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Proof step 1 from proof limit-unique (conclusion) (back to tree)

⊦ converges(f, r0) → (converges(f, r1) ↔ r0 = r1)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof limit-unique (previous step) (back to tree)

¬(converges(f, r1) ↔ r0 = r1), converges(f, r0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, converges(f, r0), converges(f, r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: convergesto-def, seq: ⊦ converges(f, r) ↔ (∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r) < ε)), subst:[f, r] → [f, r0], <all paths>

Additional information

Used terms: f, r0

Proof step 4 from proof limit-unique (previous step) (back to tree)

⊦ converges(f, r) ↔ (∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r) < ε))

Lemma convergesto-def

Proof step 5 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, converges(f, r1), ∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r0) < ε) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ r - r0 = r + ~ r0; (s) ⊦ n ≥ m ↔ m ≤ n; (s) ⊦ r > r0 ↔ r0 < r; (s)

Proof step 6 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, converges(f, r1), ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r0) < ε) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: convergesto-def, seq: ⊦ converges(f, r) ↔ (∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r) < ε)), subst:[f, r] → [f, r1], <all paths>

Additional information

Used terms: f, r1

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

⊦ converges(f, r) ↔ (∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r) < ε))

Lemma convergesto-def

Proof step 8 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r0) < ε), ∀ ε. ε > 0.r → (∃ n. ∀ n0. n0 ≥ n → abs(f(n0) - r1) < ε) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ r - r0 = r + ~ r0; (s) ⊦ n ≥ m ↔ m ≤ n; (s) ⊦ r > r0 ↔ r0 < r; (s)

Proof step 9 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r0) < ε), ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r1) < ε) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: abs(r0 - r1) * 2.r.^-1 abort: F discard: T computed: F

Additional information

Used terms: abs(r0 - r1) * 2.r.^-1

Proof step 10 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 0.r < abs(r0 - r1) * 2.r.^-1 → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r0) < abs(r0 - r1) * 2.r.^-1), ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r1) < ε) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv 0.r < r ⊦ r0 < r.^-1 * r1 ↔ r * r0 < r1; (s) ⊦ i.r < j.r ↔ i < j; (s) ⊦ r - r0 = r + ~ r0; (s) add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ r + 0.r = r; (s) ⊦ r + ~ r0 = r1 ↔ r = r1 + r0; (s) ⊦ 0.r < abs(r) ↔ r ≠ 0.r; (s) ⊦ 0.r * r = 0.r; (s)

Proof step 11 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ∀ ε. 0.r < ε → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r1) < ε), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: abs(r0 - r1) * 2.r.^-1 abort: F discard: T computed: F

Additional information

Used terms: abs(r0 - r1) * 2.r.^-1

Proof step 12 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 0.r < abs(r0 - r1) * 2.r.^-1 → (∃ n. ∀ n0. n ≤ n0 → abs(f(n0) + ~ r1) < abs(r0 - r1) * 2.r.^-1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv 0.r < r ⊦ r0 < r.^-1 * r1 ↔ r * r0 < r1; (s) ⊦ i.r < j.r ↔ i < j; (s) ⊦ r - r0 = r + ~ r0; (s) add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ r + 0.r = r; (s) ⊦ r + ~ r0 = r1 ↔ r = r1 + r0; (s) ⊦ 0.r < abs(r) ↔ r ≠ 0.r; (s) ⊦ 0.r * r = 0.r; (s)

Proof step 13 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n ≤ n0

Additional information

Used formulas: n ≤ n0

Proof step 14 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n ≤ n0, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: n0 abort: F discard: T computed: T

Additional information

Used terms: n0

Proof step 15 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n ≤ n0, n0 ≤ n0 → 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m; (s)

Proof step 16 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), n ≤ n0, ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: n0 abort: F discard: T computed: T

Additional information

Used terms: n0

Proof step 17 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), n ≤ n0, n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), n ≤ n0 ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1

Additional information

Used formulas: abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1

Proof step 19 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1, n ≤ n0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv ⊦ 1.r * r = r; (s) ⊦ r * r.^-1 = 1.r; (s) ⊦ r + r = 2.r * r; (s)

Proof step 20 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1), n ≤ n0 ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 2, left 3

Additional information

Proof step 21 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1), n ≤ n0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [r0 + ~ r1], <all paths>

Additional information

Used terms: r0 + ~ r1

Proof step 22 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 23 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), n ≤ n0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [f(n0) + ~ r1], <all paths>

Additional information

Used terms: f(n0) + ~ r1

Proof step 24 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 25 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, (f(n0) + ~ r1 < 0.r ⊃ ~(f(n0) + ~ r1);f(n0) + ~ r1) + abs(f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), n ≤ n0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [f(n0) + ~ r0], <all paths>

Additional information

Used terms: f(n0) + ~ r0

Proof step 26 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 27 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, (f(n0) + ~ r1 < 0.r ⊃ ~(f(n0) + ~ r1);f(n0) + ~ r1) + (f(n0) + ~ r0 < 0.r ⊃ ~(f(n0) + ~ r0);f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), n ≤ n0 ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 28 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ~(f(n0) + ~ r1) + (f(n0) + ~ r0 < 0.r ⊃ ~(f(n0) + ~ r0);f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), f(n0) + ~ r1 < 0.r, n ≤ n0 ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Used simplifier rules: ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ ~ ~ r = r; (s) ⊦ r + ~ r0 < r1 ↔ r < r0 + r1; (s) ⊦ r + ~ r0 < 0.r ↔ r < r0; (s) ⊦ r < r0 ↔ ¬ r0 ≤ r; (ss) forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs trans: ⊦ r < r0 ∧ r0 < r1 → r < r1; (forward) from specification real-basic ⊦ r + r = 2.r * r; (s) ⊦ r < r0 + ~ r1 ↔ r + r1 < r0; (s) ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ r + 0.r = r; (s) add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ ¬ r < r; (s)

Proof step 29 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, f(n0) + ~ r1 < 0.r, ~(f(n0) + ~ r1) + ~(f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), f(n0) + ~ r0 < 0.r, n ≤ n0 ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 3

Additional information

Used simplifier rules: ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ ~ ~ r = r; (s) ⊦ ~ r * r0 = ~(r * r0); (s) ⊦ r + r0 < r + r1 ↔ r0 < r1; (s) ⊦ r + ~ r0 < ~ r1 ↔ r + r1 < r0; (s) ⊦ r + r = 2.r * r; (s) ⊦ i > 0 → (i.r * r0 < i.r * r2 ↔ r0 < r2); (s) mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv ⊦ r + ~ r0 < 0.r ↔ r < r0; (s) ⊦ r < r0 → ¬ r0 < r; (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic

Proof step 30 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, (f(n0) + ~ r1) + (f(n0) + ~ r0 < 0.r ⊃ ~(f(n0) + ~ r0);f(n0) + ~ r0) < (r0 + ~ r1 < 0.r ⊃ ~(r0 + ~ r1);r0 + ~ r1), n ≤ n0, ¬ f(n0) + ~ r1 < 0.r ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ r < r0 ↔ ¬ r0 ≤ r; (ss) ⊦ r + ~ r0 < 0.r ↔ r < r0; (s) ⊦ (r + ~ r0) + r1 < r2 ↔ r + r1 < r0 + r2; (s) ⊦ ~ ~ r = r; (s) ⊦ ~(r + r0) = ~ r + ~ r0; (s)

Proof step 31 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, f(n0) + (f(n0) < r0 ⊃ ~ f(n0) + r0;f(n0) + ~ r0) < r1 + (r0 < r1 ⊃ ~ r0 + r1;r0 + ~ r1), n ≤ n0, r1 ≤ f(n0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Used simplifier rules: trans: ⊦ r < r0 ∧ r0 < r1 → r < r1; (forward) from specification real-basic forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ r < r0 → ¬ r0 < r; (s) ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ r + 0.r = r; (s) add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ ¬ r < r; (s)

Proof step 32 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, f(n0) + f(n0) + ~ r0 < r1 + (r0 < r1 ⊃ ~ r0 + r1;r0 + ~ r1), r1 ≤ f(n0), n ≤ n0, ¬ f(n0) < r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 33 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * f(n0) < r0 + r1 + (r0 < r1 ⊃ ~ r0 + r1;r0 + ~ r1), n ≤ n0, r1 ≤ f(n0), r0 ≤ f(n0) ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 2

Additional information

Used simplifier rules: ⊦ ¬ r < r0 → (r = r0 ↔ ¬ r0 < r); (ss) ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ r + 0.r = r; (s) ⊦ r + r = 2.r * r; (s) add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ i > 0 → (i.r * r0 < i.r * r2 ↔ r0 < r2); (s) ⊦ r ≤ r0 → ¬ r0 < r; (s) mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv

Proof step 34 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), n ≤ n0, ¬ abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0) < abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-abs, name: le-lem, seq: ⊦ r < r0 ↔ ¬ r0 ≤ r, subst:[r, r0] → [abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0), abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1], <all paths>

Additional information

Used terms: abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0), abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1

Proof step 35 from proof limit-unique (previous step) (back to tree)

⊦ r < r0 ↔ ¬ r0 ≤ r

A spec-lemma

Proof step 36 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n0) + ~ r1) < abs(r0 + ~ r1), 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), abs(r0 + ~ r1) * 2.r.^-1 + abs(r0 + ~ r1) * 2.r.^-1 ≤ abs(f(n0) + ~ r1) + abs(f(n0) + ~ r0), n ≤ n0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-abs, name: lsls-add-notge, seq: ⊦ r < r0 ∧ r1 < r2 → (r0 + r2 ≤ r + r1 ↔ false), subst:[r0, r2, r, r1] → [abs(r0 + ~ r1) * 2.r.^-1, abs(r0 + ~ r1) * 2.r.^-1, abs(f(n0) + ~ r1), abs(f(n0) + ~ r0)], <all paths>

Additional information

Used simplifier rules: ⊦ r + r = 2.r * r; (s) ⊦ r * r.^-1 = 1.r; (s) ⊦ 1.r * r = r; (s) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic ⊦ i.r < j.r ↔ i < j; (s) 0.r < r ⊦ r0 < r.^-1 * r1 ↔ r * r0 < r1; (s) mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv Used terms: abs(r0 + ~ r1) * 2.r.^-1, abs(r0 + ~ r1) * 2.r.^-1, abs(f(n0) + ~ r1), abs(f(n0) + ~ r0)

Proof step 37 from proof limit-unique (previous step) (back to tree)

⊦ r < r0 ∧ r1 < r2 → (r0 + r2 ≤ r + r1 ↔ false)

A spec-lemma

Proof step 38 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, ¬ n ≤ n0, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 39 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule apply proof lemma

And obtained the following new goals

Rule argument

path: 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1 subst: [r0, r1, n, n0, f] → [r1, r0, n0, n, f]

Additional information

Proof step 40 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n ≤ n0, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Proof step 14 used as proof-lemma

Proof step 41 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦ r1 ≠ r0 ∧ n0 ≤ n ∧ (∀ n1. n ≤ n1 → 2.r * abs(f(n1) + ~ r0) < abs(r1 + ~ r0)) ∧ (∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r1 + ~ r0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)

Proof step 42 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦ (n ≤ n1 → 2.r * abs(f(n1) + ~ r0) < abs(r1 + ~ r0)) ∧ (n0 ≤ n2 → 2.r * abs(f(n2) + ~ r1) < abs(r1 + ~ r0))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 43 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n ≤ n1, ¬ 2.r * abs(f(n1) + ~ r0) < abs(r1 + ~ r0), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ r < r0 ↔ ¬ r0 ≤ r; (ss)

Proof step 44 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n0 < n1, n ≤ n1, abs(r1 + ~ r0) ≤ 2.r * abs(f(n1) + ~ r0), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 45 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n1, n0 < n, abs(r1 + ~ r0) ≤ 2.r * abs(f(n1) + ~ r0), n ≤ n1, n ≤ n1 → 2.r * abs(f(n1) + ~ r0) < abs(r0 + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

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

Proof step 46 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n0 < n1, 2.r * abs(f(n1) + ~ r0) < abs(r0 + ~ r1), abs(r1 + ~ r0) < abs(r0 + ~ r1), n ≤ n1, abs(r1 + ~ r0) ≤ 2.r * abs(f(n1) + ~ r0), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: mult-com, seq: ⊦ r * r0 = r0 * r, subst:[r, r0] → [2.r, abs(f(n1) + ~ r0)], <all paths>

Additional information

Used terms: 2.r, abs(f(n1) + ~ r0)

Proof step 47 from proof limit-unique (previous step) (back to tree)

⊦ r * r0 = r0 * r

A spec-lemma

Proof step 48 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, abs(r1 + ~ r0) < abs(r0 + ~ r1), abs(f(n1) + ~ r0) * 2.r < abs(r0 + ~ r1), n0 < n1, n0 < n, abs(r1 + ~ r0) ≤ abs(f(n1) + ~ r0) * 2.r, n ≤ n1, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 49 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n0 < n1, abs(f(n1) + ~ r0) * 2.r < abs(r0 + ~ r1), abs(r1 + ~ r0) < abs(r0 + ~ r1), n ≤ n1, abs(r1 + ~ r0) ≤ abs(f(n1) + ~ r0) * 2.r, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [r1 + ~ r0], <all paths>

Additional information

Used terms: r1 + ~ r0

Proof step 50 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 51 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, (r1 + ~ r0 < 0.r ⊃ ~(r1 + ~ r0);r1 + ~ r0) < abs(r0 + ~ r1), abs(f(n1) + ~ r0) * 2.r < abs(r0 + ~ r1), n0 < n1, n0 < n, (r1 + ~ r0 < 0.r ⊃ ~(r1 + ~ r0);r1 + ~ r0) ≤ abs(f(n1) + ~ r0) * 2.r, n ≤ n1, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ r + ~ r0 < r1 ↔ r < r0 + r1; (s) ⊦ r < r0 → (r0 ≤ r ↔ false); (as) ⊦ r + ~ r0 ≤ r1 ↔ r ≤ r0 + r1; (s) ⊦ r < r0 + ~ r1 ↔ r + r1 < r0; (s) ⊦ r - r0 = r + ~ r0; (s) ⊦ r0 < r → abs(r + ~ r0) = r - r0; (s) forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ ~ ~ r = r; (s) ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ r + ~ r0 < 0.r ↔ r < r0; (s)

Proof step 52 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, abs(f(n1) + ~ r0) * 2.r < abs(r0 + ~ r1), n0 < n1, n0 < n, r1 < r0 + abs(r0 + ~ r1), (r1 < r0 ⊃ ~ r1 + r0;r1 + ~ r0) ≤ abs(f(n1) + ~ r0) * 2.r, n ≤ n1, ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [r0 + ~ r1], <all paths>

Additional information

Used simplifier rules: ⊦ r + ~ r0 < 0.r ↔ r < r0; (s) ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ ~ ~ r = r; (s) forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ r + 0.r = r; (s) ⊦ ¬ r < r; (s) ⊦ r + ~ r0 < r1 ↔ r < r0 + r1; (s) ⊦ r + r = 2.r * r; (s) ⊦ i > 0 → (i.r * r0 < i.r * r2 ↔ r0 < r2); (s) mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv ⊦ r < r0 → ¬ r0 < r; (s) ⊦ r < r0 + ~ r1 ↔ r + r1 < r0; (s) ⊦ r + ~ r0 ≤ r1 ↔ r ≤ r0 + r1; (s) ⊦ r < r0 → (r0 ≤ r ↔ false); (as) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used terms: r0 + ~ r1

Proof step 53 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 54 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n0 ≤ n2, ¬ 2.r * abs(f(n2) + ~ r1) < abs(r1 + ~ r0), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 55 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, n0 ≤ n2, abs(r1 + ~ r0) ≤ 2.r * abs(f(n2) + ~ r1), ∀ n1. n0 ≤ n1 → 2.r * abs(f(n1) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: n2 abort: F discard: T computed: T

Additional information

Used terms: n2

Proof step 56 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, abs(r1 + ~ r0) ≤ 2.r * abs(f(n2) + ~ r1), n0 ≤ n2, n0 ≤ n2 → 2.r * abs(f(n2) + ~ r1) < abs(r0 + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

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

Proof step 57 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, n0 < n, 2.r * abs(f(n2) + ~ r1) < abs(r0 + ~ r1), abs(r1 + ~ r0) < abs(r0 + ~ r1), n0 ≤ n2, abs(r1 + ~ r0) ≤ 2.r * abs(f(n2) + ~ r1), ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [r1 + ~ r0], <all paths>

Additional information

Used terms: r1 + ~ r0

Proof step 58 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof step 59 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, (r1 + ~ r0 < 0.r ⊃ ~(r1 + ~ r0);r1 + ~ r0) < abs(r0 + ~ r1), 2.r * abs(f(n2) + ~ r1) < abs(r0 + ~ r1), n0 < n, (r1 + ~ r0 < 0.r ⊃ ~(r1 + ~ r0);r1 + ~ r0) ≤ 2.r * abs(f(n2) + ~ r1), n0 ≤ n2, ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ r + ~ r0 < r1 ↔ r < r0 + r1; (s) ⊦ r < r0 → (r0 ≤ r ↔ false); (as) ⊦ r + ~ r0 ≤ r1 ↔ r ≤ r0 + r1; (s) ⊦ r < r0 + ~ r1 ↔ r + r1 < r0; (s) ⊦ r - r0 = r + ~ r0; (s) ⊦ r0 < r → abs(r + ~ r0) = r - r0; (s) forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ ~ ~ r = r; (s) ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ r + ~ r0 < 0.r ↔ r < r0; (s)

Proof step 60 from proof limit-unique (previous step) (back to tree)

r0 ≠ r1, 2.r * abs(f(n2) + ~ r1) < abs(r0 + ~ r1), n0 < n, r1 < r0 + abs(r0 + ~ r1), (r1 < r0 ⊃ ~ r1 + r0;r1 + ~ r0) ≤ 2.r * abs(f(n2) + ~ r1), n0 ≤ n2, ∀ n0. n ≤ n0 → 2.r * abs(f(n0) + ~ r0) < abs(r0 + ~ r1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-multdiv, name: abs-ite, seq: ⊦ abs(r) = (r < 0.r ⊃ ~ r;r), subst:[r] → [r0 + ~ r1], <all paths>

Additional information

Used simplifier rules: ⊦ r + ~ r0 < 0.r ↔ r < r0; (s) ⊦ ~(r + r0) = ~ r + ~ r0; (s) ⊦ ~ ~ r = r; (s) forward-lels: ⊦ r ≤ r0 ∧ r0 < r1 → r < r1; (forward) from specification real-abs ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ r + 0.r = r; (s) ⊦ ¬ r < r; (s) ⊦ r + ~ r0 < r1 ↔ r < r0 + r1; (s) ⊦ r + r = 2.r * r; (s) ⊦ i > 0 → (i.r * r0 < i.r * r2 ↔ r0 < r2); (s) mult-ass: ⊦ (r * r0) * r1 = r * r0 * r1; from specification real-multdiv commutativity: ⊦ r * r0 = r0 * r; from specification real-multdiv ⊦ r < r0 → ¬ r0 < r; (s) ⊦ r < r0 + ~ r1 ↔ r + r1 < r0; (s) ⊦ r + ~ r0 ≤ r1 ↔ r ≤ r0 + r1; (s) ⊦ r < r0 → (r0 ≤ r ↔ false); (as) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic Used terms: r0 + ~ r1

Proof step 61 from proof limit-unique (previous step) (back to tree)

⊦ abs(r) = (r < 0.r ⊃ ~ r;r)

A spec-lemma

Proof Tree for limit-unique (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