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Proof step 1 from proof lemma_3-15 (conclusion) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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(cstart, cstep, cext), cs, el, cs0]; ['s, 'e, 'a] → ['cs, 'e, 'ca], <all paths>

Additional information

Used terms: IOAut(cstart, cstep, cext), cs, el, cs0

Proof step 4 from proof lemma_3-15 (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)

A spec-lemma

Proof step 5 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), t(cs0, as0), ∃ al, sl. frag*(IOAut(cstart, cstep, cext))(sl, al) ∧ cs = sl[0] ∧ cs0 = sl.last ∧ IOAut(cstart, cstep, cext).ext !(al) = el ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as)

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 6 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal), as0) ∧ t(csl[0], as)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: cal

Additional information

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

Proof step 7 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, []), t(csl.last, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !([]), as0) ∧ t(csl[0], as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !([]) = []; (s) ⊦ x = x.head ' → x.last = x.head; (s) ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s)

Proof step 8 from proof lemma_3-15 (previous step) (back to tree)

csl = csl.head ', aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), t(csl.head, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(csl[0], as)

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] → [csl, cs, csl0]; ['a] → ['cs]

Additional information

Used simplifier rules: ⊦ a ' ≠ []; (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: csl, cs, csl0

Proof step 9 from proof lemma_3-15 (previous step) (back to tree)

csl = cs ', csl = cs + csl0, csl ≠ [], aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), t(cs, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(csl[0], as)

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) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws)

Proof step 10 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), t(cs, as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(cs, as)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as0

Additional information

Used simplifier rules: ⊦ move(IOAut(start, step, ext))(s, [], s); (s) Used terms: as0

Proof step 11 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca + cal0), t(csl.last, as0), ∀ aINV, as0, cstart, cstep, cext, astart, astep, aext, t, cINV, csl. aINV(as0) ∧ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ∧ cINV(csl[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl, cal0) ∧ t(csl.last, as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl[0], as)) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca + cal0), as0) ∧ t(csl[0], as)

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 ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ a + x = a ' + x; (s)

Proof step 12 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal0), t(csl.last, as0), ∀ aINV, as0, cstart, cstep, cext, astart, astep, aext, t, cINV, csl. aINV(as0) ∧ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ∧ cINV(csl[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl, cal0) ∧ t(csl.last, as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl[0], as)) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(csl[0], as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: aINV, as0, cstart, cstep, cext, astart, astep, aext, t, cINV, csl.tail abort: F discard: T computed: F

Additional information

Used terms: aINV, as0, cstart, cstep, cext, astart, astep, aext, t, cINV, csl.tail

Proof step 13 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal0), t(csl.last, as0), aINV(as0) ∧ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ∧ cINV(csl.tail[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl.tail, cal0) ∧ t(csl.tail.last, as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl.tail[0], as)) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(csl[0], as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal0), t(csl.last, as0), cINV(csl.tail[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl.tail, cal0) ∧ t(csl.tail.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl.tail[0], as) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(csl[0], as)

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] → [csl, cs, csl0]; ['a] → ['cs]

Additional information

Used simplifier rules: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: csl, cs, csl0

Proof step 15 from proof lemma_3-15 (previous step) (back to tree)

csl = cs + csl0, csl ≠ [], aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal0), t(csl.last, as0), cINV(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ t(csl0.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl0[0], as) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(csl[0], as)

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) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 16 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl0, ca ' + cal0), t((cs ' + csl0).last, as0), cINV(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ t(csl0.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl0[0], as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [ca, cal0, cstep, cs, cstart, cext, csl0]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used terms: cal0, cs, cext, csl0, ca, cstart, cstep

Proof step 17 from proof lemma_3-15 (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)

A spec-lemma

Proof step 18 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), cstep(cs, ca, csl0.head), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((cs ' + csl0).last, as0), cINV(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ t(csl0.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl0[0], as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 19 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), cstep(cs, ca, csl0.head), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((cs ' + csl0).last, as0), cINV(csl0[0]) ∧ t(csl0.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl0[0], as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

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] → [csl0, cs0, csl]; ['a] → ['cs]

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: csl0, cs0, csl

Proof step 20 from proof lemma_3-15 (previous step) (back to tree)

csl0 = cs0 + csl, csl0 ≠ [], aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((cs ' + csl0).last, as0), cINV(csl0[0]) ∧ t(csl0.last, as0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(csl0[0], as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

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) ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 21 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), t((cs0 ' + csl).last, as0), cINV(cs0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(cs0, as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-backwardsiminv, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, aINV, cINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV, cINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cstep, cext, aext, t, astart, cstart, cINV, astep, aINV

Proof step 22 from proof lemma_3-15 (previous step) (back to tree)

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)))

Lemma proofobls-backwardsiminv

Proof step 23 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), t((cs0 ' + csl).last, as0), cINV(cs0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(cs0, as), ∀ cs. cstart(cs) → cINV(cs), ∀ as. astart(as) → aINV(as), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: cs, ca, cs0 abort: F discard: T computed: T

Additional information

Used terms: cs, ca, cs0

Proof step 24 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), t((cs0 ' + csl).last, as0), cINV(cs0) → aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(cal0), as0) ∧ t(cs0, as), cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ as. astart(as) → aINV(as), ∀ cs. cstart(cs) → cINV(cs) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 25 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), cINV(cs), cINV(cs0), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal0), as0), t((cs0 ' + csl).last, as0), t(cs0, as), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs. cstart(cs) → cINV(cs), ∀ as. astart(as) → aINV(as), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: cs, ca, cs0, as abort: F discard: T computed: T

Additional information

Used terms: cs, ca, cs0, as

Proof step 26 from proof lemma_3-15 (previous step) (back to tree)

aINV(as), aINV(as0), cINV(cs0), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal0), as0), t(cs0, as), t((cs0 ' + csl).last, as0), cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as) ∧ aINV(as) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as) ∧ t(cs, as2)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ as. astart(as) → aINV(as), ∀ cs. cstart(cs) → cINV(cs) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), aINV(as2), cINV(cs), cINV(cs0), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal0), as0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), t((cs0 ' + csl).last, as0), t(cs0, as), t(cs, as2), ∀ cs. cstart(cs) → cINV(cs), ∀ as. astart(as) → aINV(as), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as1)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: as2 abort: F discard: T computed: T

Additional information

Used terms: as2

Proof step 28 from proof lemma_3-15 (previous step) (back to tree)

aINV(as2), aINV(as), aINV(as0), cINV(cs0), cINV(cs), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), move(IOAut(astart, astep, aext))(as, cext !(cal0), as0), t(cs, as2), t(cs0, as), t((cs0 ' + csl).last, as0), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ as. astart(as) → aINV(as), ∀ cs. cstart(cs) → cINV(cs) ⊦ aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0) ∧ t(cs, as2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 29 from proof lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), aINV(as2), cINV(cs), cINV(cs0), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal0), as0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), t((cs0 ' + csl).last, as0), t(cs0, as), t(cs, as2), ¬ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0), as0), ∀ cs. cstart(cs) → cINV(cs), ∀ as. astart(as) → aINV(as), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:move-trans, seq:⊦ move(IOAut(start, step, ext))(s, el1, s1) ∧ move(IOAut(start, step, ext))(s1, el2, s2) → move(IOAut(start, step, ext))(s, el1 + el2, s2), subst:[s1, s2, step, s, start, el2, ext, el1] → [as, as0, astep, as2, astart, cext !(cal0), aext, (cext(ca) = None ⊃ [];cext(ca).get ')]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: cext !(cal0), as2, aext, as0, astart, (cext(ca) = None ⊃ [];cext(ca).get '), as, astep

Proof step 30 from proof lemma_3-15 (previous step) (back to tree)

⊦ move(IOAut(start, step, ext))(s, el1, s1) ∧ move(IOAut(start, step, ext))(s1, el2, s2) → move(IOAut(start, step, ext))(s, el1 + el2, s2)

A spec-lemma

Proof Tree for lemma_3-15 (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