Proof step 1 from proof infinite-append (conclusion) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 2:

  ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 2 from proof infinite-append (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 3:

  ⊦ ∀ n. ∃ nf, ii. ii(0) = 0 ∧ (∀ m. m < n → ii(m + 1) = ii(m) + # lf(m) ∧ (nf from ii(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → ii(m0) ≤ ii(m)))

• Proof step 4:

  ∀ n. ∃ nf, ii. ii(0) = 0 ∧ (∀ m. m < n → ii(m + 1) = ii(m) + # lf(m) ∧ (nf from ii(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → ii(m0) ≤ ii(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: lf

Proof step 3 from proof infinite-append (previous step) (back to tree)

Lemma infinite-append-lem

Proof step 4 from proof infinite-append (previous step) (back to tree)

User applied rule choice left

And obtained the following new goals

• Proof step 5:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 5 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 6 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 7:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 163:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)

Proof step 7 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 8 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 9:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 144:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦ ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)

Proof step 9 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 10 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 11:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 104:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)

Proof step 11 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 12 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 13:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 98:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m) ⊦ ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)

Proof step 13 from proof infinite-append (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 14:

  ∀ n. ii(n) ≤ ii(n + 1) ⊦ (∀ n. ∃ m. ii(m) ≥ n) ∨ (∃ n. ∀ m. m ≥ n → ii(m) = ii(n))

• Proof step 15:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∀ n. iigen(n)(n) ≤ iigen(n + 1)(n + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 17:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), (∀ n. ∃ m. iigen(m)(m) ≥ n) ∨ (∃ n. ∀ m. m ≥ n → iigen(m)(m) = iigen(n)(n)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: λ n. iigen(n)(n)

Proof step 14 from proof infinite-append (previous step) (back to tree)

Lemma weakmon-lem

Proof step 15 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  iigen(n + 1)(n + 1) < iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

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 16 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n

Proof step 17 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), (∀ n. ∃ m. n ≤ iigen(m)(m)) ∨ (∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ n ≥ m ↔ m ≤ n; (s)

Proof step 18 from proof infinite-append (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 19:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ n. ∃ m. n ≤ iigen(m)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 56:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 19 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 20:

  ∀ n. ∃ m. n ≤ iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 45:

  ∀ n. ∃ m. n ≤ iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: ∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)

Proof step 20 from proof infinite-append (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 21:

  ∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 21 from proof infinite-append (previous step) (back to tree)

User applied rule choice left

And obtained the following new goals

• Proof step 22:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 22 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 23:

  ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ iigen(0)(0) = 0 ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ∧ ¬ # ⌜λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]⌝ < ⌜iigen(n + 1)(n + 1)⌝ ∧ (⌜λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]⌝ from iigen(n)(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (finp(⌜λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]⌝ ) ⊃ ∃ n. # ⌜λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]⌝ = ⌜iigen(n)(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ iigen(m)(m) = iigen(n)(n)) ;∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1) )

Rule argument

Additional information

Used terms: ⌜λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]⌝ , λ n. iigen(n)(n)

Proof step 23 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦ iigen(0)(0) = 0 ∧ iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ∧ ((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n) = lf(n) ∧ (∃ m. iigen(m)(m) ≤ n0 ∧ n0 < iigen(m + 1)(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ finp(⌜nf⌝ ); (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; (s) ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 24 from proof infinite-append (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 25:

  iigen(0)(0) ≠ 0, ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

• Proof step 26:

  iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

• Proof step 27:

  ((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n) ≠ lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

• Proof step 42:

  ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ m. iigen(m)(m) ≤ n0 ∧ n0 < iigen(m + 1)(m + 1)

Rule argument

Additional information

Proof step 25 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: 0

Proof step 26 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n

Proof step 27 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 28:

  ((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n) ≠ lf(n), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used terms: n

Proof step 28 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  ((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n) ≠ lf(n), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 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

Proof step 29 from proof infinite-append (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 30:

  ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

• Proof step 31:

  n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), #(((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n)) ≠ # lf(n) ∨ (∃ n0. ¬ ( n0 < #(((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n)) → (((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n))[n0] = lf(n)[n0])) ⊦

Rule argument

Additional information

Used terms: lf(n), ((λ n. lf(ii(n))[n - iigen(ii(n))(ii(n))]) from iigen(n)(n)) to # lf(n)

Proof step 30 from proof infinite-append (previous step) (back to tree)

A spec-lemma

Proof step 31 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 32:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], lf(n) ≠ [], n0 < # lf(n), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ((nf from n))(m) = nf(m + n); (s) m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ #(nf to n) = n; (s)

Proof step 32 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 33:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 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 Used formulas: ii(iigen(n)(n) + n0) = n

Proof step 33 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 34:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], n0 < # lf(n), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) ≤ n, iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used terms: iigen(n)(n) + n0

Proof step 34 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 35:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), n0 < # lf(n), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) ≤ n, iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 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

Proof step 35 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 36:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), n < ii(iigen(n)(n) + n0), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

• Proof step 40:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, iigen(ii(n))(ii(n)) ≤ n, ¬ n < ii(iigen(n)(n) + n0), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used formulas: n < ii(iigen(n)(n) + n0)

Proof step 36 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 37:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(ii(n))(ii(n)) < ii(iigen(n)(n) + n0), iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), n < ii(iigen(n)(n) + n0), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 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

Proof step 37 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 38:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], n < ii(iigen(n)(n) + n0), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), n0 < # lf(n), n < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < ii(iigen(n)(n) + n0), iigen(ii(n))(ii(n)) ≤ n, iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, n + 1 ≤ ii(iigen(n)(n) + n0) → iigen(n + 1)(n + 1) ≤ iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used terms: n + 1, ii(iigen(n)(n) + n0)

Proof step 38 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 39:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], iigen(ii(n))(ii(n)) < ii(iigen(n)(n) + n0), iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), n < ii(iigen(n)(n) + n0), iigen(n + 1)(n + 1) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, iigen(ii(n))(ii(n)) ≤ n, iigen(n + 1)(n + 1) ≤ iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)), iigen(n + 1)(n + 1) ≤ iigen(n)(n) + n0, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 39 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (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 Used terms: n

Proof step 40 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 41:

  lf(ii(n0 + iigen(n)(n)))[n0 + iigen(n)(n) - iigen(ii(n0 + iigen(n)(n)))(ii(n0 + iigen(n)(n)))] ≠ lf(n)[n0], ii(iigen(n)(n) + n0) ≠ n, lf(n) ≠ [], ii(iigen(n)(n) + n0) < iigen(ii(n) + 1)(ii(n) + 1), ii(iigen(n)(n) + n0) < n, iigen(n)(n) + n0 < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), iigen(ii(n))(ii(n)) < iigen(ii(n) + 1)(ii(n) + 1), n < iigen(ii(n) + 1)(ii(n) + 1), n0 < # lf(n), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) < iigen(ii(iigen(n)(n) + n0) + 1)(ii(iigen(n)(n) + n0) + 1), ii(iigen(n)(n) + n0) + 1 < iigen(ii(n) + 1)(ii(n) + 1), iigen(ii(iigen(n)(n) + n0))(ii(iigen(n)(n) + n0)) ≤ iigen(n)(n) + n0, iigen(ii(n))(ii(n)) ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n0. iigen(ii(n0))(ii(n0)) ≤ n0 ∧ n0 < iigen(ii(n0) + 1)(ii(n0) + 1) ⊦

Rule argument

Additional information

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

Proof step 41 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ii(iigen(n)(n) + n0) + 1, n

Proof step 42 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 43:

  n0 < iigen(ii(n0) + 1)(ii(n0) + 1), iigen(ii(n0))(ii(n0)) ≤ n0, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ m. iigen(m)(m) ≤ n0 ∧ n0 < iigen(m + 1)(m + 1)

Rule argument

Additional information

Used terms: n0

Proof step 43 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 44:

  iigen(ii(n0))(ii(n0)) < iigen(ii(n0) + 1)(ii(n0) + 1), n0 < iigen(ii(n0) + 1)(ii(n0) + 1), iigen(ii(n0))(ii(n0)) ≤ n0, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ m. iigen(m)(m) ≤ n0 ∧ n0 < iigen(m + 1)(m + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ 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

Proof step 44 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And closed the goal!

Rule argument

Additional information

Used terms: ii(n0)

Proof step 45 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 46:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ n. ∃ m. n ≤ iigen(m)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)

Rule argument

Additional information

Proof step 46 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 47:

  iigen(0)(0) = 0, ∀ n. ∃ m. n ≤ iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 55:

  iigen(0)(0) ≠ 0, ∀ n. ∃ m. n ≤ iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used formulas: iigen(0)(0) = 0

Proof step 47 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 48:

  iigen(0)(0) = 0, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∃ m. n + 1 ≤ iigen(m)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)

Rule argument

Additional information

Used terms: n + 1

Proof step 48 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 49:

  iigen(0)(0) = 0, n < iigen(m)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)

Rule argument

Additional information

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

Proof step 49 from proof infinite-append (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 50:

  iigen(0)(0) = 0, n < iigen(m)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m1, iigen, n, lf, nfgen. m1 < m → iigen(0)(0) = 0 ∧ n < iigen(m1)(m1) ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)) ∧ (∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ∨ (∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)

Rule argument

Additional information

Used simplifier rules: <; (wellfounded) Used terms: m

Proof step 50 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 51:

  iigen(0)(0) = 0, n < iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m1, iigen, n, lf, nfgen. m1 < m → iigen(0)(0) = 0 ∧ n < iigen(m1)(m1) ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)) ∧ (∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ∨ (∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)) ⊦ ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 51 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 52:

  iigen(0)(0) = 0, n < iigen(m)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m1, iigen, n, lf, nfgen. m1 < m → iigen(0)(0) = 0 ∧ n < iigen(m1)(m1) ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)) ∧ (∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ∨ (∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), iigen(m - 1)(m - 1) ≤ n ∧ n < iigen((m - 1) + 1)((m - 1) + 1)

Rule argument

Additional information

Used terms: m - 1

Proof step 52 from proof infinite-append (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 53:

  iigen(0)(0) = 0, m = m2 + 1, m ≠ 0, n < iigen(m)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m1, iigen, n, lf, nfgen. m1 < m → iigen(0)(0) = 0 ∧ n < iigen(m1)(m1) ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)) ∧ (∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ∨ (∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)) ⊦ ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), iigen(m2)(m2) ≤ n ∧ n < iigen(m2 + 1)(m2 + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m, m2

Proof step 53 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 54:

  iigen(0)(0) = 0, n < iigen(m2)(m2), n < iigen(m2 + 1)(m2 + 1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m1, iigen, n, lf, nfgen. m1 < m2 + 1 → iigen(0)(0) = 0 ∧ n < iigen(m1)(m1) ∧ (∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n)) ∧ (∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ∨ (∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1)) ⊦ ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

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) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 54 from proof infinite-append (previous step) (back to tree)

User applied rule apply induction

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m2, iigen, n, lf, nfgen

Proof step 55 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: 0

Proof step 56 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 57:

  ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ iigen(0)(0) = 0 ∧ (∀ n0. iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0) ∧ ¬ # ⌜nfgen(n) to iigen(n)(n)⌝ < ⌜iigen(n0 + 1)(n0 + 1)⌝ ∧ (⌜nfgen(n) to iigen(n)(n)⌝ from iigen(n0)(n0)) to # lf(n0) = ⌜lf(n0)⌝ ) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) ∧ (finp(⌜nfgen(n) to iigen(n)(n)⌝ ) ⊃ ∃ n0. # ⌜nfgen(n) to iigen(n)(n)⌝ = ⌜iigen(n0)(n0)⌝ ∧ (∀ m. n0 ≤ m → lf(m) = [] ∧ iigen(m)(m) = iigen(n0)(n0)) ;∀ n. ∃ m. iigen(m)(m) ≤ n ∧ n < iigen(m + 1)(m + 1) )

Rule argument

Additional information

Used terms: ⌜nfgen(n) to iigen(n)(n)⌝ , λ n. iigen(n)(n)

Proof step 57 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 58:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦ iigen(0)(0) = 0 ∧ iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0) ∧ ¬ iigen(n)(n) < iigen(n0 + 1)(n0 + 1) ∧ ((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0) = lf(n0) ∧ (∃ n1. iigen(n)(n) = iigen(n1)(n1) ∧ (∀ m. n1 ≤ m → lf(m) = [] ∧ iigen(m)(m) = iigen(n1)(n1)))

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ = ⌜n0⌝ ↔ n = n0; (s) ⊦ # ⌜nf to n⌝ = ⌜n⌝ ; (s) ⊦ finp(⌜nf to n⌝ ); (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; (s) ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 58 from proof infinite-append (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 59:

  iigen(0)(0) ≠ 0, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

• Proof step 60:

  iigen(n0 + 1)(n0 + 1) ≠ iigen(n0)(n0) + # lf(n0), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

• Proof step 61:

  iigen(n)(n) < iigen(n0 + 1)(n0 + 1), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

• Proof step 65:

  ((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0) ≠ lf(n0), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

• Proof step 91:

  ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ ∃ n1. iigen(n)(n) = iigen(n1)(n1) ∧ (∀ m. n1 ≤ m → lf(m) = [] ∧ iigen(m)(m) = iigen(n1)(n1))

Rule argument

Additional information

Proof step 59 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: 0

Proof step 60 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n0

Proof step 61 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 62:

  iigen(n)(n) < iigen(n0 + 1)(n0 + 1), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

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

Proof step 62 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 63:

  iigen(n)(n) < iigen(n0 + 1)(n0 + 1), n ≤ n0 + 1 → iigen(n0 + 1)(n0 + 1) = iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦

Rule argument

Additional information

Used terms: n0 + 1

Proof step 63 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 64:

  iigen(n0 + 1)(n0 + 1) ≠ iigen(n)(n), n0 + 1 < n, iigen(n)(n) < iigen(n0 + 1)(n0 + 1), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

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) ⊦ ¬ n < n; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic

Proof step 64 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ ¬ m < n → ¬ m + m0 < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n0 + 1, n

Proof step 65 from proof infinite-append (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 66:

  ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

• Proof step 67:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ (∃ n1. ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1])) ⊦

Rule argument

Additional information

Used terms: lf(n0), ((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)

Proof step 66 from proof infinite-append (previous step) (back to tree)

A spec-lemma

Proof step 67 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 68:

  ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Proof step 68 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 69:

  n ≤ n0, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

• Proof step 74:

  ¬ n ≤ n0, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used formulas: n ≤ n0

Proof step 69 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 70:

  n ≤ n0, n ≤ n0 → iigen(n0)(n0) = iigen(n)(n), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used terms: n0

Proof step 70 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 71:

  iigen(n0)(n0) = iigen(n)(n), n ≤ n0, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #([] to # lf(n0)) ≠ # lf(n0) ∨ ¬(n1 < #([] to # lf(n0)) → ([] to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ (nf to n) from n = []; (s)

Proof step 71 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 72:

  iigen(n0)(n0) = iigen(n)(n), iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), n ≤ n0, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #([] to # lf(n0)) ≠ # lf(n0) ∨ ¬(n1 < #([] to # lf(n0)) → ([] to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used terms: n0

Proof step 72 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 73:

  iigen(n0 + 1)(n0 + 1) = iigen(n)(n) + # lf(n0), iigen(n0)(n0) = iigen(n)(n), n ≤ n0, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #([] to # lf(n0)) ≠ # lf(n0) ∨ ¬(n1 < #([] to # lf(n0)) → ([] to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Proof step 73 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m = m + n ↔ n = 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m < n → n ≠ 0; (s) ⊦ ¬ m + n < m; (s) ¬ # x < n ⊦ #(x to n) = n; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # [] = 0; (s) Used terms: n0 + 1

Proof step 74 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 75:

  n0 < n, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

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 75 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 76:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), n0 < n, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used terms: n0

Proof step 76 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 77:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), n0 < n, n0 ≤ n → iigen(n0)(n0) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used terms: n0, n

Proof step 77 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 78:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

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

Proof step 78 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 79:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), n0 + 1 ≤ n → iigen(n0 + 1)(n0 + 1) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) ≠ # lf(n0) ∨ ¬ ( n1 < #(((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0)) → (((nfgen(n) to iigen(n)(n)) from iigen(n0)(n0)) to # lf(n0))[n1] = lf(n0)[n1]) ⊦

Rule argument

Additional information

Used terms: n0 + 1, n

Proof step 79 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 80:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), lf(n0) ≠ [], nfgen(n)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic m < n ⊦ (nf to n)[m] = nf(m); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss) m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; (s) ⊦ #(nf to n) = n; (s) ¬ # x < n, m < n ⊦ (x to n)[m] = x[m]; (s) ⊦ m ≤ n → (n < m ↔ false); (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ¬ # x < n ⊦ #(x from n) = # x - n; (s) ¬ # x < n ⊦ #(x to n) = n; (s) ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 80 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 81:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), lf(n0) ≠ [], nfgen(n)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n0 < n, n1 < # lf(n0), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), iigen(n0)(n0) ≤ iigen(n)(n), n0 + 1 ≤ n ∧ iigen(n0)(n0) + n1 < iigen(n0 + 1)(n0 + 1) → nfgen(n0 + 1)(iigen(n0)(n0) + n1) = nfgen(n)(iigen(n0)(n0) + n1), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦

Rule argument

Additional information

Used terms: iigen(n0)(n0) + n1, n0 + 1, n

Proof step 81 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 82:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), nfgen(n0 + 1)(iigen(n0)(n0) + n1) = nfgen(n)(iigen(n0)(n0) + n1), lf(n0) ≠ [], nfgen(n)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 82 from proof infinite-append (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 83:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n0 < n, n1 < # lf(n0), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), iigen(n0)(n0) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦

Rule argument

Additional information

Proof step 83 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 84:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), iigen(n0 + 1)(0) = 0, lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m. m < n0 + 1 → iigen(n0 + 1)(m + 1) = iigen(n0 + 1)(m) + # lf(m) ∧ (nfgen(n0 + 1) from iigen(n0 + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0 + 1)(m0) ≤ iigen(n0 + 1)(m)) ⊦

Rule argument

Additional information

Used terms: n0 + 1

Proof step 84 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 85:

  iigen(n0 + 1)(0) = 0, iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n0 < n, n1 < # lf(n0), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), iigen(n0)(n0) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m. ¬ n0 < m → iigen(n0 + 1)(m + 1) = iigen(n0 + 1)(m) + # lf(m) ∧ (nfgen(n0 + 1) from iigen(n0 + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0 + 1)(m0) ≤ iigen(n0 + 1)(m)) ⊦

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 85 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 86:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), iigen(n0 + 1)(0) = 0, lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ¬ n0 < n0 → iigen(n0 + 1)(n0 + 1) = iigen(n0 + 1)(n0) + # lf(n0) ∧ (nfgen(n0 + 1) from iigen(n0 + 1)(n0)) to # lf(n0) = lf(n0) ∧ (∀ m0. m0 ≤ n0 → iigen(n0 + 1)(m0) ≤ iigen(n0 + 1)(n0)), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

Used terms: n0

Proof step 86 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 87:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), iigen(n0 + 1)(0) = 0, iigen(n0)(n0) = iigen(n0 + 1)(n0), (nfgen(n0 + 1) from iigen(n0)(n0)) to # lf(n0) = lf(n0), lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ∀ m0. m0 ≤ n0 → iigen(n0 + 1)(m0) ≤ iigen(n0)(n0), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

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

Proof step 87 from proof infinite-append (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 88:

  ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

• Proof step 89:

  #((nfgen(n0 + 1) from iigen(n0)(n0)) to # lf(n0)) = # lf(n0), iigen(n0)(n0) = iigen(n0 + 1)(n0), iigen(n0 + 1)(0) = 0, iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n0 < n, n1 < # lf(n0), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), iigen(n0)(n0) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m0. m0 ≤ n0 → iigen(n0 + 1)(m0) ≤ iigen(n0)(n0), ∀ n. n < #((nfgen(n0 + 1) from iigen(n0)(n0)) to # lf(n0)) → ((nfgen(n0 + 1) from iigen(n0)(n0)) to # lf(n0))[n] = lf(n0)[n] ⊦

Rule argument

Additional information

Used terms: lf(n0), (nfgen(n0 + 1) from iigen(n0)(n0)) to # lf(n0)

Proof step 88 from proof infinite-append (previous step) (back to tree)

A spec-lemma

Proof step 89 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 90:

  iigen(n0 + 1)(n0 + 1) = iigen(n0)(n0) + # lf(n0), iigen(n0 + 1)(0) = 0, iigen(n0)(n0) = iigen(n0 + 1)(n0), lf(n0) ≠ [], nfgen(n0 + 1)(iigen(n0)(n0) + n1) ≠ lf(n0)[n1], n1 < # lf(n0), n0 < n, iigen(n0)(n0) ≤ iigen(n)(n), iigen(n0)(n0) + # lf(n0) ≤ iigen(n)(n), ∀ m0. m0 ≤ n0 → iigen(n0 + 1)(m0) ≤ iigen(n0)(n0), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ n2. n2 < # lf(n0) → nfgen(n0 + 1)(n2 + iigen(n0)(n0)) = lf(n0)[n2] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ((nf from n))(m) = nf(m + n); (s) m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ #(nf to n) = n; (s)

Proof step 90 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic Used terms: n1

Proof step 91 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 92:

  ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦ iigen(n)(n) = iigen(n)(n) ∧ (∀ m. n ≤ m → lf(m) = [] ∧ iigen(m)(m) = iigen(n)(n))

Rule argument

Additional information

Used terms: n

Proof step 92 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 93:

  n ≤ m, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦ lf(m) = [] ∧ iigen(m)(m) = iigen(n)(n)

Rule argument

Additional information

Proof step 93 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 94:

  n ≤ m, n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n) ⊦ lf(m) = [] ∧ iigen(m)(m) = iigen(n)(n)

Rule argument

Additional information

Used terms: m

Proof step 94 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 95:

  iigen(m)(m) = iigen(n)(n), lf(m) ≠ [], n ≤ m, ∀ m. n ≤ m → iigen(m)(m) = iigen(n)(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n) ⊦

Rule argument

Additional information

Proof step 95 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 96:

  iigen(m)(m) = iigen(n)(n), lf(m) ≠ [], n ≤ m, n ≤ m + 1 → iigen(m + 1)(m + 1) = iigen(n)(n), ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used terms: m + 1

Proof step 96 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 97:

  iigen(m)(m) = iigen(n)(n), iigen(m + 1)(m + 1) = iigen(n)(n), lf(m) ≠ [], n ≤ m, n ≤ m + 1, ∀ n. iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

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

Proof step 97 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m = m + n ↔ n = 0; (s) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m

Proof step 98 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 99:

  iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 99 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 100:

  iigen(n + 1)(0) = 0, iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m. m < n + 1 → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n + 1

Proof step 100 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 101:

  iigen(n + 1)(0) = 0, iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 101 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 102:

  iigen(n + 1)(0) = 0, iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ¬ n < n → iigen(n + 1)(n + 1) = iigen(n + 1)(n) + # lf(n) ∧ (nfgen(n + 1) from iigen(n + 1)(n)) to # lf(n) = lf(n) ∧ (∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n + 1)(n)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n

Proof step 102 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 103:

  iigen(n + 1)(0) = 0, iigen(n + 1)(n + 1) = iigen(n + 1)(n) + # lf(n), (nfgen(n + 1) from iigen(n + 1)(n)) to # lf(n) = lf(n), iigen(n + 1)(n + 1) ≠ iigen(n)(n) + # lf(n), ∀ m, n1, n2. n1 ≤ n2 ∧ m < iigen(n1)(n1) → nfgen(n1)(m) = nfgen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n + 1)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s)

Proof step 103 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

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

Proof step 104 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 105:

  nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

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

Proof step 105 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 106:

  nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 133:

  nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)

Rule argument

Additional information

Used formulas: ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)

Proof step 106 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 107:

  nfgen(n1)(m) ≠ nfgen(n2)(m), iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ 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

Proof step 107 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 108:

  iigen(n1)(0) = 0, nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) ≤ m, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n1

Proof step 108 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 109:

  iigen(n1)(0) = 0, nfgen(n1)(m) ≠ nfgen(n2)(m), iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 109 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 110:

  iigen(n1)(0) = 0, nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) ≤ m, n1 ≤ n2, n < n1 → iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n) ∧ (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n) ∧ (∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n

Proof step 110 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 111:

  iigen(n1)(0) = 0, nfgen(n1)(m) ≠ nfgen(n2)(m), iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, n < n1 → iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n) ∧ (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n) ∧ (∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n)), n1 ≤ n → iigen(n1)(n1) ≤ iigen(n)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n1, n

Proof step 111 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 112:

  iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

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 → (n ≤ m ↔ false); (s)

Proof step 112 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 113:

  iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n2 → iigen(n2)(m + 1) = iigen(n2)(m) + # lf(m) ∧ (nfgen(n2) from iigen(n2)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n2)(m0) ≤ iigen(n2)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n2

Proof step 113 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 114:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, ∀ m. m < n2 → iigen(n2)(m + 1) = iigen(n2)(m) + # lf(m) ∧ (nfgen(n2) from iigen(n2)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n2)(m0) ≤ iigen(n2)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 114 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 115:

  iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, n < n2 → iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n) ∧ (nfgen(n2) from iigen(n2)(n)) to # lf(n) = lf(n) ∧ (∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n)), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n

Proof step 115 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 116:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), (nfgen(n2) from iigen(n2)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 116 from proof infinite-append (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 117:

  iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n2) from iigen(n2)(n)) to # lf(n) = (nfgen(n1) from iigen(n1)(n)) to # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 117 from proof infinite-append (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 118:

  ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

• Proof step 119:

  #((nfgen(n2) from iigen(n2)(n)) to # lf(n)) = #((nfgen(n1) from iigen(n1)(n)) to # lf(n)), iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ n0. n0 < #((nfgen(n2) from iigen(n2)(n)) to # lf(n)) → ((nfgen(n2) from iigen(n2)(n)) to # lf(n))[n0] = ((nfgen(n1) from iigen(n1)(n)) to # lf(n))[n0] ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: (nfgen(n1) from iigen(n1)(n)) to # lf(n), (nfgen(n2) from iigen(n2)(n)) to # lf(n)

Proof step 118 from proof infinite-append (previous step) (back to tree)

A spec-lemma

Proof step 119 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 120:

  iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ n0. n0 < # lf(n) → nfgen(n2)(n0 + iigen(n2)(n)) = nfgen(n1)(n0 + iigen(n1)(n)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ ((nf from n))(m) = nf(m + n); (s) m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ #(nf to n) = n; (s)

Proof step 120 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 121:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, m - iigen(n)(n) < # lf(n) → nfgen(n2)((m - iigen(n)(n)) + iigen(n2)(n)) = nfgen(n1)((m - iigen(n)(n)) + iigen(n1)(n)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: m - iigen(n)(n)

Proof step 121 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 122:

  iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), m < iigen(n1)(n1), m < iigen(n + 1)(n + 1), n1 ≤ n2, iigen(n)(n) ≤ m, m < # lf(n) + iigen(n)(n) → nfgen(n2)((m - iigen(n)(n)) + iigen(n2)(n)) = nfgen(n1)((m - iigen(n)(n)) + iigen(n1)(n)), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 122 from proof infinite-append (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 123:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), m = n3 + iigen(n)(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n + 1)(n + 1), m < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) ≤ m, n1 ≤ n2, m < # lf(n) + iigen(n)(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: m, iigen(n)(n), n3

Proof step 123 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 124:

  iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

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

Proof step 124 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 125:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n + 1)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m. m < n + 1 → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n + 1

Proof step 125 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 126:

  iigen(n + 1)(0) = 0, iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 126 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 127:

  iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n + 1)(0) = 0, (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ¬ n < n → iigen(n + 1)(n + 1) = iigen(n + 1)(n) + # lf(n) ∧ (nfgen(n + 1) from iigen(n + 1)(n)) to # lf(n) = lf(n) ∧ (∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n + 1)(n)), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n

Proof step 127 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 128:

  iigen(n + 1)(n + 1) = iigen(n + 1)(n) + # lf(n), iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n + 1)(0) = 0, (nfgen(n + 1) from iigen(n + 1)(n)) to # lf(n) = lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n + 1)(n), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ ¬ n < n; (s)

Proof step 128 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 129:

  iigen(n + 1)(0) = 0, iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, iigen(n + 1)(n + 1) = iigen(n + 1)(n) + # lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), (nfgen(n + 1) from iigen(n + 1)(n)) to # lf(n) = lf(n), nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n < n2, n < n1, iigen(n)(n) < iigen(n1)(n1), iigen(n)(n) < iigen(n + 1)(n + 1), n3 + iigen(n)(n) < iigen(n1)(n1), n3 + iigen(n)(n) < iigen(n + 1)(n + 1), n1 ≤ n2, n3 < # lf(n) → nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), n ≤ n ∧ n ≤ n + 1 → iigen(n)(n) = iigen(n + 1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n + 1)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n, n, n + 1

Proof step 129 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 130:

  iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n + 1)(0) = 0, iigen(n)(n) = iigen(n + 1)(n), nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), (nfgen(n + 1) from iigen(n)(n)) to # lf(n) = lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), lf(n) ≠ [], nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), n3 < # lf(n), n3 + iigen(n)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n)(n) < iigen(n1)(n1), n1 ≤ n2, ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n)(n), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ m < m + n ↔ n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < n → n ≠ 0; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ m ≤ m + n; (s) ⊦ m ≤ m; (s)

Proof step 130 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 131:

  iigen(n)(n) = iigen(n + 1)(n), iigen(n + 1)(0) = 0, iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n1)(0) = 0, iigen(n2)(0) = 0, iigen(n + 1)(n + 1) = iigen(n)(n) + # lf(n), nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), (nfgen(n + 1) from iigen(n)(n)) to # lf(n) = lf(n), lf(n) ≠ [], nfgen(n1)(n3 + iigen(n)(n)) ≠ nfgen(n2)(n3 + iigen(n)(n)), iigen(n)(n) < iigen(n1)(n1), n < n2, n < n1, n3 + iigen(n)(n) < iigen(n1)(n1), n3 < # lf(n), n1 ≤ n2, n ≤ n ∧ n ≤ n1 → iigen(n)(n) = iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n, n, n1

Proof step 131 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 132:

  iigen(n + 1)(n + 1) = iigen(n1)(n + 1), iigen(n2)(0) = 0, iigen(n1)(0) = 0, iigen(n1)(n + 1) = iigen(n1)(n) + # lf(n), iigen(n2)(n + 1) = iigen(n2)(n) + # lf(n), iigen(n + 1)(0) = 0, iigen(n1)(n) = iigen(n + 1)(n), iigen(n)(n) = iigen(n1)(n), nfgen(n2)(n3 + iigen(n2)(n)) = nfgen(n1)(n3 + iigen(n1)(n)), (nfgen(n + 1) from iigen(n1)(n)) to # lf(n) = lf(n), (nfgen(n1) from iigen(n1)(n)) to # lf(n) = lf(n), lf(n) ≠ [], nfgen(n1)(n3 + iigen(n1)(n)) ≠ nfgen(n2)(n3 + iigen(n1)(n)), n3 < # lf(n), n3 + iigen(n1)(n) < iigen(n1)(n1), n < n1, n < n2, iigen(n1)(n) < iigen(n1)(n1), n1 ≤ n2, ∀ m0. m0 ≤ n → iigen(n + 1)(m0) ≤ iigen(n1)(n), ∀ m. ¬ n < m → iigen(n + 1)(m + 1) = iigen(n + 1)(m) + # lf(m) ∧ (nfgen(n + 1) from iigen(n + 1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n + 1)(m0) ≤ iigen(n + 1)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m0. m0 ≤ n → iigen(n1)(m0) ≤ iigen(n1)(n), ∀ m0. m0 ≤ n → iigen(n2)(m0) ≤ iigen(n2)(n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

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

Proof step 132 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m; (s) ⊦ m < n → m ≤ n; (s) Used terms: n, n, n2

Proof step 133 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 134:

  nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 134 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 135:

  iigen(0)(0) = 0, nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)

• Proof step 143:

  iigen(0)(0) ≠ 0, nfgen(n1)(m) ≠ nfgen(n2)(m), m < iigen(n1)(n1), n1 ≤ n2, ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)

Rule argument

Additional information

Used formulas: iigen(0)(0) = 0

Proof step 135 from proof infinite-append (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 136:

  iigen(0)(0) = 0, m < iigen(n1)(n1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 136 from proof infinite-append (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 137:

  iigen(0)(0) = 0, m < iigen(n1)(n1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n, iigen, m, lf, nfgen. n < n1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: <; (wellfounded) Used terms: n1

Proof step 137 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 138:

  iigen(0)(0) = 0, m < iigen(n1)(n1), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ n, iigen, m, lf, nfgen. n < n1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)

Rule argument

Additional information

Proof step 138 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 139:

  iigen(0)(0) = 0, n1 ≠ 0, m < iigen(n1)(n1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n, iigen, m, lf, nfgen. n < n1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) Used formulas: n1 = 0

Proof step 139 from proof infinite-append (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 140:

  iigen(0)(0) = 0, n1 ≠ 0, m < iigen(n1)(n1), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ n, iigen, m, lf, nfgen. n < n1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), iigen(n1 - 1)(n1 - 1) ≤ m ∧ m < iigen((n1 - 1) + 1)((n1 - 1) + 1)

Rule argument

Additional information

Used terms: n1 - 1

Proof step 140 from proof infinite-append (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 141:

  iigen(0)(0) = 0, n1 = m1 + 1, n1 ≠ 0, m < iigen(n1)(n1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n, iigen, m, lf, nfgen. n < n1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), iigen(m1)(m1) ≤ m ∧ m < iigen(m1 + 1)(m1 + 1)

Rule argument

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n1, m1

Proof step 141 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 142:

  iigen(0)(0) = 0, m < iigen(m1)(m1), m < iigen(m1 + 1)(m1 + 1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n), ∀ n, iigen, m, lf, nfgen. n < m1 + 1 → iigen(0)(0) = 0 ∧ m < iigen(n)(n) ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) ∧ (∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m)) ∧ (∀ m, n. m ≤ n → iigen(m)(m) ≤ iigen(n)(n)) → (∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1)) ∨ (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ n1. iigen(n1)(n1) ≤ m ∧ m < iigen(n1 + 1)(n1 + 1), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

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) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 142 from proof infinite-append (previous step) (back to tree)

User applied rule apply induction

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m1, iigen, m, lf, nfgen

Proof step 143 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: 0

Proof step 144 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 145:

  iigen(n)(n) < iigen(m)(m), m ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

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 145 from proof infinite-append (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 146:

  iigen(n)(n) < iigen(m)(m), m ≤ n, ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Proof step 146 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 147:

  iigen(n)(n) < iigen(m)(m), m ≤ n, m ≤ m ∧ m ≤ n → iigen(m)(m) = iigen(n)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦

Rule argument

Additional information

Used terms: m, m, n

Proof step 147 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 148:

  iigen(m)(m) = iigen(n)(m), iigen(n)(n) < iigen(n)(m), m ≤ n, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦

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)

Proof step 148 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 149:

  iigen(m)(m) = iigen(n)(m), iigen(n)(0) = 0, iigen(n)(n) < iigen(n)(m), m ≤ n, ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)) ⊦

Rule argument

Additional information

Used terms: n

Proof step 149 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 150:

  iigen(n)(0) = 0, iigen(m)(m) = iigen(n)(m), iigen(n)(n) < iigen(n)(m), m ≤ n, ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦

Rule argument

Additional information

Proof step 150 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 151:

  iigen(m)(m) = iigen(n)(m), iigen(n)(0) = 0, iigen(n)(n) < iigen(n)(m), m ≤ n, n - 1 < n → iigen(n)((n - 1) + 1) = iigen(n)(n - 1) + # lf(n - 1) ∧ (nfgen(n) from iigen(n)(n - 1)) to # lf(n - 1) = lf(n - 1) ∧ (∀ m0. m0 ≤ n - 1 → iigen(n)(m0) ≤ iigen(n)(n - 1)), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)) ⊦

Rule argument

Additional information

Used terms: n - 1

Proof step 151 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 152:

  iigen(n)(0) = 0, iigen(m)(m) = iigen(n)(m), iigen(n)(n) < iigen(n)(m), m ≤ n, n - 1 < n → iigen(n)((n - 1) + 1) = iigen(n)(n - 1) + # lf(n - 1) ∧ (nfgen(n) from iigen(n)(n - 1)) to # lf(n - 1) = lf(n - 1) ∧ (∀ m0. m0 < n → iigen(n)(m0) ≤ iigen(n)(n - 1)), ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m) ⊦

Rule argument

Additional information

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

Proof step 152 from proof infinite-append (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 153:

  iigen(m)(m) = iigen(n)(m), iigen(n)(0) = 0, n ≠ 0, iigen(n)(n) < iigen(n)(m), m ≤ n, n - 1 < n → iigen(n)((n - 1) + 1) = iigen(n)(n - 1) + # lf(n - 1) ∧ (nfgen(n) from iigen(n)(n - 1)) to # lf(n - 1) = lf(n - 1) ∧ (∀ m0. m0 < n → iigen(n)(m0) ≤ iigen(n)(n - 1)), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)) ⊦

Rule argument

Additional information

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

Proof step 153 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 154:

  iigen(n)(0) = 0, iigen(m)(m) = iigen(n)(m), iigen(n)((n - 1) + 1) = iigen(n)(n - 1) + # lf(n - 1), (nfgen(n) from iigen(n)(n - 1)) to # lf(n - 1) = lf(n - 1), n ≠ 0, iigen(n)(n) < iigen(n)(m), m ≤ n, ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m0. m0 < n → iigen(n)(m0) ≤ iigen(n)(n - 1) ⊦

Rule argument

Additional information

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

Proof step 154 from proof infinite-append (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 155:

  iigen(n)(m1 + 1) = iigen(n)(m1) + # lf(m1), iigen(m)(m) = iigen(n)(m), iigen(n)(0) = 0, n = m1 + 1, (nfgen(n) from iigen(n)(m1)) to # lf(m1) = lf(m1), n ≠ 0, iigen(n)(n) < iigen(n)(m), m ≤ n, ∀ m0. m0 < n → iigen(n)(m0) ≤ iigen(n)(m1), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n → iigen(n)(m + 1) = iigen(n)(m) + # lf(m) ∧ (nfgen(n) from iigen(n)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n)(m0) ≤ iigen(n)(m)) ⊦

Rule argument

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m1

Proof step 155 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 156:

  iigen(m1 + 1)(m1 + 1) = iigen(m1 + 1)(m1) + # lf(m1), iigen(m)(m) = iigen(m1 + 1)(m), iigen(m1 + 1)(0) = 0, (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1 + 1) < iigen(m)(m), m ≤ m1 + 1, ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0) ≤ iigen(m1 + 1)(m1), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 156 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 157:

  iigen(m1 + 1)(0) = 0, iigen(m)(m) = iigen(m1 + 1)(m), iigen(m1 + 1)(m1 + 1) = iigen(m1 + 1)(m1) + # lf(m1), (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1 + 1) < iigen(m)(m), m ≤ m1 + 1, ¬ m1 < m1 → iigen(m1 + 1)(m1 + 1) = iigen(m1 + 1)(m1) + # lf(m1) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1) ∧ (∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0) ≤ iigen(m1 + 1)(m1) ⊦

Rule argument

Additional information

Used terms: m1

Proof step 157 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 158:

  iigen(m1 + 1)(m1 + 1) = iigen(m1 + 1)(m1) + # lf(m1), iigen(m)(m) = iigen(m1 + 1)(m), iigen(m1 + 1)(0) = 0, (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1 + 1) < iigen(m)(m), m ≤ m1 + 1, ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0) ≤ iigen(m1 + 1)(m1), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s)

Proof step 158 from proof infinite-append (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 159:

  iigen(m1 + 1)(0) = 0, iigen(m)(m) = iigen(m1 + 1)(m), (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1) + # lf(m1) < iigen(m)(m), m ≤ m1 + 1, ∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0) ≤ iigen(m1 + 1)(m1) ⊦

Rule argument

Additional information

Proof step 159 from proof infinite-append (previous step) (back to tree)

User applied rule subst equation

And obtained the following new goals

• Proof step 160:

  iigen(m1 + 1)(0) = 0, (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1) + # lf(m1) < iigen(m1 + 1)(m), m ≤ m1 + 1, ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0) ≤ iigen(m1 + 1)(m1), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1) ⊦

Rule argument

Additional information

Proof step 160 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 161:

  iigen(m1 + 1)(0) = 0, (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1) + # lf(m1) < iigen(m1 + 1)(m), m ≤ m1 + 1, ¬ m1 < m → iigen(m1 + 1)(m) ≤ iigen(m1 + 1)(m1), ∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)) ⊦

Rule argument

Additional information

Used terms: m

Proof step 161 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 162:

  iigen(m1 + 1)(0) = 0, (nfgen(m1 + 1) from iigen(m1 + 1)(m1)) to # lf(m1) = lf(m1), iigen(m1 + 1)(m1) + # lf(m1) < iigen(m1 + 1)(m1 + 1), iigen(m1 + 1)(m1) < iigen(m1 + 1)(m1 + 1), ∀ m2. m2 ≤ m1 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m1), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m, n1, n2. m ≤ n1 ∧ n1 ≤ n2 → iigen(n1)(m) = iigen(n2)(m), ∀ m0. ¬ m1 < m0 → iigen(m1 + 1)(m0 + 1) = iigen(m1 + 1)(m0) + # lf(m0) ∧ (nfgen(m1 + 1) from iigen(m1 + 1)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m2. m2 ≤ m0 → iigen(m1 + 1)(m2) ≤ iigen(m1 + 1)(m0)) ⊦

Rule argument

Additional information

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

Proof step 162 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s) lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m +1 = m + 1; (s) ⊦ m < m + n ↔ n ≠ 0; (s) ⊦ # x = 0 ↔ x = []; (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; (s) Used terms: m1

Proof step 163 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 164:

  iigen(n1)(m) ≠ iigen(n2)(m), m ≤ n1, n1 ≤ n2, m ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat

Proof step 164 from proof infinite-append (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 165:

  iigen(n1)(0) ≠ iigen(n2)(0), 0 ≤ n2, n1 ≤ n2, 0 ≤ n1, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

• Proof step 169:

  iigen(n1)(m0 +1) ≠ iigen(n2)(m0 +1), m0 +1 ≤ n2, n1 ≤ n2, m0 +1 ≤ n1, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ iigen, n1, n2, lf, nfgen. iigen(n1)(m0) ≠ iigen(n2)(m0) ∧ m0 ≤ n1 ∧ n1 ≤ n2 ∧ m0 ≤ n2 ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: nat freely generated by 0, +1;;

Proof step 165 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 166:

  iigen(n1)(0) ≠ iigen(n2)(0), n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s)

Proof step 166 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 167:

  iigen(n1)(0) = 0, iigen(n1)(0) ≠ iigen(n2)(0), n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n1

Proof step 167 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 168:

  iigen(n1)(0) = 0, 0 ≠ iigen(n2)(0), n1 ≤ n2, ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 168 from proof infinite-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n2

Proof step 169 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 170:

  iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n2, m0 < n1, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ iigen, n1, n2, lf, nfgen. iigen(n1)(m0) ≠ iigen(n2)(m0) ∧ m0 ≤ n1 ∧ n1 ≤ n2 ∧ m0 ≤ n2 ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

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

Proof step 170 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 171:

  iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n1, m0 < n2, n1 ≤ n2, iigen(n1)(m0) ≠ iigen(n2)(m0) ∧ m0 ≤ n1 ∧ n1 ≤ n2 ∧ m0 ≤ n2 ∧ (∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m)))) → (∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: iigen, n1, n2, lf, nfgen

Proof step 171 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 172:

  iigen(n1)(m0) = iigen(n2)(m0), iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n2, m0 < n1, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

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

Proof step 172 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 173:

  iigen(n1)(m0) = iigen(n2)(m0), iigen(n2)(0) = 0, iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n1, m0 < n2, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m. m < n2 → iigen(n2)(m + 1) = iigen(n2)(m) + # lf(m) ∧ (nfgen(n2) from iigen(n2)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n2)(m0) ≤ iigen(n2)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n2

Proof step 173 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 174:

  iigen(n2)(0) = 0, iigen(n1)(m0) = iigen(n2)(m0), iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n2, m0 < n1, n1 ≤ n2, ∀ m. m < n2 → iigen(n2)(m + 1) = iigen(n2)(m) + # lf(m) ∧ (nfgen(n2) from iigen(n2)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n2)(m0) ≤ iigen(n2)(m)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 174 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 175:

  iigen(n1)(m0) = iigen(n2)(m0), iigen(n2)(0) = 0, iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n1, m0 < n2, n1 ≤ n2, m0 < n2 → iigen(n2)(m0 + 1) = iigen(n2)(m0) + # lf(m0) ∧ (nfgen(n2) from iigen(n2)(m0)) to # lf(m0) = lf(m0) ∧ (∀ m1. m1 ≤ m0 → iigen(n2)(m1) ≤ iigen(n2)(m0)), ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: m0

Proof step 175 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 176:

  iigen(n2)(0) = 0, iigen(n1)(m0) = iigen(n2)(m0), iigen(n2)(m0 + 1) = iigen(n2)(m0) + # lf(m0), (nfgen(n2) from iigen(n2)(m0)) to # lf(m0) = lf(m0), iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n2, m0 < n1, n1 ≤ n2, ∀ n0. iigen(n0)(0) = 0 ∧ (∀ m. m < n0 → iigen(n0)(m + 1) = iigen(n0)(m) + # lf(m) ∧ (nfgen(n0) from iigen(n0)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n0)(m0) ≤ iigen(n0)(m))), ∀ m1. m1 ≤ m0 → iigen(n2)(m1) ≤ iigen(n2)(m0) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 176 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 177:

  iigen(n2)(m0 + 1) = iigen(n2)(m0) + # lf(m0), iigen(n1)(m0) = iigen(n2)(m0), iigen(n2)(0) = 0, iigen(n1)(0) = 0, (nfgen(n2) from iigen(n2)(m0)) to # lf(m0) = lf(m0), iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n1, m0 < n2, n1 ≤ n2, ∀ m1. m1 ≤ m0 → iigen(n2)(m1) ≤ iigen(n2)(m0), ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Used terms: n1

Proof step 177 from proof infinite-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 178:

  iigen(n1)(0) = 0, iigen(n2)(0) = 0, iigen(n1)(m0) = iigen(n2)(m0), iigen(n2)(m0 + 1) = iigen(n2)(m0) + # lf(m0), (nfgen(n2) from iigen(n2)(m0)) to # lf(m0) = lf(m0), iigen(n1)(m0 + 1) ≠ iigen(n2)(m0 + 1), m0 < n2, m0 < n1, n1 ≤ n2, ∀ m. m < n1 → iigen(n1)(m + 1) = iigen(n1)(m) + # lf(m) ∧ (nfgen(n1) from iigen(n1)(m)) to # lf(m) = lf(m) ∧ (∀ m0. m0 ≤ m → iigen(n1)(m0) ≤ iigen(n1)(m)), ∀ m1. m1 ≤ m0 → iigen(n2)(m1) ≤ iigen(n2)(m0) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # lf(n) = ⌜lf(n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

Rule argument

Additional information

Proof step 178 from proof infinite-append (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: m0