Proof step 1 from proof prime-7 (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬ prime(7) ⊦

Rule argument

Additional information

Proof step 2 from proof prime-7 (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 < 7, ∃ i. ¬(1 < i ∧ i * i ≤ 7 → 7 % i ≠ 0) ⊦

Rule argument

Additional information

Used terms: 7

Proof step 3 from proof prime-7 (previous step) (back to tree)

Lemma prime-def-mod

Proof step 4 from proof prime-7 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  7 % i = 0, 0 < i, 1 < i, i * i ≤ 7 ⊦

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-7 (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 6:

  7 % i = 0, 1 < i, 0 < i, i * i ≤ 7, 3 ≤ i ⊦

• Proof step 10:

  7 % i = 0, 1 < i, 0 < i, i * i ≤ 7, ¬ 3 ≤ i ⊦

Rule argument

Additional information

Used formulas: 3 ≤ i

Proof step 6 from proof prime-7 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  7 % i = 0, 0 < i, 3 ≤ i, i * i ≤ 7, 0 ≤ i ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs i0 ≠ 0, i0 < i ⊦ (* i) ≤ j → (* i0) < j; (s)

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

User applied rule cut formula

And obtained the following new goals

• Proof step 8:

  7 % i = 0, 0 < i, 0 ≤ i, i * i ≤ 7, 3 ≤ i, ¬ 3 * 3 ≤ 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: 3 * 3 ≤ i * i

Proof step 8 from proof prime-7 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ i * i ≤ j * j ↔ abs(i) ≤ abs(j)

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < j → abs(j) = j; (s) ⊦ i ≤ j ↔ ¬ j < i; (ss) ⊦ i ≤ j → ¬ j < i; (s) Used terms: i, 3

Proof step 9 from proof prime-7 (previous step) (back to tree)

A spec-lemma

Proof step 10 from proof prime-7 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  7 % i = 0, 0 < i, 1 < i, i < 3, i * i ≤ 7 ⊦

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 prime-7 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ 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 12 from proof prime-7 (previous step) (back to tree)

A spec-lemma