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Proof step 1 from proof PCIOAut (conclusion) (back to tree)

⊦ IOAut?(IOAut(lstart, lstep, lext)) ∧ IOAut?(IOAut(rstart, rstep, rext)) ∧ Πok(lext, rext, lren, rren) → IOAut?(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(lstart, lstep, lext)), IOAut?(IOAut(rstart, rstep, rext)), Πok(lext, rext, lren, rren), ¬ IOAut?(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: product-def: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren));

Proof step 3 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(rstart, rstep, rext)), IOAut?(IOAut(lstart, lstep, lext)), Πok(lext, rext, lren, rren), ¬ IOAut?(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)]; ['s, 'a, 'e] → [('ls, 'rs), 'pa, 'e], <all paths>

Additional information

Used terms: Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)

Proof step 4 from proof PCIOAut (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 5 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(lstart, lstep, lext)), IOAut?(IOAut(rstart, rstep, rext)), Πok(lext, rext, lren, rren), ∀ s. ¬(Π(lstart, rstart)(s) ∧ oinj(Π(lext, rext, lren, rren))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(rstart, rstep, rext)), IOAut?(IOAut(lstart, lstep, lext)), Πok(lext, rext, lren, rren), (∀ s. ¬ Π(lstart, rstart)(s)) ∨ ¬ oinj(Π(lext, rext, lren, rren)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [rstart, rstep, rext]; ['s, 'a, 'e] → ['rs, 'ra, 'e], <all paths>

Additional information

Used terms: rstart, rstep, rext

Proof step 7 from proof PCIOAut (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 8 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(lstart, lstep, lext)), Πok(lext, rext, lren, rren), ∃ s. rstart(s) ∧ oinj(rext), (∀ s. ¬ Π(lstart, rstart)(s)) ∨ ¬ oinj(Π(lext, rext, lren, rren)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof PCIOAut (previous step) (back to tree)

IOAut?(IOAut(lstart, lstep, lext)), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), (∀ s. ¬ Π(lstart, rstart)(s)) ∨ ¬ oinj(Π(lext, rext, lren, rren)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [lstart, lstep, lext]; ['s, 'a, 'e] → ['ls, 'la, 'e], <all paths>

Additional information

Used terms: lstart, lstep, lext

Proof step 10 from proof PCIOAut (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 11 from proof PCIOAut (previous step) (back to tree)

oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∃ s. lstart(s) ∧ oinj(lext), (∀ s. ¬ Π(lstart, rstart)(s)) ∨ ¬ oinj(Π(lext, rext, lren, rren)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof PCIOAut (previous step) (back to tree)

lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), (∀ s. ¬ Π(lstart, rstart)(s)) ∨ ¬ oinj(Π(lext, rext, lren, rren)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Proof step 13 from proof PCIOAut (previous step) (back to tree)

lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ s. ¬ Π(lstart, rstart)(s) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: (ls, rs) abort: F discard: T computed: F

Additional information

Used simplifier rules: lrstart-def: ⊦ Π(lstart, rstart)((ls, rs)) ↔ lstart(ls) ∧ rstart(rs); Used terms: (ls, rs)

Proof step 14 from proof PCIOAut (previous step) (back to tree)

lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ¬ oinj(Π(lext, rext, lren, rren)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: option, name:oinj-def, seq: ⊦ oinj(of) ↔ (∀ a1, a2. of(a1) ≠ None ∧ of(a1) = of(a2) → a1 = a2), subst:[of] → [Π(lext, rext, lren, rren)]; ['a, 'b] → ['pa, 'e], <all paths>

Additional information

Used terms: Π(lext, rext, lren, rren)

Proof step 15 from proof PCIOAut (previous step) (back to tree)

⊦ oinj(of) ↔ (∀ a1, a2. of(a1) ≠ None ∧ of(a1) = of(a2) → a1 = a2)

A spec-lemma

Proof step 16 from proof PCIOAut (previous step) (back to tree)

lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∃ pa, pa0. ¬(Π(lext, rext, lren, rren)(pa) ≠ None ∧ Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0) → pa = pa0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:productok-def, seq:⊦ Πok(lext, rext, lren, rren) ↔ inj(lren) ∧ inj(rren) ∧ surj(lren, rren) ∧ sharingok(lext, rext, lren, rren), subst:[lext, rext, lren, rren] → [lext, rext, lren, rren]; ['la, 'e, 'ra, 'pa] → ['la, 'e, 'ra, 'pa]

Additional information

Used terms: lext, rext, lren, rren

Proof step 18 from proof PCIOAut (previous step) (back to tree)

⊦ Πok(lext, rext, lren, rren) ↔ inj(lren) ∧ inj(rren) ∧ surj(lren, rren) ∧ sharingok(lext, rext, lren, rren)

Lemma productok-def

Proof step 19 from proof PCIOAut (previous step) (back to tree)

Πok(lext, rext, lren, rren) ↔ inj(lren) ∧ inj(rren) ∧ surj(lren, rren) ∧ sharingok(lext, rext, lren, rren), Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 20 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), sharingok(lext, rext, lren, rren), surj(lren, rren), Πok(lext, rext, lren, rren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: surj-def, seq: ⊦ surj(lren, rren) ↔ (∀ pa. (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra))), subst:[lren, rren] → [lren, rren]; ['la, 'pa, 'ra] → ['la, 'pa, 'ra], <all paths>

Additional information

Used terms: lren, rren

Proof step 21 from proof PCIOAut (previous step) (back to tree)

⊦ surj(lren, rren) ↔ (∀ pa. (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra)))

Lemma surj-def

Proof step 22 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), ∀ pa. (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: pa abort: F discard: F computed: F

Additional information

Used terms: pa

Proof step 23 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), ∀ pa. (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra)), (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), ∀ pa. (∃ la. pa = lren(la)) ∨ (∃ ra. pa = rren(ra)), pa = lren(la) ∨ pa = rren(ra) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: pa0 abort: F discard: T computed: F

Additional information

Used terms: pa0

Proof step 25 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), pa = lren(la) ∨ pa = rren(ra), (∃ la. pa0 = lren(la)) ∨ (∃ ra. pa0 = rren(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 26 from proof PCIOAut (previous step) (back to tree)

Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), pa = lren(la) ∨ pa = rren(ra), pa0 = lren(la0) ∨ pa0 = rren(ra0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

Proof step 27 from proof PCIOAut (previous step) (back to tree)

pa = lren(la), Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), pa0 = lren(la0) ∨ pa0 = rren(ra0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lrext-def-left: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(lren(la)) = lext(la); lrext-def-right: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(rren(ra)) = rext(ra); ⊦ inj(f) → (f(a1) = f(a2) ↔ a1 = a2); (s) ⊦ oinj(of) ∧ of(a1) ≠ None → (of(a1) = of(a2) ↔ a1 = a2); (s)

Proof step 28 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: sharingok-def, seq:⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. ( lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ ( ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra))), subst:[lext, rext, lren, rren] → [lext, rext, lren, rren]; ['la, 'e, 'ra, 'pa] → ['la, 'e, 'ra, 'pa], <all paths>

Additional information

Used terms: lext, rext, lren, rren

Proof step 29 from proof PCIOAut (previous step) (back to tree)

⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)))

Lemma sharingok-def

Proof step 30 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: la, ra0 abort: F discard: F computed: T

Additional information

Used terms: la, ra0

Proof step 31 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), lren(la) = rren(ra0) → lext(la) ≠ None ∧ rext(ra0) ≠ None ∧ lext(la) = rext(ra0), ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra0)) ∧ lext(la) ≠ None ∧ rext(ra0) ≠ None → lext(la) ≠ rext(ra0), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 32 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦ ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra0))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: cod(rren)(lren(la))

Additional information

Used formulas: cod(rren)(lren(la))

Proof step 33 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), cod(rren)(lren(la)), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦ ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as) ⊦ ¬ cod(f)(b) → b ≠ f(a); (as)

Proof step 34 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), rren(ra) = lren(la), rren(ra1) = lren(la), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: la, ra abort: F discard: F computed: T

Additional information

Used simplifier rules: ⊦ oinj(of) ∧ of(a1) ≠ None → (of(a1) = of(a2) ↔ a1 = a2); (s) Used terms: la, ra

Proof step 35 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦ ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 36 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la1) = rren(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: la1, ra0 abort: F discard: F computed: T

Additional information

Used terms: la1, ra0

Proof step 37 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lren(la1) = rren(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), lren(la1) = rren(ra0) → lext(la1) ≠ None ∧ rext(ra0) ≠ None ∧ lext(la1) = rext(ra0), ¬ cod(rren)(lren(la1)) ∧ ¬ cod(lren)(rren(ra0)) ∧ lext(la1) ≠ None ∧ rext(ra0) ≠ None → lext(la1) ≠ rext(ra0), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(f(a)); (s)

Proof step 38 from proof PCIOAut (previous step) (back to tree)

lext(la1) = rext(ra0), lext(la) = rext(ra0), lren(la1) = rren(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: option, name:oinj-def, seq: ⊦ oinj(of) ↔ (∀ a1, a2. of(a1) ≠ None ∧ of(a1) = of(a2) → a1 = a2), subst:[of] → [lext]; ['a, 'b] → ['la, 'e], <all paths>

Additional information

Used terms: lext

Proof step 39 from proof PCIOAut (previous step) (back to tree)

⊦ oinj(of) ↔ (∀ a1, a2. of(a1) ≠ None ∧ of(a1) = of(a2) → a1 = a2)

A spec-lemma

Proof step 40 from proof PCIOAut (previous step) (back to tree)

lext(la) = rext(ra0), lext(la1) = rext(ra0), lren(la1) = rren(ra0), lren(la) ≠ rren(ra0), rext(ra0) ≠ None, rren(ra0) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)), ∀ la, la0. lext(la) ≠ None ∧ lext(la) = lext(la0) → la = la0 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 15 terms: la, la1 abort: F discard: T computed: F

Additional information

Used terms: la, la1

Proof step 41 from proof PCIOAut (previous step) (back to tree)

pa = rren(ra), Π(lext, rext, lren, rren)(pa) = Π(lext, rext, lren, rren)(pa0), pa ≠ pa0, Π(lext, rext, lren, rren)(pa0) ≠ None, inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren), pa0 = lren(la0) ∨ pa0 = rren(ra0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lrext-def-right: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(rren(ra)) = rext(ra); lrext-def-left: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(lren(la)) = lext(la); ⊦ inj(f) → (f(a1) = f(a2) ↔ a1 = a2); (s) ⊦ oinj(of) ∧ of(a1) ≠ None → (of(a1) = of(a2) ↔ a1 = a2); (s)

Proof step 42 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), sharingok(lext, rext, lren, rren), Πok(lext, rext, lren, rren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: sharingok-def, seq:⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. ( lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ ( ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra))), subst:[lext, rext, lren, rren] → [lext, rext, lren, rren]; ['la, 'e, 'ra, 'pa] → ['la, 'e, 'ra, 'pa], <all paths>

Additional information

Used terms: lext, rext, lren, rren

Proof step 43 from proof PCIOAut (previous step) (back to tree)

⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)))

Lemma sharingok-def

Proof step 44 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: la0, ra abort: F discard: F computed: T

Additional information

Used terms: la0, ra

Proof step 45 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), lren(la0) = rren(ra) → lext(la0) ≠ None ∧ rext(ra) ≠ None ∧ lext(la0) = rext(ra), ¬ cod(rren)(lren(la0)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la0) ≠ None ∧ rext(ra) ≠ None → lext(la0) ≠ rext(ra), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 46 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦ ¬ cod(rren)(lren(la0)) ∧ ¬ cod(lren)(rren(ra))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 47 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), cod(rren)(lren(la0)), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 48 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), rren(ra1) = lren(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: la0, ra1 abort: F discard: T computed: T

Additional information

Used simplifier rules: ⊦ oinj(of) ∧ of(a1) ≠ None → (of(a1) = of(a2) ↔ a1 = a2); (s) Used terms: la0, ra1

Proof step 49 from proof PCIOAut (previous step) (back to tree)

rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), cod(lren)(rren(ra)), inj(lren), inj(rren), lstart(ls), oinj(rext), oinj(lext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 50 from proof PCIOAut (previous step) (back to tree)

lren(la) = rren(ra), rext(ra) = lext(la0), lext(la0) ≠ None, lren(la0) ≠ rren(ra0), rren(ra) ≠ lren(la0), inj(rren), inj(lren), lstart(ls), oinj(lext), oinj(rext), rstart(rs), Πok(lext, rext, lren, rren), ∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: la, ra abort: F discard: F computed: T

Additional information

Used simplifier rules: ⊦ oinj(of) ∧ of(a1) ≠ None → (of(a1) = of(a2) ↔ a1 = a2); (s) Used terms: la, ra

Proof Tree for PCIOAut (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