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Proof step 1 from proof moveone-splitstep (conclusion) (back to tree)

move(IOAut(start, step, ext))(s, e ', s2) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], s2)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(start, step, ext), s, e ', s2]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s, e ', s2

Proof step 2 from proof moveone-splitstep (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

Lemma move

Proof step 3 from proof moveone-splitstep (previous step) (back to tree)

∃ al, sl. frag*(IOAut(start, step, ext))(sl, al) ∧ s = sl[0] ∧ s2 = sl.last ∧ IOAut(start, step, ext).ext !(al) = e ' ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], s2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 4 from proof moveone-splitstep (previous step) (back to tree)

ext !(al) = e ', frag*(IOAut(start, step, ext))(sl, al) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: al

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ ext !([]) = []; (s) ⊦ a ' ≠ []; (s)

Proof step 5 from proof moveone-splitstep (previous step) (back to tree)

ext !(a + al0) = e ', frag*(IOAut(start, step, ext))(sl, a + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s)

Proof step 6 from proof moveone-splitstep (previous step) (back to tree)

ext !(a ') + ext !(al0) = e ', frag*(IOAut(start, step, ext))(sl, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: sl

Additional information

Used simplifier rules: List('a) freely generated by [], +;;

Proof step 7 from proof moveone-splitstep (previous step) (back to tree)

ext !(a ') + ext !(al0) = e ', sl = [], frag*(IOAut(start, step, ext))(sl, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof moveone-splitstep (previous step) (back to tree)

ext !(a ') + ext !(al0) = e ', frag*(IOAut(start, step, ext))([], a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))([][0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], [].last)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: [][0], [].last, a

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); Used terms: [][0], [].last, a

Proof step 9 from proof moveone-splitstep (previous step) (back to tree)

ext !(a ') + ext !(al0) = e ', sl = s + sl0, frag*(IOAut(start, step, ext))(sl, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s)

Proof step 10 from proof moveone-splitstep (previous step) (back to tree)

ext !(a ') + ext !(al0) = e ', frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-one, seq: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '), subst:[ext, a] → [ext, a]; ['a, 'e] → ['a, 'e], <all paths>

Additional information

Used terms: ext, a

Proof step 11 from proof moveone-splitstep (previous step) (back to tree)

⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get ')

A spec-lemma

Proof step 12 from proof moveone-splitstep (previous step) (back to tree)

(ext(a) = None ⊃ [];ext(a).get ') + ext !(al0) = e ', frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 13 from proof moveone-splitstep (previous step) (back to tree)

[] + ext !(al0) = e ', ext(a) = None, frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s)

Proof step 14 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [a, al0, step, s, start, ext, sl0]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: al0, s, ext, sl0, a, start, step

Proof step 15 from proof moveone-splitstep (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

Lemma finfrag-rec

Proof step 16 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, frag*(IOAut(start, step, ext))(sl0, al0), step(s, a, sl0.head), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl0, s2, sl1]; ['a] → ['s]

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl0, s2, sl1

Proof step 17 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, sl0 = s2 + sl1, sl0 ≠ [], frag*(IOAut(start, step, ext))(sl0, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 18 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: ext, e, start, step, s2 ' + sl1 abort: F discard: T computed: T

Additional information

Used terms: ext, e, start, step, s2 ' + sl1

Proof step 19 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(s2 ' + sl1, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))((s2 ' + sl1)[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s)

Proof step 20 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s2, [], s0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), step(s, a, s2), step(s0, a0, s1) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: s0, s1, a0 abort: F discard: T computed: T

Additional information

Used terms: s0, s1, a0

Proof step 21 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a0) = Some(e), ext(a) = None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), move(IOAut(start, step, ext))(s2, [], s0), step(s0, a0, s1), step(s, a, s2) ⊦ ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s2, [], s0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), step(s, a, s2), step(s0, a0, s1), ¬ move(IOAut(start, step, ext))(s, [], s0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(start, step, ext), s, [], s0]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s, [], s0

Proof step 23 from proof moveone-splitstep (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

Lemma move

Proof step 24 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a0) = Some(e), ext(a) = None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), move(IOAut(start, step, ext))(s2, [], s0), step(s0, a0, s1), step(s, a, s2), ∀ al, sl. ¬(frag*(IOAut(start, step, ext))(sl, al) ∧ s = sl[0] ∧ s0 = sl.last ∧ IOAut(start, step, ext).ext !(al) = []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 25 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s2, [], s0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), step(s, a, s2), step(s0, a0, s1), ∀ al, sl. ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ s ≠ sl[0] ∨ s0 ≠ sl.last ∨ ext !(al) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(start, step, ext), s2, [], s0]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s2, [], s0

Proof step 26 from proof moveone-splitstep (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

Lemma move

Proof step 27 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext(a0) = Some(e), ext(a) = None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last), step(s0, a0, s1), step(s, a, s2), ∀ al, sl. ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ s ≠ sl[0] ∨ s0 ≠ sl.last ∨ ext !(al) ≠ [], ∃ al, sl. frag*(IOAut(start, step, ext))(sl, al) ∧ s2 = sl[0] ∧ s0 = sl.last ∧ IOAut(start, step, ext).ext !(al) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 28 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext !(al) = [], ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl[0] ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (sl[0] ' + sl1).last), step(s, a, sl[0]), step(sl.last, a0, s1), ∀ al, sl0. ¬ frag*(IOAut(start, step, ext))(sl0, al) ∨ s ≠ sl0[0] ∨ sl.last ≠ sl0.last ∨ ext !(al) ≠ [] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: a ' + al, s ' + sl abort: F discard: T computed: F

Additional information

Used terms: a ' + al, s ' + sl

Proof step 29 from proof moveone-splitstep (previous step) (back to tree)

ext !(al) = [], ext !(al0) = e ', ext(a0) = Some(e), ext(a) = None, frag*(IOAut(start, step, ext))(sl[0] ' + sl1, al0), frag*(IOAut(start, step, ext))(sl, al), move(IOAut(start, step, ext))(s1, [], (sl[0] ' + sl1).last), step(sl.last, a0, s1), step(s, a, sl[0]), ¬ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ∨ s ≠ (s ' + sl)[0] ∨ sl.last ≠ (s ' + sl).last ∨ ext !(a ' + al) ≠ [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ ext(a) = None → ext !(a ') = []; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ (a ' + x)[0] = a; (s)

Proof step 30 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext !(al) = [], ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl[0] ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (sl[0] ' + sl1).last), step(s, a, sl[0]), step(sl.last, a0, s1), ¬ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ∨ sl.last ≠ (s ' + sl).last ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [a, al, step, s, start, ext, sl]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: al, s, ext, sl, a, start, step

Proof step 31 from proof moveone-splitstep (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

Lemma finfrag-rec

Proof step 32 from proof moveone-splitstep (previous step) (back to tree)

ext !(al) = [], ext !(al0) = e ', ext(a0) = Some(e), ext(a) = None, frag*(IOAut(start, step, ext))(sl[0] ' + sl1, al0), frag*(IOAut(start, step, ext))(sl, al), move(IOAut(start, step, ext))(s1, [], (sl[0] ' + sl1).last), step(sl.last, a0, s1), step(s, a, sl[0]), ¬(step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)) ∨ sl.last ≠ (s ' + sl).last ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 33 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = e ', ext !(al) = [], ext(a) = None, ext(a0) = Some(e), frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl[0] ' + sl1, al0), move(IOAut(start, step, ext))(s1, [], (sl[0] ' + sl1).last), step(s, a, sl[0]), step(sl.last, a0, s1), ¬ step(s, a, sl.head) ∨ sl.last ≠ (s ' + sl).last ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl, s0, sl0]; ['a] → ['s]

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl, s0, sl0

Proof step 34 from proof moveone-splitstep (previous step) (back to tree)

ext(a).get ' + ext !(al0) = e ', ext(a) ≠ None, frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(e) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws)

Proof step 35 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) ≠ None, frag*(IOAut(start, step, ext))(s ' + sl0, a ' + al0), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [a, al0, step, s, start, ext, sl0]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: al0, s, ext, sl0, a, start, step

Proof step 36 from proof moveone-splitstep (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

Lemma finfrag-rec

Proof step 37 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) ≠ None, frag*(IOAut(start, step, ext))(sl0, al0), step(s, a, sl0.head), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl0, s2, sl1]; ['a] → ['s]

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl0, s2, sl1

Proof step 38 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], sl0 = s2 + sl1, ext(a) ≠ None, sl0 ≠ [], frag*(IOAut(start, step, ext))(sl0, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s ' + sl0).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 39 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ∃ s0, s1, a0. ext(a0) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s, [], s0) ∧ step(s0, a0, s1) ∧ move(IOAut(start, step, ext))(s1, [], (s2 ' + sl1).last)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: s, s2, a abort: F discard: T computed: T

Additional information

Used terms: s, s2, a

Proof step 40 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ext(a) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s, [], s) ∧ step(s, a, s2) ∧ move(IOAut(start, step, ext))(s2, [], (s2 ' + sl1).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: move-empty: ⊦ move(IOAut(start, step, ext))(s, [], s);

Proof step 41 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ext(a) = Some(ext(a).get) ∧ move(IOAut(start, step, ext))(s2, [], (s2 ' + sl1).last)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: option, name:elim-Some, seq: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)), subst:[opt, get] → [ext(a), e0]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)); (elim) Used terms: ext(a), e0

Proof step 42 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) = Some(e0), ext(a) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦ ext(a) = Some(e0) ∧ move(IOAut(start, step, ext))(s2, [], (s2 ' + sl1).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 43 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) = Some(e0), Some(e0) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ¬ move(IOAut(start, step, ext))(s2, [], (s2 ' + sl1).last), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(start, step, ext), s2, [], (s2 ' + sl1).last]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s2, [], (s2 ' + sl1).last

Proof step 44 from proof moveone-splitstep (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

Lemma move

Proof step 45 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) = Some(e0), Some(e0) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)), ∀ al, sl. ¬(frag*(IOAut(start, step, ext))(sl, al) ∧ s2 = sl[0] ∧ (s2 ' + sl1).last = sl.last ∧ IOAut(start, step, ext).ext !(al) = []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 46 from proof moveone-splitstep (previous step) (back to tree)

ext !(al0) = [], ext(a) = Some(e0), Some(e0) ≠ None, frag*(IOAut(start, step, ext))(s2 ' + sl1, al0), step(s, a, s2), ∀ ext, e, start, step, sl. ext !(al0) = e ' ∧ frag*(IOAut(start, step, ext))(sl, al0) → (∃ s0, s1, a. ext(a) = Some(e) ∧ move(IOAut(start, step, ext))(sl[0], [], s0) ∧ step(s0, a, s1) ∧ move(IOAut(start, step, ext))(s1, [], sl.last)), ∀ al, sl. ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ s2 ≠ sl[0] ∨ (sl[0] ' + sl1).last ≠ sl.last ∨ ext !(al) ≠ [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: al0, s2 ' + sl1

Additional information

Used simplifier rules: ⊦ (a ' + x)[0] = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: al0, s2 ' + sl1

Proof Tree for moveone-splitstep (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