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Proof step 1 from proof monls-exists-maxleq (conclusion) (back to tree)

mon<(nσ), nσ[0] < n, n ≤ nσ[m], ⌜m⌝ < # nσ ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof monls-exists-maxleq (previous step) (back to tree)

⌜m⌝ < # nσ, nσ[0] < n, mon<(nσ), n ≤ nσ[m] ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

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

Proof step 3 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[0] < n, ⌜m⌝ < # nσ, nσ[0] < nσ[m], mon<(nσ), n ≤ nσ[m] ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n = nσ[m]

Additional information

Used formulas: n = nσ[m]

Proof step 4 from proof monls-exists-maxleq (previous step) (back to tree)

n = nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < n, mon<(nσ), n ≤ nσ[m] ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m; (s)

Proof step 5 from proof monls-exists-maxleq (previous step) (back to tree)

⌜m⌝ < # nσ, nσ[0] < nσ[m], mon<(nσ) ⊦ ∃ m0. nσ[m0] < nσ[m] ∧ nσ[m] ≤ nσ[m0 + 1] ∧ m0 ≤ m

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: m

Additional information

Used simplifier rules: nat freely generated by 0, +1;; ⊦ ⌜0⌝ < # σ ↔ σ ≠ []; (s) ⊦ ¬ n < n; (s)

Proof step 6 from proof monls-exists-maxleq (previous step) (back to tree)

m = m1 +1, nσ[0] < nσ[m], ⌜m⌝ < # nσ, mon<(nσ) ⊦ ∃ m0. nσ[m0] < nσ[m] ∧ nσ[m] ≤ nσ[m0 + 1] ∧ m0 ≤ m

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 +1 = m + 1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 7 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[0] < nσ[m1 + 1], ⌜m1⌝ + ⌜1⌝ < # nσ, mon<(nσ) ⊦ ∃ m0. nσ[m0] < nσ[m1 + 1] ∧ nσ[m1 + 1] ≤ nσ[m0 + 1] ∧ m0 ≤ m1 + 1

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: m1 abort: F discard: T computed: F

Additional information

Used terms: m1

Proof step 8 from proof monls-exists-maxleq (previous step) (back to tree)

⌜m1⌝ + ⌜1⌝ < # nσ, nσ[0] < nσ[m1 + 1], mon<(nσ) ⊦ nσ[m1] < nσ[m1 + 1] ∧ nσ[m1 + 1] ≤ nσ[m1 + 1] ∧ m1 ≤ m1 + 1

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 ⊦ m < n ↔ ¬ n ≤ m; (ss) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s) ⊦ m ≤ m; (s)

Proof step 9 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[0] < nσ[m1 + 1], ⌜m1⌝ + ⌜1⌝ < # nσ, nσ[0] < nσ[m1], mon<(nσ), nσ[m1 + 1] ≤ nσ[m1] ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monls-ls, seq: ⊦ mon<(nσ) ∧ m < n ∧ ⌜n⌝ < # nσ → nσ[m] < nσ[n], subst:[nσ, m, n] → [nσ, m1, m1 + 1]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: nσ, m1, m1 + 1

Proof step 10 from proof monls-exists-maxleq (previous step) (back to tree)

⊦ mon<(nσ) ∧ m < n ∧ ⌜n⌝ < # nσ → nσ[m] < nσ[n]

Lemma monls-ls

Proof step 11 from proof monls-exists-maxleq (previous step) (back to tree)

n ≠ nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < n, mon<(nσ), n ≤ nσ[m] ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)

Proof step 12 from proof monls-exists-maxleq (previous step) (back to tree)

n ≠ nσ[m], n < nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < n, nσ[0] + 1 < nσ[m], mon<(nσ) ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monls-exists-maxls, seq:mon<(nσ), nσ[0] ≤ n, n < nσ[m], ⌜m⌝ < # nσ ⊦ ∃ m0. nσ[m0] ≤ n ∧ n < nσ[m0 + 1] ∧ m0 ≤ m, subst:[nσ, n, m] → [nσ, n, m]

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m < n → ¬ n < m; (s) Used terms: nσ, n, m

Proof step 13 from proof monls-exists-maxleq (previous step) (back to tree)

mon<(nσ), nσ[0] ≤ n, n < nσ[m], ⌜m⌝ < # nσ ⊦ ∃ m0. nσ[m0] ≤ n ∧ n < nσ[m0 + 1] ∧ m0 ≤ m

Lemma monls-exists-maxls

Proof step 14 from proof monls-exists-maxleq (previous step) (back to tree)

n ≠ nσ[m], nσ[0] + 1 < nσ[m], nσ[0] < n, ⌜m⌝ < # nσ, nσ[0] < nσ[m], n < nσ[m], mon<(nσ), ∃ m0. nσ[m0] ≤ n ∧ n < nσ[m0 + 1] ∧ m0 ≤ m ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic

Proof step 15 from proof monls-exists-maxleq (previous step) (back to tree)

n ≠ nσ[m], nσ[0] < nσ[m0 + 1], nσ[m0] < nσ[m], nσ[m0] < nσ[m0 + 1], nσ[0] + 1 < nσ[m], nσ[0] < n, ⌜m⌝ < # nσ, nσ[0] < nσ[m], n < nσ[m], n < nσ[m0 + 1], nσ[0] + 1 < nσ[m0 + 1], mon<(nσ), m0 ≤ m, nσ[m0] ≤ n ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: m0 abort: F discard: F computed: F

Additional information

Used terms: m0

Proof step 16 from proof monls-exists-maxleq (previous step) (back to tree)

n ≠ nσ[m], nσ[0] + 1 < nσ[m0 + 1], n < nσ[m0 + 1], n < nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < n, nσ[0] + 1 < nσ[m], nσ[m0] < nσ[m0 + 1], nσ[m0] < nσ[m], nσ[0] < nσ[m0 + 1], mon<(nσ), nσ[m0] ≤ n, m0 ≤ m ⊦ ∃ m0. nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m, nσ[m0] < n ∧ n ≤ nσ[m0 + 1] ∧ m0 ≤ m

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 ≤ m; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ m ≤ n → (n ≤ m ↔ m = n); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → m ≤ n; (s)

Proof step 17 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[m0] ≠ nσ[m], nσ[0] < nσ[m0 + 1], nσ[0] + 1 < nσ[m], nσ[0] < nσ[m0], ⌜m⌝ < # nσ, nσ[0] < nσ[m], nσ[0] + 1 < nσ[m0 + 1], nσ[m0] < nσ[m], nσ[m0] < nσ[m0 + 1], mon<(nσ), m0 ≤ m ⊦ ∃ m1. nσ[m1] < nσ[m0] ∧ nσ[m0] ≤ nσ[m1 + 1] ∧ m1 ≤ m

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: m0

Additional information

Used simplifier rules: nat freely generated by 0, +1;; ⊦ 0 ≤ m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ¬ n < n; (s)

Proof step 18 from proof monls-exists-maxleq (previous step) (back to tree)

m0 = m2 +1, nσ[m0] ≠ nσ[m], nσ[m0] < nσ[m0 + 1], nσ[m0] < nσ[m], nσ[0] + 1 < nσ[m0 + 1], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < nσ[m0], nσ[0] + 1 < nσ[m], nσ[0] < nσ[m0 + 1], mon<(nσ), m0 ≤ m ⊦ ∃ m1. nσ[m1] < nσ[m0] ∧ nσ[m0] ≤ nσ[m1 + 1] ∧ m1 ≤ m

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 19 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[m2 + 1] ≠ nσ[m], nσ[0] < nσ[2 + m2], nσ[m2 + 1] < nσ[2 + m2], nσ[m2 + 1] < nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < nσ[m2 + 1], nσ[0] + 1 < nσ[m], m2 < m, nσ[0] + 1 < nσ[2 + m2], mon<(nσ) ⊦ ∃ m1. nσ[m1] < nσ[m2 + 1] ∧ nσ[m2 + 1] ≤ nσ[m1 + 1] ∧ m1 ≤ m

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: m2 abort: F discard: T computed: F

Additional information

Used terms: m2

Proof step 20 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[m2 + 1] ≠ nσ[m], nσ[0] + 1 < nσ[2 + m2], m2 < m, nσ[0] + 1 < nσ[m], nσ[0] < nσ[m2 + 1], ⌜m⌝ < # nσ, nσ[0] < nσ[m], nσ[m2 + 1] < nσ[m], nσ[m2 + 1] < nσ[2 + m2], nσ[0] < nσ[2 + m2], mon<(nσ) ⊦ nσ[m2] < nσ[m2 + 1] ∧ nσ[m2 + 1] ≤ nσ[m2 + 1] ∧ m2 ≤ m

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 ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → m ≤ n; (s) ⊦ m ≤ m; (s)

Proof step 21 from proof monls-exists-maxleq (previous step) (back to tree)

nσ[m2 + 1] ≠ nσ[m], nσ[0] < nσ[2 + m2], nσ[m2 + 1] < nσ[2 + m2], nσ[m2 + 1] < nσ[m], nσ[0] < nσ[m], ⌜m⌝ < # nσ, nσ[0] < nσ[m2 + 1], nσ[0] + 1 < nσ[m], m2 < m, nσ[0] + 1 < nσ[2 + m2], nσ[0] < nσ[m2], mon<(nσ), nσ[m2 + 1] ≤ nσ[m2] ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monls-ls, seq: ⊦ mon<(nσ) ∧ m < n ∧ ⌜n⌝ < # nσ → nσ[m] < nσ[n], subst:[nσ, m, n] → [nσ, m2, m2 + 1]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n → (n < m ↔ false); (s) lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: nσ, m2, m2 + 1

Proof step 22 from proof monls-exists-maxleq (previous step) (back to tree)

⊦ mon<(nσ) ∧ m < n ∧ ⌜n⌝ < # nσ → nσ[m] < nσ[n]

Lemma monls-ls

Proof Tree for monls-exists-maxleq (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