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

r * (r + 1.r) ≤ r0 * (r0 + 1.r) + 2.r * r2, r0 * (r0 + 1.r) + 2.r * r2 < (r + 1.r) * (r + 2.r), r2 ≤ r0, 0.r ≤ r, 0.r ≤ r2 ⊦ r < r0 + 1.r ∧ r0 < r + 1.r

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

0.r ≤ r2, 0.r ≤ r, r * (r + 1.r) ≤ r0 * (r0 + 1.r) + 2.r * r2, r0 * (r0 + 1.r) + 2.r * r2 < (r + 1.r) * (r + 2.r), r2 ≤ r0 ⊦ r < r0 + 1.r ∧ r0 < r + 1.r

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 ⊦ r < r + r0 ↔ 0.r < r0; (s) ⊦ r + r0 < r + r1 ↔ r0 < r1; (s) forward-lele: ⊦ r ≤ r0 ∧ r0 ≤ r1 → r ≤ r1; (forward) from specification real-abs forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs

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

0.r ≤ r0, 0.r ≤ r2, 0.r ≤ r, 0.r < r * 2.r + 2.r, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r2 ≤ r0 ⊦ r < r0 + 1.r ∧ r0 < r + 1.r

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 < r + 1.r

Additional information

Used formulas: r0 < r + 1.r

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

0.r < r * 2.r + 2.r, 0.r ≤ r, 0.r ≤ r2, 0.r ≤ r0, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r0 < r + 1.r, r2 ≤ r0 ⊦ r < r0 + 1.r ∧ r0 < r + 1.r

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ r < r0 ↔ ¬ r0 ≤ r; (ss)

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

0.r ≤ r0, 0.r ≤ r2, 0.r ≤ r, 0.r < r * 2.r + 2.r, 0.r < r + 1.r, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, r0 < r + 1.r, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r0 + 1.r ≤ r, r2 ≤ r0, r2 < r + 1.r ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:square-le-04, seq: ⊦ 0.r ≤ r ∧ 0.r ≤ r0 → (r * (r + 1.r) ≤ r0 * (r0 + 1.r) ↔ r ≤ r0), subst:[r, r0] → [r0 + 1.r, r]

Additional information

Used terms: r0 + 1.r, r

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

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

Lemma square-le-04

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

0.r < r + 1.r, 0.r < r * 2.r + 2.r, 0.r ≤ r, 0.r ≤ r2, 0.r ≤ r0, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, r0 + 1.r ≤ r, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r0 < r + 1.r, r2 < r + 1.r, r2 ≤ r0, 0.r ≤ r0 + 1.r ∧ 0.r ≤ r → ((r0 + 1.r) * ((r0 + 1.r) + 1.r) ≤ r * (r + 1.r) ↔ r0 + 1.r ≤ r) ⊦

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 ⊦ r + r0 ≤ r + r2 ↔ r0 ≤ r2; (s) forward-lele: ⊦ r ≤ r0 ∧ r0 ≤ r1 → r ≤ r1; (forward) from specification real-abs ⊦ i.r < j.r ↔ i < j; (s) ⊦ r + r0 < r0 ↔ r < 0.r; (s) ⊦ i < j → ((* i).r + r0 < (* j).r + r1 ↔ r0 < (j - i).r + r1); (s) ⊦ r + r0 < r + r1 ↔ r0 < r1; (s) forward-lsle: ⊦ r < r0 ∧ r0 ≤ r1 → r < r1; (forward) from specification real-abs trans: ⊦ r < r0 ∧ r0 < r1 → r < r1; (forward) from specification real-basic ⊦ r < 0.r ∧ 0.r ≤ r0 → r ≤ r0; (s) ⊦ r ≤ r0 ↔ ¬ r0 < r; (ss) add-c: ⊦ i + j = j + i; from specification int-basic2 add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 ⊦ (* i).r + (* j).r = (i + j).r; (s)

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

0.r ≤ r0 + 1.r, 0.r ≤ r0, 0.r ≤ r2, 0.r ≤ r, 0.r < r * 2.r + 2.r, 0.r < r + 1.r, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, (r0 * 2.r + r0 * r0) + 2.r + r0 ≤ r * r + r, r0 < r + 1.r, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r0 + 1.r ≤ r, r0 * 2.r + 2.r ≤ 2.r * r2, r2 ≤ r0, r2 < r + 1.r ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (r0 + 1.r) * 2.r ≤ 2.r * r2

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-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic ⊦ r ≤ r0 ↔ ¬ r0 < r; (ss) ⊦ r ≤ r0 → ¬ r0 < r; (s) Used formulas: (r0 + 1.r) * 2.r ≤ 2.r * r2

Proof step 9 from proof square-interval (previous step) (back to tree)

0.r < r + 1.r, 0.r < r * 2.r + 2.r, 0.r ≤ r, 0.r ≤ r2, 0.r ≤ r0, 0.r ≤ r0 + 1.r, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, r0 * 2.r + 2.r ≤ 2.r * r2, r0 + 1.r ≤ r, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r0 < r + 1.r, (r0 * 2.r + r0 * r0) + 2.r + r0 ≤ r * r + r, (r0 + 1.r) * 2.r ≤ 2.r * r2, r2 < r + 1.r, r2 ≤ r0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: le-posint, seq: ⊦ i > 0 → (i.r * r0 ≤ i.r * r2 ↔ r0 ≤ r2), subst:[i, r0, r2] → [2, r0 + 1.r, r2], <all paths>

Additional information

Used simplifier rules: mult-com: ⊦ r * r0 = r0 * r; 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; forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs forward-lele: ⊦ r ≤ r0 ∧ r0 ≤ r1 → r ≤ r1; (forward) from specification real-abs ⊦ r + r0 < r + r1 ↔ r0 < r1; (s) ⊦ r + r0 ≤ r ↔ r0 ≤ 0.r; (s) ⊦ i.r ≤ j.r ↔ i ≤ j; (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used terms: 2, r0 + 1.r, r2

Proof step 10 from proof square-interval (previous step) (back to tree)

⊦ i > 0 → (i.r * r0 ≤ i.r * r2 ↔ r0 ≤ r2)

Lemma le-posint

Proof step 11 from proof square-interval (previous step) (back to tree)

0.r < r * 2.r + 2.r, 0.r ≤ r, 0.r ≤ r2, 0.r ≤ r0, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r2 ≤ r0, ¬ r0 < r + 1.r ⊦ r < r0 + 1.r ∧ r0 < r + 1.r

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ r ≤ r0 → ¬ r0 < r; (s) ⊦ r < r0 ↔ ¬ r0 ≤ r; (ss)

Proof step 12 from proof square-interval (previous step) (back to tree)

0.r ≤ r0, 0.r ≤ r2, 0.r ≤ r, 0.r < r * 2.r + 2.r, r * r + r ≤ (r0 * r0 + r0) + 2.r * r2, r + 1.r ≤ r0, (r0 * r0 + r0) + 2.r * r2 < (r * r + r * 2.r) + r + 2.r, r2 ≤ r0 ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:square-le-04, seq: ⊦ 0.r ≤ r ∧ 0.r ≤ r0 → (r * (r + 1.r) ≤ r0 * (r0 + 1.r) ↔ r ≤ r0), subst:[r, r0] → [r + 1.r, r0]

Additional information

Used simplifier rules: distr: ⊦ r * (r0 + r1) = r * r0 + r * r1; mult-one: ⊦ 1.r * r = r; zero-ls-02: ⊦ 0.r ≤ r → (r * r0 < 0.r ↔ r0 < 0.r ∧ r ≠ 0.r); mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; mult-com: ⊦ r * r0 = r0 * r; ⊦ (* i).r + (* j).r = (i + j).r; (s) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 ⊦ r ≤ r0 ↔ ¬ r0 < r; (ss) ⊦ r < 0.r ∧ 0.r ≤ r0 → r ≤ r0; (s) trans: ⊦ r < r0 ∧ r0 < r1 → r < r1; (forward) from specification real-basic ⊦ i < j → ((* i).r + r0 < (* j).r + r1 ↔ r0 < (j - i).r + r1); (s) forward-lsle: ⊦ r < r0 ∧ r0 ≤ r1 → r < r1; (forward) from specification real-abs ⊦ r + r0 < r + r1 ↔ r0 < r1; (s) ⊦ r + r0 < r0 ↔ r < 0.r; (s) ⊦ i.r < j.r ↔ i < j; (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used terms: r + 1.r, r0

Proof step 13 from proof square-interval (previous step) (back to tree)

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

Lemma square-le-04

Proof Tree for square-interval (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