[Imprint][Data Privacy Statement][E-Mail]



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Proof step 1 from proof min-skolemf (conclusion) (back to tree)

⊦ mon<(nσ) → (∃ m0. (nσ = [] ⊃ m0 = 0 ;(# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n ;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m) ) ))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof min-skolemf (previous step) (back to tree)

mon<(nσ) ⊦ ∃ m0. (nσ = [] ⊃ m0 = 0 ;(# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n ;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m) ) )

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof min-skolemf (previous step) (back to tree)

mon<(nσ) ⊦ ∃ m0. (nσ = [] ⊃ m0 = 0 ;(# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n ;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m) ) )

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: nσ = []

Additional information

Used formulas: nσ = []

Proof step 4 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ) ⊦ ∃ m0. (nσ = [] ⊃ m0 = 0 ;(# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n ;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m) ) )

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ) ⊦ ∃ m0. (# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Additional information

Used formulas: # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Proof step 6 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ) ⊦ ∃ m0. (# nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n ⊃ m0 = (# nσ).n;⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m)), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 7 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ) ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

User applied rule induction

And obtained the following new goals

Rule argument

indvar: n wfpred: <

Additional information

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

Proof step 8 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < n → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < n → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

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

Proof step 10 from proof min-skolemf (previous step) (back to tree)

n = 0, nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < n → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s) ⊦ ¬ n < 0; (s)

Proof step 11 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < 0 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ → m0 ≤ m)

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: 0 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ ⌜0⌝ < # σ ↔ σ ≠ []; (s) ⊦ 0 ≤ m; (s) Used terms: 0

Proof step 12 from proof min-skolemf (previous step) (back to tree)

n = n0.+1, nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < n → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n ≤ nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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) ⊦ m.+1 = m + 1; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 13 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [n1, nσ] → [n0, nσ]

Additional information

Used simplifier rules: ⊦ m.+1 = m + 1; (s) ⊦ 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: n0, nσ

Proof step 14 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), (∃ m0. ⌜m0⌝ < # nσ ∧ n0 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 ≤ nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

User applied rule exists right

And obtained the following new goals

Rule argument

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

Additional information

Used terms: n

Proof step 16 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 ≤ nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → n ≤ m), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 ≤ nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (⌜n1⌝ < # nσ ∧ n0 < nσ(n1) → n ≤ n1), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n0 ≠ nσ(n)

Additional information

Used formulas: n0 ≠ nσ(n)

Proof step 18 from proof min-skolemf (previous step) (back to tree)

n0 ≠ nσ(n), nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 ≤ nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (⌜n1⌝ < # nσ ∧ n0 < nσ(n1) → n ≤ n1), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 19 from proof min-skolemf (previous step) (back to tree)

n0 ≠ nσ(n), nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (⌜n1⌝ < # nσ ∧ n0 < nσ(n1) → n ≤ n1), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 4

Additional information

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

Proof step 20 from proof min-skolemf (previous step) (back to tree)

n0 ≠ nσ(n), nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ, n0 < nσ(n), ∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (⌜n1⌝ < # nσ ∧ n0 < nσ(n1) → n ≤ n1), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 21 from proof min-skolemf (previous step) (back to tree)

n0 ≠ nσ(n), nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ, n0 < nσ(n1), n0 < nσ(n), n1 < n, ⌜n1⌝ < # nσ, ∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: n1

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: n1

Proof step 22 from proof min-skolemf (previous step) (back to tree)

n0 = nσ(n), nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ n0 ≤ nσ(n) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n0, ∀ n1, nσ. n1 < n0.+1 → nσ ≠ [] ∧ mon<(nσ) → (∃ m0. ⌜m0⌝ < # nσ ∧ n1 ≤ nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n1 ≤ nσ(m) → m0 ≤ m)) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ n0 < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ n0 < nσ(m) → m0 ≤ m), ⌜n⌝ < # nσ ∧ n0 < nσ(n) ∧ (⌜n1⌝ < # nσ ∧ n0 < nσ(n1) → n ≤ n1), # nσ ≠ ∞ ∧ ¬ n0 < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 23 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: n + 1 abort: F discard: F computed: F

Additional information

Used terms: n + 1

Proof step 24 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), ⌜n + 1⌝ < # nσ ∧ nσ(n) < nσ(n + 1) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → n + 1 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 25 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), ⌜n⌝ + ⌜1⌝ < # nσ ∧ nσ(n) < nσ(n + 1) ∧ (⌜n0⌝ < # nσ ∧ nσ(n) < nσ(n0) → n < n0), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ⌜n⌝ + ⌜1⌝ < # nσ

Additional information

Used formulas: ⌜n⌝ + ⌜1⌝ < # nσ

Proof step 26 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), ⌜n⌝ + ⌜1⌝ < # nσ ∧ nσ(n) < nσ(n + 1) ∧ (⌜n0⌝ < # nσ ∧ nσ(n) < nσ(n0) → n < n0), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1), nσ(n) < nσ(n + 1) ∧ (⌜n0⌝ < # nσ ∧ nσ(n) < nσ(n0) → n < n0)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 3

Additional information

Proof step 28 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, ¬ nσ(n) < nσ(n + 1), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 29 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, nσ(n + 1) ≤ nσ(n), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: monls-def, seq: ⊦ mon<(nσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # nσ → nσ(n) < nσ(n + 1)), subst:[nσ] → [nσ], <all paths>

Additional information

Used terms: nσ

Proof step 30 from proof min-skolemf (previous step) (back to tree)

⊦ mon<(nσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # nσ → nσ(n) < nσ(n + 1))

Lemma monls-def

Proof step 31 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], ⌜n⌝ + ⌜1⌝ < # nσ, nσ(n + 1) ≤ nσ(n), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n. ⌜n⌝ + ⌜1⌝ < # nσ → nσ(n) < nσ(n + 1), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: n

Additional information

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

Proof step 32 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n0⌝ < # nσ, nσ(n) < nσ(n0), ¬ n < n0, ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 33 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, n0 ≤ n, ⌜n0⌝ < # nσ, nσ(n) < nσ(n0), ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

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

Proof step 34 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n⌝ < # nσ, ⌜n0⌝ < # nσ, n0 ≤ n, nσ(n) < nσ(n0), ∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m, ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 35 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ, ⌜n⌝ + ⌜1⌝ < # nσ, n0 ≤ n, ⌜n0⌝ < # nσ, nσ(n) < nσ(n0), ∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m, ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: n0

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) ⊦ m ≤ n → (n ≤ m ↔ m = n); (s) ⊦ ¬ n < n; (s) Used terms: n0

Proof step 36 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ¬ ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), ⌜n⌝ + ⌜1⌝ < # nσ ∧ nσ(n) < nσ(n + 1) ∧ (⌜n0⌝ < # nσ ∧ nσ(n) < nσ(n0) → n < n0), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 37 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ¬ ⌜n⌝ + ⌜1⌝ < # nσ, ⌜n⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m) ∨ # nσ ≠ ∞ ∧ nσ((# nσ).n - 1) < nσ(n), ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 4

Additional information

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

Proof step 38 from proof min-skolemf (previous step) (back to tree)

nσ ≠ [], mon<(nσ), ⌜n⌝ < # nσ, ¬ ⌜n⌝ + ⌜1⌝ < # nσ, ∀ m. ⌜m⌝ < # nσ ∧ nσ(n) ≤ nσ(m) → n ≤ m, ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ⌜m0⌝ < # nσ ∧ nσ(n) < nσ(m0) ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ(n) < nσ(m) → m0 ≤ m), # nσ ≠ ∞ ∧ ¬ nσ(n) < nσ((# nσ).n - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic 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) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) ⊦ n∞ = ⌜m⌝ + ⌜n⌝ → (n∞ = ∞ ↔ false); (s) ⊦ ¬ n < n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ < n∞ → (⌜m⌝ + ⌜1⌝ < n∞ ↔ ⌜m⌝ + ⌜1⌝ ≠ n∞); (ss)

Proof step 39 from proof min-skolemf (previous step) (back to tree)

⌜n⌝ + ⌜1⌝ = # nσ, nσ ≠ [], mon<(nσ), nσ(n) < nσ((# nσ).n - 1), ∀ m. ¬ n < m ∧ nσ(n) ≤ nσ(m) → n ≤ m, ∀ n1, nσ0. n1 < (nσ(n)).+1 → nσ0 ≠ [] ∧ mon<(nσ0) → (∃ m0. ⌜m0⌝ < # nσ0 ∧ n1 ≤ nσ0(m0) ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n1 ≤ nσ0(m) → m0 ≤ m)) ∨ # nσ0 ≠ ∞ ∧ nσ0((# nσ0).n - 1) < n1 ⊦ ∃ m0. ¬ n < m0 ∧ nσ(n) < nσ(m0) ∧ (∀ m. ¬ n < m ∧ nσ(n) < nσ(m) → m0 ≤ m)

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 1, flags: rotate, discard

Additional information

Used simplifier rules: ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .n = n; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s) ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ¬ n < n; (s)

Proof Tree for min-skolemf (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree