

(i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, (i1 + 1) * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (i1 + 1) * (i1 + 1) * 4 = 4 * i1 * i1 + 8 * i1 + 4

Additional information

Used formulas: (i1 + 1) * (i1 + 1) * 4 = 4 * i1 * i1 + 8 * i1 + 4

Proof step 16 from proof pot-square-est-02 (previous step) (back to tree)

(i1 + 1) * (i1 + 1) * 4 = 4 * i1 * i1 + 8 * i1 + 4, (i1 + 1) * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 17 from proof pot-square-est-02 (previous step) (back to tree)

4 * i1 * i1 + 8 * i1 + 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, 4 * i1 * i1 + 8 * i1 + 4 < i1 * i1 + 4 * i1 + 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2 add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 commutativity: ⊦ i * j = j * i; from specification int-mult associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult ⊦ i < 0 ∧ 0 ≤ j → i < j; (s) ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s) ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0; (s) ⊦ (* i2) * i + j < (* i1) * i + j0 ↔ (i2 - i1) * i + j < j0; (s) ⊦ i +1 = i + 1; (s) forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1 lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1

Proof step 18 from proof pot-square-est-02 (previous step) (back to tree)

3 * i1 * i1 + 4 * i1 < 0, 4 * i1 * i1 + 8 * i1 + 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, (3 * i1 * i1 + 4 * i1) + 1 < i1, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic2, name:pls-01, seq:¬ i < 0, ¬ j < 0 ⊦ ¬ i + j < 0, subst:[i, j] → [3 * i1 * i1, 4 * i1], <all paths>

Additional information

Used simplifier rules: ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s) ⊦ i < j → ¬ j < i; (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 3 * i1 * i1, 4 * i1

Proof step 19 from proof pot-square-est-02 (previous step) (back to tree)

¬ i < 0, ¬ j < 0 ⊦ ¬ i + j < 0

A spec-lemma

Proof step 20 from proof pot-square-est-02 (previous step) (back to tree)

(i1 + 1) * (i1 + 1) * 4 ≠ 4 * i1 * i1 + 8 * i1 + 4, (i1 + 1) * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[j, i0, i] → [(i1 + 1) * 4, 1, i1], <all paths>

Additional information

Used terms: (i1 + 1) * 4, 1, i1

Proof step 21 from proof pot-square-est-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

A spec-lemma

Proof step 22 from proof pot-square-est-02 (previous step) (back to tree)

i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 ≠ 4 * i1 * i1 + 8 * i1 + 4, i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, i1 * (i1 + 1) * 4 + 1 * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[j, i0, i] → [4 * i1, 1, i1], <all paths>

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 4 * i1, 1, i1

Proof step 23 from proof pot-square-est-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

A spec-lemma

Proof step 24 from proof pot-square-est-02 (previous step) (back to tree)

(i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 ≠ 4 * i1 * i1 + 8 * i1 + 4, (i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 * 4 * i1 + 1 * 4 * i1) + 1 * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:one, seq: ⊦ 1 * i = i, subst:[i] → [(i1 + 1) * 4], <all paths>

Additional information

Used terms: (i1 + 1) * 4

Proof step 25 from proof pot-square-est-02 (previous step) (back to tree)

⊦ 1 * i = i

A spec-lemma

Proof step 26 from proof pot-square-est-02 (previous step) (back to tree)

(i1 * 4 * i1 + 1 * 4 * i1) + (i1 + 1) * 4 ≠ 4 * i1 * i1 + 8 * i1 + 4, (i1 * 4 * i1 + 1 * 4 * i1) + (i1 + 1) * 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, (i1 * 4 * i1 + 1 * 4 * i1) + (i1 + 1) * 4 < i1 * i1 + 4 * i1 + 4, 4 ^ i1 * 4 ≤ i1 * i1 + 4 * i1 + 4, 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[j, i0, i] → [4, 1, i1], <all paths>

Additional information

Used simplifier rules: ⊦ i * j + i * j0 = i * (j + j0); (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult ⊦ i + j = i + i0 ↔ j = i0; (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: 4, 1, i1

Proof step 27 from proof pot-square-est-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

A spec-lemma

Proof step 28 from proof pot-square-est-02 (previous step) (back to tree)

(2 + i1) * (2 + i1) ≠ i1 * i1 + 4 * i1 + 4, (i1 + 1) * (i1 + 1) * 4 < (2 + i1) * (2 + i1), 0 < i1, 1 < i1, (i1 + 1) * (i1 + 1) < 4 ^ i1, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, 0 ≤ i1, 4 ^ i1 * 4 ≤ (2 + i1) * (2 + i1), ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[j, i0, i] → [2 + i1, i1, 2], <all paths>

Additional information

Used terms: 2 + i1, i1, 2

Proof step 29 from proof pot-square-est-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

A spec-lemma

Proof step 30 from proof pot-square-est-02 (previous step) (back to tree)

2 * (2 + i1) + i1 * (2 + i1) ≠ i1 * i1 + 4 * i1 + 4, (i1 + 1) * (i1 + 1) * 4 < 4 ^ i1 * 4, (i1 + 1) * (i1 + 1) < 4 ^ i1, 1 < i1, 0 < i1, (i1 + 1) * (i1 + 1) * 4 < 2 * (2 + i1) + i1 * (2 + i1), 4 ^ i1 * 4 ≤ 2 * (2 + i1) + i1 * (2 + i1), 0 ≤ i1, ∀ i0. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 1 < i0 ∧ 4 ^ i0 ≤ (i0 + 1) * (i0 + 1) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[j, i0, i] → [i1, i1, 2], <all paths>

Additional information

Used simplifier rules: ⊦ i + i = 2 * i; (s) ⊦ i * j + i = i * (j + 1); (s) ⊦ i * (i0 + 1) + j = i + j0 ↔ i * i0 + j = j0; (s) ⊦ i * j + i * j0 = i * (j + j0); (s) ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i1, i1, 2

Proof step 31 from proof pot-square-est-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

A spec-lemma

Proof Tree for pot-square-est-02 (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