Proof step 1 from proof prime-23 (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬ prime(23) ⊦

Rule argument

Additional information

Proof step 2 from proof prime-23 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 3:

  ⊦ 1 < j → (prime(j) ↔ (∀ i. 1 < i ∧ i * i ≤ j → j % i ≠ 0))

• Proof step 4:

  1 < 23, ∃ i. ¬(1 < i ∧ i * i ≤ 23 → 23 % i ≠ 0) ⊦

Rule argument

Additional information

Used terms: 23

Proof step 3 from proof prime-23 (previous step) (back to tree)

Lemma prime-def-mod

Proof step 4 from proof prime-23 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  23 % i = 0, 0 < i, 1 < i, i * i ≤ 23 ⊦

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 5 from proof prime-23 (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 6:

  23 % i = 0, 1 < i, 0 < i, i * i ≤ 23, ¬ 5 * 5 ≤ i * i ⊦

Rule argument

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 forward-le: ⊦ 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 forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs Used formulas: 5 * 5 ≤ i * i

Proof step 6 from proof prime-23 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ i * i ≤ j * j ↔ abs(i) ≤ abs(j)

• Proof step 8:

  23 % i = 0, 0 < i, 1 < i, i * i ≤ 23, ¬ abs(5) ≤ abs(i) ⊦

Rule argument

Additional information

Used terms: i, 5

Proof step 7 from proof prime-23 (previous step) (back to tree)

A spec-lemma

Proof step 8 from proof prime-23 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  23 % i = 0, 1 < i, 0 < i, i < 5, i * i ≤ 23 ⊦

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) ⊦ 1 < i → abs(i) = i; (s)

Proof step 9 from proof prime-23 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 10:

  ⊦ i ≠ j -1 → (i < j ↔ i < j -1)

• Proof step 11:

  23 % i = 0, i ≠ 5 -1, i < 5 -1, 0 < i, 1 < i, i * i ≤ 23 ⊦

Rule argument

Additional information

Used terms: 5, i

Proof step 10 from proof prime-23 (previous step) (back to tree)

A spec-lemma

Proof step 11 from proof prime-23 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  23 % i = 0, i ≠ 4, 1 < i, 0 < i, i < 4, i * i ≤ 23 ⊦

Rule argument

Additional information

Proof step 12 from proof prime-23 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ i ≠ j -1 → (i < j ↔ i < j -1)

• Proof step 14:

  23 % i = 0, i ≠ 4 -1, i ≠ 4, i < 4 -1, 0 < i, 1 < i, i * i ≤ 23 ⊦

Rule argument

Additional information

Used terms: 4, i

Proof step 13 from proof prime-23 (previous step) (back to tree)

A spec-lemma

Proof step 14 from proof prime-23 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  23 % i = 0, i ≠ 4, i ≠ 3, 1 < i, 0 < i, i < 3, i * i ≤ 23 ⊦

Rule argument

Additional information

Proof step 15 from proof prime-23 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 16:

  ⊦ i ≠ j -1 → (i < j ↔ i < j -1)

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < j → (j < 2 ↔ j = 1); (s) ⊦ i = 1 → j % i = 0; (s) ⊦ i = 1 → i * j = j; (s) associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult commutativity: ⊦ i * j = j * i; from specification int-mult forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs Used terms: 3, i

Proof step 16 from proof prime-23 (previous step) (back to tree)

A spec-lemma