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



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof finfrag-split (conclusion) (back to tree)

⊦ ¬ # al < n ∧ n < # sl → ( frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al.take(n)) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al.drop(n)))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof finfrag-split (previous step) (back to tree)

¬ ( frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al.take(n)) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al.drop(n))), n < # sl, ¬ # al < n ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 3 from proof finfrag-split (previous step) (back to tree)

¬ ( frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al.take(n)) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al.drop(n))), sl ≠ [], n < # sl, n ≤ # al ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-del, name:elim-firstn-restn, seq: ⊦ n ≤ # x → (y = x.take(n) ∧ z = x.drop(n) ↔ x = y + z ∧ # y = n), subst:[x, n, y, z] → [al, n, al0, al1]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ n ≤ # x → (y = x.take(n) ∧ z = x.drop(n) ↔ x = y + z ∧ # y = n); (elim) Used terms: al, n, al0, al1

Proof step 4 from proof finfrag-split (previous step) (back to tree)

# al0 = n, al = al0 + al1, ¬(frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al0) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al1)), sl ≠ [], n < # sl, n ≤ # al ⊦

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 ≤ m + n; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 5 from proof finfrag-split (previous step) (back to tree)

¬ ( frag*(IOAut(start, step, ext))(sl, al0 + al1) ↔ frag*(IOAut(start, step, ext))(sl.take(# al0 + 1), al0) ∧ frag*(IOAut(start, step, ext))(sl.drop(# al0), al1)), sl ≠ [], # al0 < # sl ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:frag-split, seq:⊦ ¬ # α < ⌜n⌝ ∧ ⌜n⌝ < # σ → ( frag(IOAut(start, step, ext))(σ, α) ↔ frag*(IOAut(start, step, ext))(σ.take(n + 1), α.take(n)) ∧ frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n))), subst:[step, α, start, ext, n, σ] → [step, ⌜al0 + al1⌝ , start, ext, # al0, ⌜sl⌝ ]; ['s, 'a, 'e] → ['s, 'a, 'e]

Additional information

Used simplifier rules: frag-finite-sl: ⊦ frag(ioa)(⌜sl⌝ , α) ↔ frag*(ioa)(sl, α.l) ∧ finp(α); fragstar-def: ⊦ frag(ioa)(⌜sl⌝ , ⌜al⌝ ) ↔ frag*(ioa)(sl, al); ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ (⌜x⌝ + σ).take(# x) = x; (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (⌜x⌝ + σ).drop(# x) = σ; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ext, # al0, ⌜al0 + al1⌝ , step, start, ⌜sl⌝

Proof step 6 from proof finfrag-split (previous step) (back to tree)

⊦ ¬ # α < ⌜n⌝ ∧ ⌜n⌝ < # σ → ( frag(IOAut(start, step, ext))(σ, α) ↔ frag*(IOAut(start, step, ext))(σ.take(n + 1), α.take(n)) ∧ frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)))

Lemma frag-split

Proof Tree for finfrag-split (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