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

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ¬ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name: proofobls-branchingforwardsim, seq:⊦ bfwsim(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) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, 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], <all paths>

Additional information

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

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

⊦ bfwsim(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) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ))

A spec-lemma

Proof step 4 from proof branchinfw-fw-aux (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) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, 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], <all paths>

Additional information

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

Proof step 5 from proof branchinfw-fw-aux (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)))

A spec-lemma

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

∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), (∃ 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)))) ⊦

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 7 from proof branchinfw-fw-aux (previous step) (back to tree)

∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ), ¬(cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∨ ¬(r(as, cs0) ∧ cstep(cs0, ca, cs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1))) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

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

cstart(cs), ∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as. r(as, cs) ∧ astart(as)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: cs

Additional information

Used terms: cs

Proof step 9 from proof branchinfw-fw-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), r(as, cs0), ∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (cext(ca) = None ⊃ r(as, cs0) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs0) ), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: cs0, ca, cs1, as abort: F discard: T computed: T

Additional information

Used terms: cs0, ca, cs1, as

Proof step 10 from proof branchinfw-fw-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), r(as, cs0), r(as, cs0) ∧ cstep(cs0, ca, cs1) → (cext(ca) = None ⊃ r(as, cs1) ;∃ asl, aal, aa, as1. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs1) ), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cstep(cs0, ca, cs1), r(as, cs0), (cext(ca) = None ⊃ r(as, cs1) ; frag*(IOAut(astart, astep, aext))(asl, aal) ∧ aext !(aal) = [] ∧ asl.head = as ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ astep(asl.last, aa, as1) ∧ cext(ca) = aext(aa) ∧ r(as1, cs1) ), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

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

cext(ca) = None, cstep(cs0, ca, cs1), r(as, cs0), r(as, cs1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext(ca) = None, cstep(cs0, ca, cs1), r(as, cs1), r(as, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, [], as0) ∧ r(as0, cs1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as

Additional information

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

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

aext !(aal) = [], asl.head = as, cext(ca) = aext(aa), cext(ca) ≠ None, astep(asl.last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), r(as, cs0), r(as1, cs1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl → r(as, cs0) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ r(as0, cs1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), aext(aa) ≠ None, astep(asl.last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), r(as1, cs1), r(asl.head, cs0), ∀ as. as ∈ asl → r(as, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(asl.head, aext(aa).get ', as0) ∧ r(as0, cs1)

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] → [asl, as2, asl0]; ['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: asl, as2, asl0

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

aext !(aal) = [], asl = as2 + asl0, cext(ca) = aext(aa), aext(aa) ≠ None, asl ≠ [], astep(asl.last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), r(as2, cs0), r(as1, cs1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl → r(as, cs0) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as0) ∧ r(as0, cs1)

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

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

aext !(aal) = [], cext(ca) = aext(aa), aext(aa) ≠ None, astep((as2 ' + asl0).last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(as2 ' + asl0, aal), r(as1, cs1), r(as2, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl0 → r(as, cs0) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as0) ∧ r(as0, cs1)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: as1 abort: F discard: T computed: T

Additional information

Used terms: as1

Proof step 18 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), aext(aa) ≠ None, astep((as2 ' + asl0).last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(as2 ' + asl0, aal), r(as2, cs0), r(as1, cs1), ∀ as. as ∈ asl0 → r(as, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)) ⊦ move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as1) ∧ r(as1, cs1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 19 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), aext(aa) ≠ None, astep((as2 ' + asl0).last, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(as2 ' + asl0, aal), r(as1, cs1), r(as2, cs0), ¬ move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as1), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl0 → r(as, cs0) ⊦

Heuristic elimination 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] → [as2 ' + asl0, as0, asl]; ['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: as2 ' + asl0, as0, asl

Proof step 20 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], as2 ' + asl0 = asl + as0 ', cext(ca) = aext(aa), as2 ' + asl0 ≠ [], aext(aa) ≠ None, astep(as0, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(as2 ' + asl0, aal), r(as2, cs0), r(as1, cs1), ¬ move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as1), ∀ as. as ∈ asl0 → r(as, cs0), ∀ 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 21 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], as2 ' + asl0 = asl + as0 ', cext(ca) = aext(aa), aext(aa) ≠ None, astep(as0, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl + as0 ', aal), r(as2, cs0), r(as1, cs1), ¬ move(IOAut(astart, astep, aext))(as2, aext(aa).get ', as1), ∀ as. as ∈ asl0 → r(as, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), as2 ∈ asl ∨ as2 = as0, as0 = as2 ∨ as0 ∈ asl0 ⊦

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), as2, aext(aa).get ', as1]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(astart, astep, aext), as2, aext(aa).get ', as1

Proof step 22 from proof branchinfw-fw-aux (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 23 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], as2 ' + asl0 = asl + as0 ', cext(ca) = aext(aa), aext(aa) ≠ None, astep(as0, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl + as0 ', aal), r(as1, cs1), r(as2, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl0 → r(as, cs0), ∀ al, sl. ¬(frag*(IOAut(astart, astep, aext))(sl, al) ∧ as2 = sl[0] ∧ as1 = sl.last ∧ IOAut(astart, astep, aext).ext !(al) = aext(aa).get '), as0 = as2 ∨ as0 ∈ asl0, as2 ∈ asl ∨ as2 = as0 ⊦

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 24 from proof branchinfw-fw-aux (previous step) (back to tree)

aext !(aal) = [], as2 ' + asl0 = asl + as0 ', cext(ca) = aext(aa), aext(aa) ≠ None, astep(as0, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl + as0 ', aal), r(as2, cs0), r(as1, cs1), ∀ as. as ∈ asl0 → r(as, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ al, sl. ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ as2 ≠ sl[0] ∨ as1 ≠ sl.last ∨ aext !(al) ≠ aext(aa).get ', as2 ∈ asl ∨ as2 = as0, as0 = as2 ∨ as0 ∈ asl0 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: aal + aa ', as2 ' + asl0 + as1 ' abort: F discard: T computed: F

Additional information

Used terms: aal + aa ', as2 ' + asl0 + as1 '

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

aext !(aal) = [], as2 ' + asl0 = asl + as0 ', cext(ca) = aext(aa), aext(aa) ≠ None, astep(as0, aa, as1), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl + as0 ', aal), r(as1, cs1), r(as2, cs0), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl0 → r(as, cs0), as0 = as2 ∨ as0 ∈ asl0, as2 ∈ asl ∨ as2 = as0, ¬ frag*(IOAut(astart, astep, aext))(as2 ' + asl0 + as1 ', aal + aa ') ∨ as2 ≠ (as2 ' + asl0 + as1 ')[0] ∨ as1 ≠ (as2 ' + asl0 + as1 ').last ∨ aext !(aal + aa ') ≠ aext(aa).get ' ⊦

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] → [aa, aal, astep, as1, astart, aext, as2 ' + asl0]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ (a ' + x)[0] = a; (s) ⊦ (a ' + x + b ').last = b; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ ext(a) ≠ None → ext !(a ') = ext(a).get '; (s) ⊦ a ' = x + a0 ' ↔ a = a0 ∧ x = []; (ws) ⊦ (x + a ').last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: aal, as1, aext, as2 ' + asl0, aa, astart, astep

Proof step 26 from proof branchinfw-fw-aux (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 Tree for branchinfw-fw-aux (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