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Proof step 1 from proof zero-ge-add-02 (conclusion) (back to tree)

⊦ 0 ≤ i + j ∧ 0 < j0 → 0 ≤ i * j0 + j * j0

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof zero-ge-add-02 (previous step) (back to tree)

0 < j0, 0 ≤ i + j, ¬ 0 ≤ i * j0 + j * j0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) ⊦ i < 0 ∧ 0 ≤ j → i < j; (s) ⊦ i +1 = i + 1; (s) forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (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 ⊦ i ≤ j ↔ ¬ j < i; (ss)

Proof step 3 from proof zero-ge-add-02 (previous step) (back to tree)

i * j0 + j * j0 < j0, i * j0 + j * j0 < 0, 0 < j0, (i * j0 + j * j0) + 1 < j0, 0 ≤ i + j ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:dist-lsplit, seq: ⊦ (i + i0) * j = i * j + i0 * j, subst:[i0, j, i] → [j, j0, i], <rotated> <all paths>

Additional information

Used terms: j, j0, i

Proof step 4 from proof zero-ge-add-02 (previous step) (back to tree)

⊦ (i + i0) * j = i * j + i0 * j

Lemma dist-lsplit

Proof step 5 from proof zero-ge-add-02 (previous step) (back to tree)

(i + j) * j0 + 1 < j0, 0 < j0, (i + j) * j0 < 0, (i + j) * j0 < j0, 0 ≤ i + j ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:zero-ls, seq: ⊦ 0 < i → (i * j < 0 ↔ j < 0), subst:[j, i] → [j0, i + j], <all paths>

Additional information

Used simplifier rules: zero-left: ⊦ 0 * i = 0; associativity: ⊦ i * j * j0 = (i * j) * j0; commutativity: ⊦ i * j = j * i; ⊦ i < j → ¬ j < i; (s) ⊦ i < j ↔ ¬ j ≤ i; (ss) ⊦ i ≤ j → (j ≤ i ↔ i = j); (s) lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1 trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1 lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1 forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs Used terms: j0, i + j

Proof step 6 from proof zero-ge-add-02 (previous step) (back to tree)

⊦ 0 < i → (i * j < 0 ↔ j < 0)

Lemma zero-ls

Proof Tree for zero-ge-add-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