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

weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), exec*(IOAut(cstart, cstep, cext))(csl0, cal0), csl0.last = cs, move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as) ∧ aINV(as)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

csl0.last = cs, aINV(as0), exec*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as) ∧ aINV(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

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

aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(cstart, cstep, cext))(csl0.last, el, cs0), t(cs0, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(csl0.last, as) ∧ aINV(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), csl0.last, el, cs0]; ['s, 'e, 'a] → ['cs, 'e, 'ca], <all paths>

Additional information

Used terms: IOAut(cstart, cstep, cext), csl0.last, el, cs0

Proof step 4 from proof weak-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 weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∃ al, sl. frag*(IOAut(cstart, cstep, cext))(sl, al) ∧ csl0.last = sl[0] ∧ cs0 = sl.last ∧ IOAut(cstart, cstep, cext).ext !(al) = el ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(csl0.last, as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, cext !(cal), as0) ∧ t(csl[0], as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl, []), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, cext !([]), as0) ∧ t(csl[0], as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

csl0.last = csl[0], csl = csl.head ', aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(csl.head, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(csl[0], as) ∧ aINV(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, csl1]; ['a] → ['cs]

Additional information

Used simplifier rules: ⊦ a ' ≠ []; (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: csl, cs, csl1

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

csl0.last = csl[0], csl = cs ', csl = cs + csl1, csl ≠ [], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(csl[0], as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(csl0.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, [], as0) ∧ t(csl0.last, as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl, ca + cal1), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ csl0, csl, aINV, as0, cstart, cstep, cext, cal0, t, astart, astep, aext. csl0.last = csl[0] ∧ aINV(as0) ∧ cstart(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ t(csl.last, as0) ∧ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) → (∃ as. move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(as)) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, cext !(ca + cal1), as0) ∧ t(csl[0], as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal1), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ csl0, csl, aINV, as0, cstart, cstep, cext, cal0, t, astart, astep, aext. csl0.last = csl[0] ∧ aINV(as0) ∧ cstart(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ t(csl.last, as0) ∧ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) → (∃ as. move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(as)) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: csl0 + csl.tail.head, csl.tail, aINV, as0, cstart, cstep, cext, cal0 + ca ', t, astart, astep, aext abort: F discard: T computed: F

Additional information

Used terms: csl0 + csl.tail.head, csl.tail, aINV, as0, cstart, cstep, cext, cal0 + ca ', t, astart, astep, aext

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

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), (csl0 + csl.tail.head).last = csl.tail[0] ∧ aINV(as0) ∧ cstart((csl0 + csl.tail.head)[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl.tail, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl.tail.head, cal0 + ca ') ∧ t(csl.tail.last, as0) ∧ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) → (∃ as. move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl.tail[0], as) ∧ aINV(as)) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x + a = x + a '; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + a ').last = a; (s)

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

csl0.last = csl[0], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal1), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), csl.tail.head = csl.tail[0] ∧ cstart((csl0 + csl.tail.head ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl.tail, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl.tail.head ', cal0 + ca ') ∧ t(csl.tail.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl.tail[0], as) ∧ aINV(as) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(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, csl1]; ['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, csl1

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

csl0.last = csl[0], csl = cs + csl1, csl ≠ [], aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl, ca ' + cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(csl.last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), csl1.head = csl1[0] ∧ cstart((csl0 + csl1.head ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl1.head ', cal0 + ca ') ∧ t(csl1.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl1[0], as) ∧ aINV(as) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl[0], as) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), frag*(IOAut(cstart, cstep, cext))(csl0.last ' + csl1, ca ' + cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((csl0.last ' + csl1).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), csl1.head = csl1[0] ∧ cstart((csl0 + csl1.head ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl1.head ', cal0 + ca ') ∧ t(csl1.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl1[0], as) ∧ aINV(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(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, cal1, cstep, csl0.last, cstart, cext, csl1]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used terms: cal1, csl0.last, cext, csl1, ca, cstart, cstep

Proof step 17 from proof weak-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 weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, csl1.head), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), t((csl0.last ' + csl1).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), csl1.head = csl1[0] ∧ cstart((csl0 + csl1.head ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl1.head ', cal0 + ca ') ∧ t(csl1.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl1[0], as) ∧ aINV(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, csl1.head), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((csl0.last ' + csl1).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), csl1.head = csl1[0] ∧ cstart((csl0 + csl1.head ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + csl1.head ', cal0 + ca ') ∧ t(csl1.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl1[0], as) ∧ aINV(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(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] → [csl1, cs, 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: csl1, cs, csl

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

csl1 = cs + csl, csl1 ≠ [], aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), t((csl0.last ' + csl1).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), cs = csl1[0] ∧ cstart((csl0 + cs ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ') ∧ t(csl1.last, as0) → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(csl1[0], as) ∧ aINV(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(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 weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), cstart((csl0 + cs ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ') → move(IOAut(astart, astep, aext))(as, cext !(cal1), as0) ∧ t(cs, as) ∧ aINV(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

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

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1), cstart((csl0 + cs ')[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ')

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[n, x, y] → [0, csl0, cs ']; ['a] → ['cs], <all paths>

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) Used terms: 0, csl0, cs '

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

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

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

0 < # csl0, aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1), cstart(csl0[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ')

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

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

csl0 ≠ [], aINV(as0), cstart(csl0.head), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ¬ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ') ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [csl0, cs0, csl1]; ['a] → ['cs]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: csl0, cs0, csl1

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

csl0 = cs0 + csl1, csl0 ≠ [], aINV(as0), cstart(cs0), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ¬ frag*(IOAut(cstart, cstep, cext))(csl0 + cs ', cal0 + ca ') ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

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

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

aINV(as0), cstart(cs0), cstep((cs0 ' + csl1).last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl1, cal0), t((cs ' + csl).last, as0), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ¬ frag*(IOAut(cstart, cstep, cext))((cs0 ' + csl1) + cs ', cal0 + ca ') ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t((cs0 ' + csl1).last, as1) ∧ aINV(as1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec-end, seq:⊦ frag*(IOAut(start, step, ext))(sl + s ', al + a ') ↔ step(sl.last, a, s) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [ca, cal0, cstep, cs, cstart, cext, cs0 ' + csl1]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: cal0, cs, cext, cs0 ' + csl1, ca, cstart, cstep

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

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

A spec-lemma

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

aINV(as0), aINV(as), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), t((cs ' + csl).last, as0), t(cs, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-weakbackwardsim, seq:⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as))), subst:[cstep, astep, aext, t, cext, aINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

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

⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)))

Lemma proofobls-weakbackwardsim

Proof step 31 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as), aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), t(cs, as), t((cs ' + csl).last, as0), ∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last), ∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s) ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 32 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), t((cs ' + csl).last, as0), t(cs, as), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ csl1, cal, ca0, cs0, as1. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) ∧ cstep(csl1.last, ca0, cs0) ∧ t(cs0, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(csl1.last, as2)), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ csl1, cal. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) → (∃ as. t(csl1.last, as) ∧ aINV(as)) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: csl0, cal0, ca, cs, as abort: F discard: T computed: T

Additional information

Used terms: csl0, cal0, ca, cs, as

Proof step 33 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as), aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), t(cs, as), t((cs ' + csl).last, as0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs) ∧ t(cs, as) ∧ aINV(as) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as) ∧ t(csl0.last, as2)), ∀ csl1, cal. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) → (∃ as. t(csl1.last, as) ∧ aINV(as)), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), aINV(as2), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), t((cs ' + csl).last, as0), t(cs, as), t(csl0.last, as2), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ csl1, cal. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) → (∃ as. t(csl1.last, as) ∧ aINV(as)) ⊦ ∃ as1. move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as1) ∧ aINV(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 35 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as2), aINV(as), aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), t(csl0.last, as2), t(cs, as), t((cs ' + csl).last, as0), ∀ csl1, cal. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) → (∃ as. t(csl1.last, as) ∧ aINV(as)), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as) ⊦ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0) ∧ t(csl0.last, as2) ∧ aINV(as2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 36 from proof weak-lemma_3-15 (previous step) (back to tree)

aINV(as0), aINV(as), aINV(as2), cstart(csl0[0]), cstep(csl0.last, ca, cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(astart, astep, aext))(as, cext !(cal1), as0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as), t((cs ' + csl).last, as0), t(cs, as), t(csl0.last, as2), ¬ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal1), as0), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ csl1, cal. frag*(IOAut(cstart, cstep, cext))(csl1, cal) ∧ cstart(csl1[0]) → (∃ as. t(csl1.last, as) ∧ aINV(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 !(cal1), 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 !(cal1), as2, aext, as0, astart, (cext(ca) = None ⊃ [];cext(ca).get '), as, astep

Proof step 37 from proof weak-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 weak-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