[Imprint][Data Privacy Statement][E-Mail]



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Proof step 1 from proof reorder-step-readPWB (conclusion) (back to tree)

⊦ reorder(l, fowl0, fowl1, rfo) → reorder(l, fowl0 + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) '))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof reorder-step-readPWB (previous step) (back to tree)

reorder(l, fowl0, fowl1, rfo), ¬ reorder(l, fowl0 + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: fowl0

Additional information

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

Proof step 3 from proof reorder-step-readPWB (previous step) (back to tree)

reorder(l, [], fowl1, rfo), ¬ reorder(l, [] + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-base: ⊦ reorder(l, [], fowl0, rfo) ↔ fowl0 = [] ∧ (∀ t. propFOs(t, l, rfo(t)) = []); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s)

Proof step 4 from proof reorder-step-readPWB (previous step) (back to tree)

¬ reorder(l, FO(t) ', [], rfo(t := rfo(t) + FO(l) ')), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:reorder-rec-FO, seq:⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l))), subst:[fowl0, t, l, fowl, rfo] → [[], t, l, [], rfo(t := rfo(t) + FO(l) ')]

Additional information

Used terms: [], l, [], rfo(t := rfo(t) + FO(l) '), t

Proof step 5 from proof reorder-step-readPWB (previous step) (back to tree)

⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l)))

Lemma reorder-rec-FO

Proof step 6 from proof reorder-step-readPWB (previous step) (back to tree)

¬ reorder(l, FO(t) ', [], rfo(t := rfo(t) + FO(l) ')), reorder(l, FO(t) ' + [], [], rfo(t := rfo(t) + FO(l) ')) ↔ (∃ fowl1, fowl2. [] = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, [], fowl1 + fowl2, rfo(t := rfo(t) + FO(l) '))) ∨ [] | t = [] ∧ FO(l) ∈ rfo(t := rfo(t) + FO(l) ')(t) ∧ reorder(l, [], [], rfo(t := rfo(t) + FO(l) ')(t := rfo(t := rfo(t) + FO(l) ')(t) -1l FO(l))), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-base: ⊦ reorder(l, [], fowl0, rfo) ↔ fowl0 = [] ∧ (∀ t. propFOs(t, l, rfo(t)) = []); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ [] | t = []; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 7 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, (t = t0 ⊃ (rfo(t0) + FO(l) ') -1l FO(l);rfo(t0))) ≠ [], ¬ reorder(l, FO(t) ', [], rfo(t := rfo(t) + FO(l) ')), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 8 from proof reorder-step-readPWB (previous step) (back to tree)

t = t0, propFOs(t0, l, (rfo(t0) + FO(l) ') -1l FO(l)) ≠ [], ¬ reorder(l, FO(t) ', [], rfo(t := rfo(t) + FO(l) ')), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, (rfo(t0) + FO(l) ') -1l FO(l)) ≠ [], ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: t0 abort: F discard: T computed: T

Additional information

Used terms: t0

Proof step 10 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) = [], propFOs(t0, l, (rfo(t0) + FO(l) ') -1l FO(l)) ≠ [], ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:app, seq: ⊦ a ∈ x → (x + y) -1l a = (x -1l a) + y, subst:[a, x, y] → [FO(l), rfo(t0), FO(l) ']; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ x → (x + y) -1l a = x + (y -1l a); (s) ⊦ a ' -1l a = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: FO(l), rfo(t0), FO(l) '

Proof step 11 from proof reorder-step-readPWB (previous step) (back to tree)

⊦ a ∈ x → (x + y) -1l a = (x -1l a) + y

A spec-lemma

Proof step 12 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) = [], propFOs(t0, l, (rfo(t0) -1l FO(l)) + FO(l) ') ≠ [], FO(l) ∈ rfo(t0), ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: propFOs-append: ⊦ propFOs(t, l, bel0 + bel1) = propFOs(t, l, bel0) + propFOs(t, l, bel1); propFOs-one-fo-same: ⊦ propFOs(t, l, FO(l) ') = FO(t) '; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 13 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) = [], FO(l) ∈ rfo(t0), ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:in-def, seq: ⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z)), subst:[a, x] → [FO(l), rfo(t0)]; ['a] → [Bevent], <all paths>

Additional information

Used terms: FO(l), rfo(t0)

Proof step 14 from proof reorder-step-readPWB (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

A spec-lemma

Proof step 15 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) = [], ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')), ∃ y, z. rfo(t0) = y + (FO(l) + z) ⊦

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)

Proof step 16 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) = [], rfo(t0) = bel + FO(l) ' + bel0, ¬ reorder(l, FO(t0) ', [], rfo(t0 := rfo(t0) + FO(l) ')) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 2, flags: discard

Additional information

Used simplifier rules: propFOs-append: ⊦ propFOs(t, l, bel0 + bel1) = propFOs(t, l, bel0) + propFOs(t, l, bel1); propFOs-one-fo-same: ⊦ propFOs(t, l, FO(l) ') = FO(t) '; ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 17 from proof reorder-step-readPWB (previous step) (back to tree)

propFOs(t0, l, rfo(t0)) ≠ [], t ≠ t0, ¬ reorder(l, FO(t) ', [], rfo(t := rfo(t) + FO(l) ')), ∀ t. propFOs(t, l, rfo(t)) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: t0

Additional information

Used terms: t0

Proof step 18 from proof reorder-step-readPWB (previous step) (back to tree)

reorder(l, fow + fowl, fowl1, rfo), ¬ reorder(l, (fow + fowl) + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')), ∀ l, fowl1, rfo, t. ¬(reorder(l, fowl, fowl1, rfo) ∧ ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) '))) ⊦

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)

Proof step 19 from proof reorder-step-readPWB (previous step) (back to tree)

reorder(l, fow ' + fowl, fowl1, rfo), ¬ reorder(l, (fow ' + fowl) + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fow

Additional information

Used simplifier rules: FOW freely generated by FO, W;;

Proof step 20 from proof reorder-step-readPWB (previous step) (back to tree)

fow = FO(t1), reorder(l, fow ' + fowl, fowl1, rfo), ¬ reorder(l, (fow ' + fowl) + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-rec-FO: ⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l))); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 21 from proof reorder-step-readPWB (previous step) (back to tree)

∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')), fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl, fowl2 + fowl3, rfo) ∨ fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')), fowl1 | t1 = [] ∧ FO(l) ∈ (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) ∧ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) -1l FO(l)))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 22 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 = fowl2 + FO(t1) ' + fowl3, fowl2 | t1 = [], reorder(l, fowl, fowl2 + fowl3, rfo), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')), fowl1 | t1 = [] ∧ FO(l) ∈ (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) ∧ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) -1l FO(l)))

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 ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ FO(tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s)

Proof step 23 from proof reorder-step-readPWB (previous step) (back to tree)

fowl2 | t1 = [], reorder(l, fowl, fowl2 + fowl3, rfo), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl0, fowl4. fowl2 + FO(t1) ' + fowl3 = fowl0 + FO(t1) ' + fowl4 ∧ fowl0 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl0 + fowl4, rfo(t := rfo(t) + FO(l) '))

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: l, fowl2 + fowl3, rfo, t abort: F discard: T computed: F

Additional information

Used terms: l, fowl2 + fowl3, rfo, t

Proof step 24 from proof reorder-step-readPWB (previous step) (back to tree)

fowl2 | t1 = [], reorder(l, fowl, fowl2 + fowl3, rfo), ¬ reorder(l, fowl, fowl2 + fowl3, rfo) ∨ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')) ⊦ ∃ fowl0, fowl4. fowl2 + FO(t1) ' + fowl3 = fowl0 + FO(t1) ' + fowl4 ∧ fowl0 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl0 + fowl4, rfo(t := rfo(t) + FO(l) '))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 25 from proof reorder-step-readPWB (previous step) (back to tree)

fowl2 | t1 = [], reorder(l, fowl, fowl2 + fowl3, rfo), reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')) ⊦ ∃ fowl0, fowl4. fowl2 + FO(t1) ' + fowl3 = fowl0 + FO(t1) ' + fowl4 ∧ fowl0 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl0 + fowl4, rfo(t := rfo(t) + FO(l) '))

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: fowl2, fowl3

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl2, fowl3

Proof step 26 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')), fowl1 | t1 = [] ∧ FO(l) ∈ (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) ∧ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) -1l FO(l)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) ')), FO(l) ∈ (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) ∧ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) -1l FO(l)))

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 2

Additional information

Proof step 28 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ FO(l) ∈ (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

Heuristic nonsplitting cut applied rule cut formula

And closed the goal!

Rule argument

fma: t = t1

Additional information

Used simplifier rules: ⊦ a ∈ x → a ∈ x + y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used formulas: t = t1

Proof step 29 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (t = t1 ⊃ rfo(t1) + FO(l) ';rfo(t1)) -1l FO(l))), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 30 from proof reorder-step-readPWB (previous step) (back to tree)

t = t1, fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := (rfo(t1) + FO(l) ') -1l FO(l))), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

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 → (x + y) -1l a = (x -1l a) + y; (s)

Proof step 31 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')

Additional information

Used formulas: rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')

Proof step 32 from proof reorder-step-readPWB (previous step) (back to tree)

rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 33 from proof reorder-step-readPWB (previous step) (back to tree)

rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: l, fowl1, rfo(t1 := rfo(t1) -1l FO(l)), t1 abort: F discard: F computed: T

Additional information

Used terms: l, fowl1, rfo(t1 := rfo(t1) -1l FO(l)), t1

Proof step 34 from proof reorder-step-readPWB (previous step) (back to tree)

rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')), ¬ reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))) ∨ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t1 := rfo(t1 := rfo(t1) -1l FO(l))(t1) + FO(l) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 35 from proof reorder-step-readPWB (previous step) (back to tree)

rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), fowl1 | t1 = [], reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: rfo(t1 := rfo(t1) -1l FO(l))(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')

Additional information

Used formulas: rfo(t1 := rfo(t1) -1l FO(l))(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')

Proof step 36 from proof reorder-step-readPWB (previous step) (back to tree)

rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') = rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), fowl1 | t1 = [], rfo(t1 := rfo(t1) -1l FO(l))(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') ≠ rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

User applied rule expand left

And closed the goal!

Rule argument

pos: left 3

Additional information

Proof step 37 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ') ≠ rfo(t1 := (rfo(t1) -1l FO(l)) + FO(l) '), reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t1) ', fowl1, rfo(t1 := rfo(t1) + FO(l) ')(t1 := (rfo(t1) -1l FO(l)) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t1) ', fowl2 + fowl3, rfo(t1 := rfo(t1) + FO(l) '))

User applied rule expand left

And closed the goal!

Rule argument

pos: left 2

Additional information

Proof step 38 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], t ≠ t1, reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: l, fowl1, rfo(t1 := rfo(t1) -1l FO(l)), t abort: F discard: T computed: F

Additional information

Used terms: l, fowl1, rfo(t1 := rfo(t1) -1l FO(l)), t

Proof step 39 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], t ≠ t1, reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))), ¬ reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))) ∨ reorder(l, fowl + FO(t) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t1 := rfo(t1) -1l FO(l))(t) + FO(l) ')) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 40 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], t ≠ t1, reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), reorder(l, fowl + FO(t) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ')), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ') = rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))

Additional information

Used formulas: rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ') = rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))

Proof step 41 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ') ≠ rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l)), t ≠ t1, reorder(l, fowl + FO(t) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ')), reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

User applied rule expand left

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 42 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ')(t0) ≠ rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))(t0), t ≠ t1, reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), reorder(l, fowl + FO(t) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ')), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 43 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1 | t1 = [], (t = t0 ⊃ rfo(t0) + FO(l) ';(t1 = t0 ⊃ rfo(t0) -1l FO(l);rfo(t0))) ≠ (t1 = t0 ⊃ rfo(t0) -1l FO(l);(t = t0 ⊃ rfo(t0) + FO(l) ';rfo(t0))), t ≠ t1, reorder(l, fowl + FO(t) ', fowl1, rfo(t1 := rfo(t1) -1l FO(l))(t := rfo(t) + FO(l) ')), reorder(l, fowl, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl + FO(t) ', fowl2 + fowl3, rfo(t := rfo(t) + FO(l) '))

Heuristic nonsplitting cut applied rule cut formula

And closed the goal!

Rule argument

fma: t = t0

Additional information

Used simplifier rules: ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used formulas: t = t0

Proof step 44 from proof reorder-step-readPWB (previous step) (back to tree)

fow = W(v, t1), reorder(l, fow ' + fowl, fowl1, rfo), ¬ reorder(l, (fow ' + fowl) + FO(t) ', fowl1, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-rec-W: ⊦ reorder(l, W(v, t) ' + fowl0, fowl1, rfo) ↔ fowl1 ≠ [] ∧ fowl1.head = W(v, t) ∧ reorder(l, fowl0, fowl1.tail, rfo); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 45 from proof reorder-step-readPWB (previous step) (back to tree)

fowl1.head = W(v, t1), fowl1 ≠ [], reorder(l, fowl, fowl1.tail, rfo), ¬ reorder(l, fowl + FO(t) ', fowl1.tail, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

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] → [fowl1, fow, fowl2]; ['a] → [FOW]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: fowl1, fow, fowl2

Proof step 46 from proof reorder-step-readPWB (previous step) (back to tree)

fow = W(v, t1), fowl1 = fow + fowl2, fowl1 ≠ [], reorder(l, fowl, fowl2, rfo), ¬ reorder(l, fowl + FO(t) ', fowl2, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

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

Proof step 47 from proof reorder-step-readPWB (previous step) (back to tree)

reorder(l, fowl, fowl2, rfo), ¬ reorder(l, fowl + FO(t) ', fowl2, rfo(t := rfo(t) + FO(l) ')), ∀ l0, fowl0, rfo0, t0. ¬ reorder(l0, fowl, fowl0, rfo0) ∨ reorder(l0, fowl + FO(t0) ', fowl0, rfo0(t0 := rfo0(t0) + FO(l0) ')) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: l, fowl2, rfo, t

Additional information

Used terms: l, fowl2, rfo, t

Proof Tree for reorder-step-readPWB (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