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Proof step 1 from proof fw-frag (conclusion) (back to tree)

fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), frag*(IOAut(cstart, cstep, cext))(csl, cal), r(as, csl[0]) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl.last)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof fw-frag (previous step) (back to tree)

frag*(IOAut(cstart, cstep, cext))(csl, cal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, csl[0]) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl.last)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: csl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 3 from proof fw-frag (previous step) (back to tree)

frag*(IOAut(cstart, cstep, cext))(cs + csl0, cal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, (cs + csl0)[0]), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, (cs + csl0).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 4 from proof fw-frag (previous step) (back to tree)

frag*(IOAut(cstart, cstep, cext))(cs ' + csl0, cal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).last)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cal

Additional information

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

Proof step 5 from proof fw-frag (previous step) (back to tree)

cal = [], frag*(IOAut(cstart, cstep, cext))(cs ' + csl0, cal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) ⊦ ext !([]) = []; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ (a ' + x).head = a; (s) ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s)

Proof step 6 from proof fw-frag (previous step) (back to tree)

fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ [] = aext !(aal) ∧ r(asl.last, cs)

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as ', [] abort: F discard: T computed: F

Additional information

Used simplifier rules: ⊦ a '[0] = a; (s) ⊦ frag*(IOAut(start, step, ext))(s ', []); (s) ⊦ ext !([]) = []; (s) ⊦ a '.last = a; (s) Used terms: as ', []

Proof step 7 from proof fw-frag (previous step) (back to tree)

cal = ca + cal0, frag*(IOAut(cstart, cstep, cext))(cs ' + csl0, cal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).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 ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ a + x = a ' + x; (s)

Proof step 8 from proof fw-frag (previous step) (back to tree)

frag*(IOAut(cstart, cstep, cext))(cs ' + csl0, ca ' + cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).last)

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 9 from proof fw-frag (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 10 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, csl0.head), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).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] → [csl0, cs0, csl]; ['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: csl0, cs0, csl

Proof step 11 from proof fw-frag (previous step) (back to tree)

csl0 = cs0 + csl, csl0 ≠ [], cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart, cstep, cext, cal, astart, astep, aext, r, as. frag*(IOAut(cstart, cstep, cext))(csl0, cal) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(as, csl0[0]) → (∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl0.last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs ' + csl0).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 ' + x)[0] = a; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 12 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, astep, r, aext, cext, astart, cstart] → [cstep, astep, r, aext, cext, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e]

Additional information

Used terms: cstep, cext, aext, r, astart, cstart, astep

Proof step 13 from proof fw-frag (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

Lemma proofobls-forwardsim

Proof step 14 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

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 15 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cs1, ca0, cs2, as0. r(as0, cs1) ∧ cstep(cs1, ca0, cs2) → (∃ as1. move(IOAut(astart, astep, aext))(as0, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ r(as1, cs2)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: cs, ca, cs0, as abort: F discard: T computed: T

Additional information

Used terms: cs, ca, cs0, as

Proof step 16 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as, cs), r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as1. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as1) ∧ r(as1, cs0)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as1), r(as, cs), r(as1, cs0), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

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(astart, astep, aext), as, (cext(ca) = None ⊃ [];cext(ca).get '), as1]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(astart, astep, aext), as, (cext(ca) = None ⊃ [];cext(ca).get '), as1

Proof step 18 from proof fw-frag (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 19 from proof fw-frag (previous step) (back to tree)

cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as1, cs0), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∃ al, sl. frag*(IOAut(astart, astep, aext))(sl, al) ∧ as = sl[0] ∧ as1 = sl.last ∧ IOAut(astart, astep, aext).ext !(al) = (cext(ca) = None ⊃ [];cext(ca).get ') ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)

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 20 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, astep, r, aext, cext, astart, cstart] → [cstep, astep, r, aext, cext, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e]

Additional information

Used terms: cstep, cext, aext, r, astart, cstart, astep

Proof step 21 from proof fw-frag (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

Lemma proofobls-forwardsim

Proof step 22 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl.last, cs0), r(asl[0], cs), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

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 23 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cs1, ca0, cs2, as. r(as, cs1) ∧ cstep(cs1, ca0, cs2) → (∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca0) = None ⊃ [];cext(ca0).get '), as0) ∧ r(as0, cs2)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: cs, ca, cs0, asl[0] abort: F discard: T computed: T

Additional information

Used terms: cs, ca, cs0, asl[0]

Proof step 24 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl.last, cs0), r(asl[0], cs), r(asl[0], cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(asl[0], (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs0)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 25 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), move(IOAut(astart, astep, aext))(asl[0], aext !(aal), as0), r(asl[0], cs), r(asl.last, cs0), r(as0, cs0), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

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(astart, astep, aext), asl[0], aext !(aal), as0]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(astart, astep, aext), asl[0], aext !(aal), as0

Proof step 26 from proof fw-frag (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 27 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(as0, cs0), r(asl.last, cs0), r(asl[0], cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∃ al, sl. frag*(IOAut(astart, astep, aext))(sl, al) ∧ asl[0] = sl[0] ∧ as0 = sl.last ∧ IOAut(astart, astep, aext).ext !(al) = aext !(aal) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

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 fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl0.last, cs0), ∀ cstart0, cstep0, cext0, cal, astart0, astep0, aext0, r0, as0. frag*(IOAut(cstart0, cstep0, cext0))(cs0 ' + csl, cal) ∧ fwsim(IOAut(cstart0, cstep0, cext0), IOAut(astart0, astep0, aext0), r0) ∧ r0(as0, cs0) → (∃ asl, aal. asl[0] = as0 ∧ frag*(IOAut(astart0, astep0, aext0))(asl, aal) ∧ cext0 !(cal) = aext0 !(aal) ∧ r0(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: cstart, cstep, cext, cal0, astart, astep, aext, r, asl0.last abort: F discard: T computed: T

Additional information

Used terms: cstart, cstep, cext, cal0, astart, astep, aext, r, asl0.last

Proof step 29 from proof fw-frag (previous step) (back to tree)

aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0.last, cs0), r(asl.last, cs0), r(asl0[0], cs), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0) ∧ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ∧ r(asl0.last, cs0) → (∃ asl, aal. asl[0] = asl0.last ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal0) = aext !(aal) ∧ r(asl.last, (cs0 ' + csl).last)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + cext !(cal0) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 30 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = asl0[0], asl1[0] = asl0.last, cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl1, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl1[0], cs0), r(asl1.last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ asl0, aal. asl0[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + aext !(aal1) = aext !(aal) ∧ r(asl0.last, (cs0 ' + csl).last)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: asl0 + asl1.tail, aal0 + aal1 abort: F discard: T computed: F

Additional information

Used terms: asl0 + asl1.tail, aal0 + aal1

Proof step 31 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl1[0] = asl0.last, asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl1, aal1), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl1.last, (cs0 ' + csl).last), r(asl1[0], cs0), r(asl.last, cs0), r(asl0[0], cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ (asl0 + asl1.tail)[0] = asl[0] ∧ frag*(IOAut(astart, astep, aext))(asl0 + asl1.tail, aal0 + aal1) ∧ (cext(ca) = None ⊃ [];cext(ca).get ') + aext !(aal1) = aext !(aal0 + aal1) ∧ r((asl0 + asl1.tail).last, (cs0 ' + csl).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 ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s)

Proof step 32 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl1[0] = asl0.last, asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl1, aal1), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl1.last, (cs0 ' + csl).last), r(asl1[0], cs0), r(asl.last, cs0), r(asl0[0], cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ (asl0 + asl1.tail)[0] = asl0[0] ∧ frag*(IOAut(astart, astep, aext))(asl0 + asl1.tail, aal0 + aal1) ∧ r((asl0 + asl1.tail).last, (cs0 ' + csl).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] → [asl1, as0, asl2]; ['a] → ['as]

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: asl1, as0, asl2

Proof step 33 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = asl0[0], asl1[0] = asl0.last, asl1 = as0 + asl2, asl1 ≠ [], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl1, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl1[0], cs0), r(asl1.last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ (asl0 + asl2)[0] = asl0[0] ∧ frag*(IOAut(astart, astep, aext))(asl0 + asl2, aal0 + aal1) ∧ r((asl0 + asl2).last, (cs0 ' + csl).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) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 34 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl0.last, cs0), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ (asl0 + asl2)[0] = asl0[0] ∧ frag*(IOAut(astart, astep, aext))(asl0 + asl2, aal0 + aal1) ∧ r((asl0 + asl2).last, (cs0 ' + csl).last)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 35 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0[0], (asl0 + asl2)[0] ≠ asl0[0], cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), r(asl0.last, cs0), r(asl.last, cs0), r(asl0[0], cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

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, asl0, asl2]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x]; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ # [] = 0; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) Used terms: 0, asl0, asl2

Proof step 36 from proof fw-frag (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof step 37 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), r(asl0.last, cs0), r(asl.last, cs0), r(asl0[0], cs), ¬ frag*(IOAut(astart, astep, aext))(asl0 + asl2, aal0 + aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-split, seq:⊦ ¬ # 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))), subst:[n, al, step, start, ext, sl] → [# aal0, aal0 + aal1, astep, astart, aext, asl0 + asl2]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ m + n < m; (s) ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# sl + n ≤ # al ↔ false); (as) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: aal0 + aal1, aext, asl0 + asl2, astart, # aal0, astep

Proof step 38 from proof fw-frag (previous step) (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)))

A spec-lemma

Proof step 39 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = asl0[0], # aal0 < #(asl0 + asl2), cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl0.last, cs0), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), ¬ #(aal0 + aal1) < # aal0, ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).take(# aal0 + 1), (aal0 + aal1).take(# aal0)) ∨ ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).drop(# aal0), (aal0 + aal1).drop(# aal0)) ⊦

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 a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + y).drop(# x) = y; (s) ⊦ (x + y).take(# x) = x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ m + n < m; (s) ⊦ #(x + y) = # x + # y; (s)

Proof step 40 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = asl0[0], # aal0 < # asl0 + # asl2, cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl0[0], cs), r(asl.last, cs0), r(asl0.last, cs0), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).take(# aal0 + 1), aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).drop(# aal0), aal1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-to-le, seq: ⊦ n ≤ # x → (x + y).take(n) = x.take(n), subst:[n, x, y] → [# aal0 + 1, asl0, asl2]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# al < # sl ↔ true); (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: # aal0 + 1, asl0, asl2

Proof step 41 from proof fw-frag (previous step) (back to tree)

⊦ n ≤ # x → (x + y).take(n) = x.take(n)

A spec-lemma

Proof step 42 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0[0], # aal0 < # asl0 + # asl2, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), r(asl0.last, cs0), r(asl.last, cs0), r(asl0[0], cs), # aal0 + 1 ≤ # asl0, ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))(asl0.take(# aal0 + 1), aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).drop(# aal0), aal1) ⊦

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) 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 + 1 ≤ n ↔ m < n; (s)

Proof step 43 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0.head, asl0 ≠ [], # aal0 < # asl0, # aal0 < # asl0 + # asl2, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), r(asl0.last, cs0), r(asl.last, cs0), r(asl[0], cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))(asl0.take(# aal0 + 1), aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).drop(# aal0), aal1) ⊦

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] → [asl0, as0, asl1]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: asl0, as0, asl1

Proof step 44 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), asl[0] = as0, asl0 = as0 + asl1, asl0 ≠ [], # aal0 < # asl0 + # asl2, # aal0 < # asl0, cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r(asl0.last, cs0), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))(asl0.take(# aal0 + 1), aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))((asl0 + asl2).drop(# aal0), aal1) ⊦

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 = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x).take(n + 1) = a + x.take(n); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl < # al ↔ false); (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 45 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(((asl[0] ' + asl1).last ' + asl2).last, (cs0 ' + csl).last), r((asl[0] ' + asl1).last, cs0), r(asl.last, cs0), r(asl[0], cs), # aal0 ≤ # asl1 + # asl2, ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1.take(# aal0), aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))(((asl[0] ' + asl1) + asl2).drop(# aal0), aal1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:first-all-01, seq: ⊦ # x = n → x.take(n) = x, subst:[x, n] → [asl1, # aal0]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ true); (ss) Used terms: asl1, # aal0

Proof step 46 from proof fw-frag (previous step) (back to tree)

⊦ # x = n → x.take(n) = x

A spec-lemma

Proof step 47 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r((asl[0] ' + asl1).last, cs0), r(((asl[0] ' + asl1).last ' + asl2).last, (cs0 ' + csl).last), # aal0 ≤ # asl1 + # asl2, ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ¬ frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0) ∨ ¬ frag*(IOAut(astart, astep, aext))(((asl[0] ' + asl1) + asl2).drop(# aal0), aal1) ⊦

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)

Proof step 48 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).last ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r((asl[0] ' + asl1).last, cs0), r(((asl[0] ' + asl1).last ' + asl2).last, (cs0 ' + csl).last), ¬ frag*(IOAut(astart, astep, aext))(((asl[0] ' + asl1) + asl2).drop(# aal0), aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:restN-append, seq: ⊦ n ≤ # x → (x + y).drop(n) = x.drop(n) + y, subst:[n, x, y] → [# aal0, asl[0] ' + asl1, asl2]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≤ n ↔ ¬ n < m; (ss) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ # x = 0 ↔ x = []; (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 Used terms: # aal0, asl[0] ' + asl1, asl2

Proof step 49 from proof fw-frag (previous step) (back to tree)

⊦ n ≤ # x → (x + y).drop(n) = x.drop(n) + y

A spec-lemma

Proof step 50 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # asl1 = # aal0, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(((asl[0] ' + asl1).last ' + asl2).last, (cs0 ' + csl).last), r((asl[0] ' + asl1).last, cs0), r(asl.last, cs0), r(asl[0], cs), # aal0 ≤ #(asl[0] ' + asl1), ¬ frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).drop(# aal0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

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 ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 51 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # asl1 = # aal0, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(((asl[0] ' + asl1).last ' + asl2).last, (cs0 ' + csl).last), r((asl[0] ' + asl1).last, cs0), r(asl.last, cs0), r(asl[0], cs), ¬ frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).drop(# aal0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-last, name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [asl[0] ' + asl1, as0, asl0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: asl[0] ' + asl1, as0, asl0

Proof step 52 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', asl[0] ' + asl1 ≠ [], cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl[0] ' + asl1, aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r(as0, cs0), r((as0 ' + asl2).last, (cs0 ' + csl).last), ¬ frag*(IOAut(astart, astep, aext))((asl[0] ' + asl1).drop(# aal0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(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 ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 53 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r(as0, cs0), r((as0 ' + asl2).last, (cs0 ' + csl).last), ¬ frag*(IOAut(astart, astep, aext))((asl0 + as0 ').drop(# aal0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), asl[0] ∈ asl0 ∨ asl[0] = as0, as0 = asl[0] ∨ as0 ∈ asl1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:rest-app, seq: ⊦ # x ≤ m → (x + y).drop(m) = y.drop(m - # x), subst:[m, x, y] → [# aal0, asl0, as0 ']; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: # aal0, asl0, as0 '

Proof step 54 from proof fw-frag (previous step) (back to tree)

⊦ # x ≤ m → (x + y).drop(m) = y.drop(m - # x)

A spec-lemma

Proof step 55 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((as0 ' + asl2).last, (cs0 ' + csl).last), r(as0, cs0), r(asl.last, cs0), r(asl[0], cs), ¬ frag*(IOAut(astart, astep, aext))((asl0 + as0 ').drop(# aal0) + asl2, aal1), ¬ # asl0 ≤ # aal0, ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), as0 = asl[0] ∨ as0 ∈ asl1, asl[0] ∈ asl0 ∨ asl[0] = as0 ⊦

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 inapp-08: ⊦ a ' + x = y + a0 ' ∧ y ≠ [] → a ∈ y ∧ a0 ∈ x; (forward) from specification list ⊦ a ' + x = x0 + x1 → (a ∈ x0 ↔ x0 ≠ []); (ss) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ni: ⊦ x + a ' = y + z ∧ ¬ a ∈ z → z = []; (forward) from specification list ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 56 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', asl0 ≠ [], # aal0 < # asl0, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((as0 ' + asl2).last, (cs0 ' + csl).last), r(as0, cs0), r(asl.last, cs0), r(asl[0], cs), as0 ∈ asl1, asl[0] ∈ asl0, ¬ frag*(IOAut(astart, astep, aext))((asl0 + as0 ').drop(# aal0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(astart, astep, aext), asl0 + as0 ', aal0]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) ⊦ # a ' = 0 +1; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # x = n → (x + x0).drop(n) = x0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: IOAut(astart, astep, aext), asl0 + as0 ', aal0

Proof step 57 from proof fw-frag (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

A spec-lemma

Proof step 58 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((as0 ' + asl2).last, (cs0 ' + csl).last), r(as0, cs0), r(asl.last, cs0), r(asl[0], cs), # asl0 ≤ # aal0, ¬ frag*(IOAut(astart, astep, aext))(as0 '.drop(# aal0 - # asl0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), as0 = asl[0] ∨ as0 ∈ asl1, asl[0] ∈ asl0 ∨ asl[0] = as0 ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [# aal0, # asl0, n0]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: # aal0, # asl0, n0

Proof step 59 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, # aal0 = n0 + # asl0, asl[0] ' + asl1 = asl0 + as0 ', cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r(as0, cs0), r((as0 ' + asl2).last, (cs0 ' + csl).last), # asl0 ≤ # aal0, ¬ frag*(IOAut(astart, astep, aext))(as0 '.drop(n0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), asl[0] ∈ asl0 ∨ asl[0] = as0, as0 = asl[0] ∨ as0 ∈ asl1 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n0

Additional information

Used simplifier rules: Nat freely generated by 0, +1;; ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ x.drop(0) = x; (s)

Proof step 60 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), # aal0 = n0 + # asl0, # asl1 = # aal0, asl[0] ' + asl1 = asl0 + as0 ', n0 = n +1, cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((as0 ' + asl2).last, (cs0 ' + csl).last), r(as0, cs0), r(asl.last, cs0), r(asl[0], cs), # asl0 ≤ # aal0, ¬ frag*(IOAut(astart, astep, aext))(as0 '.drop(n0) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), as0 = asl[0] ∨ as0 ∈ asl1, asl[0] ∈ asl0 ∨ asl[0] = as0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic

Proof step 61 from proof fw-frag (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), aext !(aal0) = aext !(aal), cext !(cal0) = aext !(aal1), # asl1 = # aal0, # aal0 = (n + 1) + # asl0, asl[0] ' + asl1 = asl0 + as0 ', cstep(cs, ca, cs0), frag*(IOAut(astart, astep, aext))(as0 ' + asl2, aal1), frag*(IOAut(astart, astep, aext))(asl0 + as0 ', aal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r(asl[0], cs), r(asl.last, cs0), r(as0, cs0), r((as0 ' + asl2).last, (cs0 ' + csl).last), # asl0 ≤ # aal0, ¬ frag*(IOAut(astart, astep, aext))(as0 '.drop(n + 1) + asl2, aal1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), asl[0] ∈ asl0 ∨ asl[0] = as0, as0 = asl[0] ∨ as0 ∈ asl1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(astart, astep, aext), asl0 + as0 ', aal0]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) ⊦ # a ' = 0 +1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m = m0 + 1 + n → m ≠ n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: IOAut(astart, astep, aext), asl0 + as0 ', aal0

Proof step 62 from proof fw-frag (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

A spec-lemma

Proof step 63 from proof fw-frag (previous step) (back to tree)

cext !(cal0) = aext !(aal1), aext !(aal0) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), asl[0] = asl0[0], cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(cs0 ' + csl, cal0), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(astart, astep, aext))(asl0, aal0), frag*(IOAut(astart, astep, aext))(asl0.last ' + asl2, aal1), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), r((asl0.last ' + asl2).last, (cs0 ' + csl).last), r(asl0.last, cs0), r(asl.last, cs0), r(asl0[0], cs), ¬ r((asl0 + asl2).last, (cs0 ' + csl).last), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[x, y] → [asl0.last ', asl2]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ y ≠ [] → (x + y).last = y.last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ frag*(IOAut(start, step, ext))(s ', al) ↔ al = []; (s) ⊦ x + [] = x; (s) Used terms: asl0.last ', asl2

Proof step 64 from proof fw-frag (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof Tree for fw-frag (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