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Proof step 1 from proof square-eq-01 (conclusion) (back to tree)

⊦ 0.r ≤ r ∧ 0.r ≤ r0 → (r * (r + 1.r) = r0 * (r0 + 1.r) ↔ r = r0)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof square-eq-01 (previous step) (back to tree)

¬(r * (r + 1.r) = r0 * (r0 + 1.r) ↔ r = r0), 0.r ≤ r, 0.r ≤ r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: distr: ⊦ r * (r0 + r1) = r * r0 + r * r1; mult-one: ⊦ 1.r * r = r; mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; mult-com: ⊦ r * r0 = r0 * r; add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic

Proof step 3 from proof square-eq-01 (previous step) (back to tree)

r * r + r = r0 * r0 + r0, r ≠ r0, 0.r ≤ r0, 0.r ≤ r ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-basic, name: blst-01, seq: ⊦ ¬ r < r0 → (r = r0 ↔ ¬ r0 < r), subst:[r, r0] → [r, r0], <all paths>

Additional information

Used terms: r, r0

Proof step 4 from proof square-eq-01 (previous step) (back to tree)

⊦ ¬ r < r0 → (r = r0 ↔ ¬ r0 < r)

A spec-lemma

Proof step 5 from proof square-eq-01 (previous step) (back to tree)

r * r + r = r0 * r0 + r0, r ≠ r0, r < r0, 0.r ≤ r, 0.r ≤ r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ r < r0 ∧ 0.r ≤ r → 0.r < r0; (forward) from specification real-abs

Proof step 6 from proof square-eq-01 (previous step) (back to tree)

r * r + r = r0 * r0 + r0, r ≠ r0, r < r0, 0.r < r0, 0.r ≤ r0, 0.r ≤ r ⊦

User applied rule cut formula

And closed the goal!

Rule argument

fma: r * r < r0 * r0

Additional information

Used simplifier rules: mult-gt-both: ⊦ 0.r ≤ r ∧ 0.r ≤ r0 ∧ 0.r ≤ r1 ∧ 0.r ≤ r2 ∧ r < r1 ∧ r0 < r2 → r * r0 < r1 * r2; mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; mult-com: ⊦ r * r0 = r0 * r; ⊦ r < r0 ∧ r1 < r2 → (r + r1 = r0 + r2 ↔ false); (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used formulas: r * r < r0 * r0

Proof step 7 from proof square-eq-01 (previous step) (back to tree)

r * r + r = r0 * r0 + r0, r0 < r, 0.r ≤ r, 0.r ≤ r0, ¬ r < r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ r < r0 ∧ 0.r ≤ r → 0.r < r0; (forward) from specification real-abs ⊦ r < r0 → ¬ r0 < r; (s)

Proof step 8 from proof square-eq-01 (previous step) (back to tree)

r * r + r = r0 * r0 + r0, r0 < r, 0.r < r, 0.r ≤ r0, 0.r ≤ r ⊦

User applied rule cut formula

And closed the goal!

Rule argument

fma: r0 * r0 < r * r

Additional information

Used simplifier rules: mult-gt-both: ⊦ 0.r ≤ r ∧ 0.r ≤ r0 ∧ 0.r ≤ r1 ∧ 0.r ≤ r2 ∧ r < r1 ∧ r0 < r2 → r * r0 < r1 * r2; mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; mult-com: ⊦ r * r0 = r0 * r; ⊦ r < r0 ∧ r1 < r2 → (r + r1 = r0 + r2 ↔ false); (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used formulas: r0 * r0 < r * r

Proof Tree for square-eq-01 (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