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Proof step 1 from proof enc-invert (conclusion) (back to tree)

⊦ encode(m.1, m.2) = m

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof enc-invert (previous step) (back to tree)

encode(m.1, m.2) ≠ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:dec-fst, seq: ⊦ m.1 = maxenc(m) - m.2, subst:[m] → [m], <all paths>

Additional information

Used terms: m

Proof step 3 from proof enc-invert (previous step) (back to tree)

⊦ m.1 = maxenc(m) - m.2

Lemma dec-fst

Proof step 4 from proof enc-invert (previous step) (back to tree)

encode(maxenc(m) - m.2, m.2) ≠ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:dec-snd, seq: ⊦ m.2 = m - (maxenc(m) * (maxenc(m) + 1)) / 2, subst:[m] → [m], <all paths>

Additional information

Used terms: m

Proof step 5 from proof enc-invert (previous step) (back to tree)

⊦ m.2 = m - (maxenc(m) * (maxenc(m) + 1)) / 2

Lemma dec-snd

Proof step 6 from proof enc-invert (previous step) (back to tree)

encode(maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2), m - (maxenc(m) * (maxenc(m) + 1)) / 2) ≠ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:encode-def, seq: ⊦ encode(m, n) = ((m + n) * (m + n + 1)) / 2 + n, subst:[n, m] → [m - (maxenc(m) * (maxenc(m) + 1)) / 2, maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)], <all paths>

Additional information

Used terms: m - (maxenc(m) * (maxenc(m) + 1)) / 2, maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)

Proof step 7 from proof enc-invert (previous step) (back to tree)

⊦ encode(m, n) = ((m + n) * (m + n + 1)) / 2 + n

Lemma encode-def

Proof step 8 from proof enc-invert (previous step) (back to tree)

( ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) * ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) + 1)) / 2 + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) ≠ m ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: nat-div, name:exmult, seq: ⊦ m % n = 0 ∧ n ≠ 0 → (∃ n0. m = n * n0), subst:[m, n] → [maxenc(m) * (maxenc(m) + 1), 2]

Additional information

Used terms: maxenc(m) * (maxenc(m) + 1), 2

Proof step 9 from proof enc-invert (previous step) (back to tree)

⊦ m % n = 0 ∧ n ≠ 0 → (∃ n0. m = n * n0)

A spec-lemma

Proof step 10 from proof enc-invert (previous step) (back to tree)

( ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) * ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) + 1)) / 2 + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) ≠ m, (maxenc(m) * (maxenc(m) + 1)) % 2 = 0 ∧ 2 ≠ 0 → (∃ n0. maxenc(m) * (maxenc(m) + 1) = 2 * n0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Used simplifier rules: max-nls: ⊦ ¬ m < (maxenc(m) * maxenc(m) + maxenc(m)) / 2; ⊦ m ≤ n ↔ ¬ n < m; (ss) n ≤ m ⊦ (m - n) + n0 = n1 ↔ m + n0 = n + n1; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m * (n + 1) = m * n + m; (s) ⊦ (m * m + m) % 2 = 0; (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 11 from proof enc-invert (previous step) (back to tree)

( ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) * ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) + 1)) / 2 + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) ≠ m, ∃ n0. maxenc(m) * (maxenc(m) + 1) = 2 * n0 ⊦

User applied rule exists left

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 12 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, ( ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) * ((maxenc(m) - (m - (maxenc(m) * (maxenc(m) + 1)) / 2)) + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) + 1)) / 2 + (m - (maxenc(m) * (maxenc(m) + 1)) / 2) ≠ m ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 13 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - (2 * n0) / 2)) + (m - (2 * n0) / 2)) * ((maxenc(m) - (m - (2 * n0) / 2)) + (m - (2 * n0) / 2) + 1)) / 2 + (m - (2 * n0) / 2) ≠ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-div, name:two-mdiv2-02, seq: ⊦ (2 * m) / 2 = m, subst:[m] → [n0], <all paths>

Additional information

Used terms: n0

Proof step 14 from proof enc-invert (previous step) (back to tree)

⊦ (2 * m) / 2 = m

A spec-lemma

Proof step 15 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat, name:prop-01, seq:n ≤ m ⊦ (m - n) + n = m, subst:[n, m] → [m - n0, maxenc(m)], <all paths>

Additional information

Used terms: m - n0, maxenc(m)

Proof step 16 from proof enc-invert (previous step) (back to tree)

n ≤ m ⊦ (m - n) + n = m

A spec-lemma

Proof step 17 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, ¬ m - n0 ≤ maxenc(m) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: m < n0

Additional information

Used formulas: m < n0

Proof step 18 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, m < n0, ¬ m - n0 ≤ maxenc(m) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:max-size-01, seq:⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), subst:[m] → [m]

Additional information

Used simplifier rules: ⊦ m * (n + 1) = m * n + m; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m < n → n ≠ 0; (s) 0 ≠ m ⊦ (* m) * n ≤ m * n0 ↔ n ≤ n0; (s) ⊦ m < n → (n ≤ m ↔ false); (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: m

Proof step 19 from proof enc-invert (previous step) (back to tree)

⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2)

Lemma max-size-01

Proof step 20 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, ¬ m < n0, ¬ m - n0 ≤ maxenc(m) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:max-size-01, seq:⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), subst:[m] → [m]

Additional information

Used terms: m

Proof step 21 from proof enc-invert (previous step) (back to tree)

⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2)

Lemma max-size-01

Proof step 22 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), maxenc(m) * (maxenc(m) + 1) ≤ 2 * m, ¬ m < n0, ¬ m - n0 ≤ maxenc(m) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:max-size, seq:⊦ (maxenc(m) * (maxenc(m) + 1)) / 2 ≤ m ∧ m < ((maxenc(m) + 1) * (maxenc(m) + 2)) / 2, subst:[m] → [m]

Additional information

Used terms: m

Proof step 23 from proof enc-invert (previous step) (back to tree)

⊦ (maxenc(m) * (maxenc(m) + 1)) / 2 ≤ m ∧ m < ((maxenc(m) + 1) * (maxenc(m) + 2)) / 2

Lemma max-size

Proof step 24 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), m < ((maxenc(m) + 1) * (maxenc(m) + 2)) / 2, maxenc(m) * (maxenc(m) + 1) ≤ 2 * m, (maxenc(m) * (maxenc(m) + 1)) / 2 ≤ m, ¬ m < n0, ¬ m - n0 ≤ maxenc(m) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat, name:ls-03, seq:n0 ≤ n ⊦ n - n0 ≤ m ↔ n ≤ m + n0, subst:[m, n0, n] → [maxenc(m), n0, m], <all paths>

Additional information

Used simplifier rules: ⊦ m * (n + 1) = m * n + m; (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult ⊦ (m + 2) / 2 = m / 2 + 1; (s) ⊦ m < n + 1 ↔ ¬ n < m; (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; (ss) Used terms: maxenc(m), n0, m

Proof step 25 from proof enc-invert (previous step) (back to tree)

n0 ≤ n ⊦ n - n0 ≤ m ↔ n ≤ m + n0

A spec-lemma

Proof step 26 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, m < ((maxenc(m) + 1) * (maxenc(m) + 2)) / 2, 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), (maxenc(m) * (maxenc(m) + 1)) / 2 ≤ m, maxenc(m) * (maxenc(m) + 1) ≤ 2 * m, n0 ≤ m, ¬ m < n0, ¬ m ≤ maxenc(m) + n0 ⊦

User applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: mult-c: ⊦ m * n = n * m; from specification nat-mult mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult 0 ≠ m ⊦ (* m) * n ≤ m * n0 ↔ n ≤ n0; (s) ⊦ (2 * m) / 2 = m; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ m < n → n ≠ 0; (s) let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic m0 ≠ 0 ⊦ n ≤ m / m0 ↔ n * m0 ≤ m; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ (m + 2) / 2 = m / 2 + 1; (s) ⊦ m * (n + 1) = m * n + m; (s)

Proof step 27 from proof enc-invert (previous step) (back to tree)

maxenc(m) * maxenc(m) + maxenc(m) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0)) + (maxenc(m) - (m - n0)) + (m - n0)) / 2 + (m - n0) ≠ m, 2 * n0 < (maxenc(m) + 2) * maxenc(m) + maxenc(m) + 2, maxenc(m) + n0 < m, 2 * m < (maxenc(m) + 2) * maxenc(m) + maxenc(m) + 2, 2 * n0 ≤ (maxenc(m) + 2) * maxenc(m) + maxenc(m), m * 2 ≤ (maxenc(m) + 2) * maxenc(m) + maxenc(m), n0 ≤ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-mult, name:dist, seq: ⊦ m0 * (m + n) = m0 * m + m0 * n, subst:[m0, n, m] → [maxenc(m), 2, maxenc(m)], <all paths>

Additional information

Used simplifier rules: mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: maxenc(m), 2, maxenc(m)

Proof step 28 from proof enc-invert (previous step) (back to tree)

⊦ m0 * (m + n) = m0 * m + m0 * n

A spec-lemma

Proof step 29 from proof enc-invert (previous step) (back to tree)

maxenc(m) * maxenc(m) + maxenc(m) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0)) + (maxenc(m) - (m - n0)) + (m - n0)) / 2 + (m - n0) ≠ m, 2 * m < (maxenc(m) * maxenc(m) + maxenc(m) * 2) + maxenc(m) + 2, maxenc(m) + n0 < m, 2 * n0 < (maxenc(m) * maxenc(m) + maxenc(m) * 2) + maxenc(m) + 2, n0 ≤ m, m * 2 ≤ (maxenc(m) * maxenc(m) + maxenc(m) * 2) + maxenc(m), 2 * n0 ≤ (maxenc(m) * maxenc(m) + maxenc(m) * 2) + maxenc(m) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-basic, name:c, seq: ⊦ m + n = n + m, subst:[n, m] → [maxenc(m) * 2, maxenc(m) * maxenc(m)], <all paths>

Additional information

Used terms: maxenc(m) * 2, maxenc(m) * maxenc(m)

Proof step 30 from proof enc-invert (previous step) (back to tree)

⊦ m + n = n + m

A spec-lemma

Proof step 31 from proof enc-invert (previous step) (back to tree)

maxenc(m) * maxenc(m) + maxenc(m) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0)) + (maxenc(m) - (m - n0)) + (m - n0)) / 2 + (m - n0) ≠ m, 2 * n0 < (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m) + 2, maxenc(m) + n0 < m, 2 * m < (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m) + 2, 2 * n0 ≤ (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m), m * 2 ≤ (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m), n0 ≤ m ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-basic, name:a, seq: ⊦ (m + n0) + n1 = m + n0 + n1, subst:[n1, n0, m] → [maxenc(m), maxenc(m) * maxenc(m), maxenc(m) * 2], <all paths>

Additional information

Used terms: maxenc(m), maxenc(m) * maxenc(m), maxenc(m) * 2

Proof step 32 from proof enc-invert (previous step) (back to tree)

⊦ (m + n0) + n1 = m + n0 + n1

A spec-lemma

Proof step 33 from proof enc-invert (previous step) (back to tree)

maxenc(m) * maxenc(m) + maxenc(m) = 2 * n0, (((maxenc(m) - (m - n0)) + (m - n0)) * ((maxenc(m) - (m - n0)) + (m - n0)) + (maxenc(m) - (m - n0)) + (m - n0)) / 2 + (m - n0) ≠ m, 2 * m < (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m) + 2, maxenc(m) + n0 < m, 2 * n0 < (maxenc(m) * 2 + maxenc(m) * maxenc(m)) + maxenc(m) + 2, n0 ≤ m, m * 2 ≤ maxenc(m) * 2 + maxenc(m) * maxenc(m) + maxenc(m), 2 * n0 ≤ maxenc(m) * 2 + maxenc(m) * maxenc(m) + maxenc(m) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 1, flags: discard

Additional information

Used simplifier rules: ⊦ m * n ≤ m1 * n + m2 * n ↔ m ≤ m1 + m2 ∨ n = 0; (s) ⊦ m < n → (n ≤ m ↔ false); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult

Proof step 34 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (maxenc(m) * ((maxenc(m) - (m - n0)) + (m - n0) + 1)) / 2 + (m - n0) ≠ m, m - n0 ≤ maxenc(m) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat, name:sub-add, seq: ⊦ n ≤ n0 → (m + (n0 - n)) + n = m + n0, subst:[m, n, n0] → [1, m - n0, maxenc(m)], <all paths>

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: 1, m - n0, maxenc(m)

Proof step 35 from proof enc-invert (previous step) (back to tree)

⊦ n ≤ n0 → (m + (n0 - n)) + n = m + n0

A spec-lemma

Proof step 36 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (maxenc(m) * (1 + maxenc(m))) / 2 + (m - n0) ≠ m, m - n0 ≤ maxenc(m) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic

Proof step 37 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, (2 * n0) / 2 + (m - n0) ≠ m, m - n0 ≤ maxenc(m) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-div, name:two-mdiv2-02, seq: ⊦ (2 * m) / 2 = m, subst:[m] → [n0], <all paths>

Additional information

Used terms: n0

Proof step 38 from proof enc-invert (previous step) (back to tree)

⊦ (2 * m) / 2 = m

A spec-lemma

Proof step 39 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, n0 + (m - n0) ≠ m, m - n0 ≤ maxenc(m) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat, name:prop-01, seq:n ≤ m ⊦ (m - n) + n = m, subst:[n, m] → [n0, m], <all paths>

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic Used terms: n0, m

Proof step 40 from proof enc-invert (previous step) (back to tree)

n ≤ m ⊦ (m - n) + n = m

A spec-lemma

Proof step 41 from proof enc-invert (previous step) (back to tree)

maxenc(m) * (maxenc(m) + 1) = 2 * n0, n0 + (m - n0) ≠ m, m - n0 ≤ maxenc(m), ¬ n0 ≤ m ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:max-size-01, seq:⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2), subst:[m] → [m]

Additional information

Used simplifier rules: ⊦ m * (n + 1) = m * n + m; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m < n → n ≠ 0; (s) n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 0 ≠ m ⊦ (* m) * n ≤ m * n0 ↔ n ≤ n0; (s) ⊦ m < n → (n ≤ m ↔ false); (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: m

Proof step 42 from proof enc-invert (previous step) (back to tree)

⊦ maxenc(m) * (maxenc(m) + 1) ≤ 2 * m ∧ 2 * m < (maxenc(m) + 1) * (maxenc(m) + 2)

Lemma max-size-01

Proof Tree for enc-invert (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