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Proof step 1 from proof triang-fin (conclusion) (back to tree)

nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), cstep(cs, ca, cs'), r(as, cs) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule induction

And obtained the following new goals

Rule argument

indvar: size01f(cs, ca, cs', as) wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: size01f(cs, ca, cs', as)

Proof step 3 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name:proofobls-normedforwardsim, seq:⊦ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs', as. r(as, cs) ∧ cstep(cs, ca, cs') → r(as, cs') ∧ cext(ca) = None ∨ (∃ as', aa. aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs')) ∨ (∃ as', aa. astep(as, aa, as') ∧ aext(aa) = None ∧ r(as', cs) ∧ size01f(cs, ca, cs', as') < size01f(cs, ca, cs', as))), subst:[cstep, astep, r, size01f, aext, cext, astart, cstart] → [cstep, astep, r, size01f, 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, size01f

Proof step 4 from proof triang-fin (previous step) (back to tree)

⊦ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs', as. r(as, cs) ∧ cstep(cs, ca, cs') → r(as, cs') ∧ cext(ca) = None ∨ (∃ as', aa. aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs')) ∨ (∃ as', aa. astep(as, aa, as') ∧ aext(aa) = None ∧ r(as', cs) ∧ size01f(cs, ca, cs', as') < size01f(cs, ca, cs', as)))

A spec-lemma

Proof step 5 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs', as. r(as, cs) ∧ cstep(cs, ca, cs') → r(as, cs') ∧ cext(ca) = None ∨ (∃ as', aa. aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs')) ∨ (∃ as', aa. astep(as, aa, as') ∧ aext(aa) = None ∧ r(as', cs) ∧ size01f(cs, ca, cs', as') < size01f(cs, ca, cs', as))), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 6 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ cs, ca, cs', as. r(as, cs) ∧ cstep(cs, ca, cs') → r(as, cs') ∧ cext(ca) = None ∨ (∃ as', aa. aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs')) ∨ (∃ as', aa. astep(as, aa, as') ∧ aext(aa) = None ∧ r(as', cs) ∧ size01f(cs, ca, cs', as') < size01f(cs, ca, cs', as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: cs, ca, cs', as abort: F discard: T computed: T

Additional information

Used terms: cs, ca, cs', as

Proof step 7 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as, cs) ∧ cstep(cs, ca, cs') → r(as, cs') ∧ cext(ca) = None ∨ (∃ as', aa. aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs')) ∨ (∃ as', aa. astep(as, aa, as') ∧ aext(aa) = None ∧ r(as', cs) ∧ size01f(cs, ca, cs', as') < size01f(cs, ca, cs', as)), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof triang-fin (previous step) (back to tree)

cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), r(as, cs') ∧ cext(ca) = None ∨ aext(aa) = cext(ca) ∧ astep(as, aa, as') ∧ r(as', cs') ∨ astep(as, aa0, as'0) ∧ aext(aa0) = None ∧ r(as'0, cs) ∧ size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

Proof step 9 from proof triang-fin (previous step) (back to tree)

cext(ca) = None, cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as, cs'), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof triang-fin (previous step) (back to tree)

cext(ca) = None, cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs'), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = None ∧ r(as', cs')) ∨ 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 '.head = a; (s) ⊦ ext !([]) = []; (s) ⊦ frag*(IOAut(start, step, ext))(s ', []); (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a '.last = a; (s) Used terms: as ', []

Proof step 11 from proof triang-fin (previous step) (back to tree)

aext(aa) = cext(ca), astep(as, aa, as'), cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as', cs'), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule exists right

And obtained the following new goals

Rule argument

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

Additional information

Used terms: as ', []

Proof step 12 from proof triang-fin (previous step) (back to tree)

aext(aa) = cext(ca), astep(as, aa, as'), cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as', cs'), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ as '.head = as ∧ aext !([]) = [] ∧ frag*(IOAut(astart, astep, aext))(as ', []) ∧ (∀ as0. as0 ∈ as ' → r(as0, cs)) ∧ ((∃ as', aa. astep(as '.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(as '.last, cs'))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.last = a; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ frag*(IOAut(start, step, ext))(s ', []); (s) ⊦ ext !([]) = []; (s) ⊦ a '.head = a; (s)

Proof step 13 from proof triang-fin (previous step) (back to tree)

aext(aa) = cext(ca), astep(as, aa, as'), cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as', cs'), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ as'0, aa0. astep(as, aa0, as'0) ∧ aext(aa0) = cext(ca) ∧ r(as'0, cs'), cext(ca) = None ∧ r(as, cs')

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as', aa

Additional information

Used terms: as', aa

Proof step 14 from proof triang-fin (previous step) (back to tree)

aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r] → [size01f, cs, ca, cs', as'0, cstep, cstart, cext, astart, astep, aext, r]

Additional information

Used terms: size01f, cs, ca, cs', as'0, cstep, cstart, cext, astart, astep, aext, r

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

aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∃ asl, aal. asl.head = as'0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs')), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

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 ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 16 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', asl.head) < size01f(cs, ca, cs', as), astep(as, aa0, asl.head), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(asl, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(asl.head, cs), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as. as ∈ asl → r(as, cs), astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r(asl.last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

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, as'0, 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, as'0, asl0

Proof step 17 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, asl = as'0 + asl0, asl ≠ [], size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(asl, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), ∀ as. as ∈ asl → r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r(asl.last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

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 18 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as0. as0 ∈ asl0 → r(as0, cs), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ asl, aal. asl.head = as ∧ aext !(aal) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r(asl.last, cs'))

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: as + as'0 + asl0, aa0 + aal abort: F discard: T computed: F

Additional information

Used terms: as + as'0 + asl0, aa0 + aal

Proof step 19 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), ∀ as0. as0 ∈ asl0 → r(as0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ (as + as'0 + asl0).head = as ∧ aext !(aa0 + aal) = [] ∧ frag*(IOAut(astart, astep, aext))(as + as'0 + asl0, aa0 + aal) ∧ (∀ as0. as0 ∈ as + as'0 + asl0 → r(as0, cs)) ∧ ((∃ as', aa. astep((as + as'0 + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs')) ∨ cext(ca) = None ∧ r((as + as'0 + asl0).last, cs'))

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 ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ [] + x = x; (s) ⊦ ext(a) = None → ext !(a ') = []; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ (a ' + x).head = a; (s)

Proof step 20 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as0. as0 ∈ asl0 → r(as0, cs), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ frag*(IOAut(astart, astep, aext))(as ' + as'0 ' + asl0, aa0 ' + aal) ∧ (as0 ∈ asl0 → r(as0, cs)) ∧ ((∃ as'1, aa1. astep((as'0 ' + asl0).last, aa1, as'1) ∧ aext(aa1) = cext(ca) ∧ r(as'1, cs')) ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'))

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] → [aa0, aal, astep, as, astart, aext, as'0 ' + asl0]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: aal, as, aext, as'0 ' + asl0, aa0, astart, astep

Proof step 21 from proof triang-fin (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 22 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), ∀ as0. as0 ∈ asl0 → r(as0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ (astep(as, aa0, (as'0 ' + asl0).head) ∧ frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal)) ∧ (as0 ∈ asl0 → r(as0, cs)) ∧ ((∃ as'1, aa1. astep((as'0 ' + asl0).last, aa1, as'1) ∧ aext(aa1) = cext(ca) ∧ r(as'1, cs')) ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'))

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).head = a; (s)

Proof step 23 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as0. as0 ∈ asl0 → r(as0, cs), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ (as0 ∈ asl0 → r(as0, cs)) ∧ ((∃ as'1, aa1. astep((as'0 ' + asl0).last, aa1, as'1) ∧ aext(aa1) = cext(ca) ∧ r(as'1, cs')) ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 24 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), as0 ∈ asl0, ¬ r(as0, cs), ∀ as0. as0 ∈ asl0 → r(as0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦

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 → (b ' + x).last = x.last; (s)

Proof step 25 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), as0 ∈ asl0, ¬ r(as0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as0. as0 ∈ asl0 → r(as0, cs), astep(asl0.last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r(asl0.last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: as0

Additional information

Used terms: as0

Proof step 26 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as, cs), r(as'0, cs), ∀ as0. as0 ∈ asl0 → r(as0, cs), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), astep((as'0 ' + asl0).last, aa, as') ∧ aext(aa) = cext(ca) ∧ r(as', cs') ∨ cext(ca) = None ∧ r((as'0 ' + asl0).last, cs'), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ as'1, aa1. astep((as'0 ' + asl0).last, aa1, as'1) ∧ aext(aa1) = cext(ca) ∧ r(as'1, cs'), cext(ca) = None ∧ r((as'0 ' + asl0).last, cs')

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof triang-fin (previous step) (back to tree)

aext !(aal) = [], aext(aa0) = None, aext(aa) = cext(ca), size01f(cs, ca, cs', as'0) < size01f(cs, ca, cs', as), astep(as, aa0, as'0), astep((as'0 ' + asl0).last, aa, as'), cstep(cs, ca, cs'), frag*(IOAut(astart, astep, aext))(as'0 ' + asl0, aal), nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), r(as'0, cs), r(as, cs), r(as', cs'), ∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as)), ∀ as0. as0 ∈ asl0 → r(as0, cs), ∀ size01f0, cs0, ca0, cs'0, as0, cstep, cstart, cext, astart, astep, aext, r. size01f0(cs0, ca0, cs'0, as0) < size01f(cs, ca, cs', as) → cstep(cs0, ca0, cs'0) ∧ nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f0) ∧ r(as0, cs0) → (∃ asl, aal. asl.head = as0 ∧ aext !(⌜aal⌝ ) = [] ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ (∀ as. as ∈ asl → r(as, cs0)) ∧ ((∃ as', aa. astep(asl.last, aa, as') ∧ aext(aa) = cext(ca0) ∧ r(as', cs'0)) ∨ cext(ca0) = None ∧ r(asl.last, cs'0))) ⊦ ∃ as'1, aa1. astep((as'0 ' + asl0).last, aa1, as'1) ∧ aext(aa1) = cext(ca) ∧ r(as'1, cs'), cext(ca) = None ∧ r((as'0 ' + asl0).last, cs')

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as', aa

Additional information

Used terms: as', aa

Proof Tree for triang-fin (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