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Proof step 1 from proof FOWs-FO-split (conclusion) (back to tree)

⊦ bel.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 ↔ (∃ bel0, bel1. bel = bel0 + FO(l) ' + bel1 ∧ fowl0 = bel0.FOWs(t, l) ∧ fowl1 = bel1.FOWs(t, l))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 2 from proof FOWs-FO-split (previous step) (back to tree)

bel.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 ⊦ ∃ bel0, bel1. bel = bel0 + FO(l) ' + bel1 ∧ fowl0 = bel0.FOWs(t, l) ∧ fowl1 = bel1.FOWs(t, l)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: bel

Additional information

Used simplifier rules: FOWs-base: ⊦ [].FOWs(t, l) = []; List('a) freely generated by [], +;; ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 3 from proof FOWs-FO-split (previous step) (back to tree)

(be + bel0).FOWs(t, l) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. be + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 4 from proof FOWs-FO-split (previous step) (back to tree)

be '.FOWs(t, l) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. be ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: be

Additional information

Used simplifier rules: Bevent freely generated by SF, W, FO;;

Proof step 5 from proof FOWs-FO-split (previous step) (back to tree)

be '.FOWs(t, l) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, be = SF, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. be ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-one-SF: ⊦ SF '.FOWs(t, l) = []; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s)

Proof step 6 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. SF ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: t, l, fowl0, fowl1 abort: F discard: F computed: T

Additional information

Used terms: t, l, fowl0, fowl1

Proof step 7 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. SF ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 8 from proof FOWs-FO-split (previous step) (back to tree)

∀ t0, l0, fowl0, fowl1. bel1.FOWs(t0, l0) + (l0 = l ⊃ FO(t0) ';[]) + bel2.FOWs(t0, l0) = fowl0 + FO(t0) ' + fowl1 → (∃ bel, bel3. bel1 + FO(l) ' + bel2 = bel + FO(l0) ' + bel3 ∧ fowl0 = bel.FOWs(t0, l0) ∧ fowl1 = bel3.FOWs(t0, l0)) ⊦ ∃ bel, bel3. SF ' + bel1 + FO(l) ' + bel2 = bel + FO(l) ' + bel3 ∧ bel1.FOWs(t, l) = bel.FOWs(t, l) ∧ bel2.FOWs(t, l) = bel3.FOWs(t, l)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: SF ' + bel1, bel2

Additional information

Used simplifier rules: FOWs-SF: ⊦ (SF ' + bel).FOWs(t, l) = bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: SF ' + bel1, bel2

Proof step 9 from proof FOWs-FO-split (previous step) (back to tree)

be '.FOWs(t, l) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, be = W(l0, v), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. be ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-one-W: ⊦ W(l0, v) '.FOWs(t, l) = (l = l0 ⊃ W(v, t) ';[]);

Proof step 10 from proof FOWs-FO-split (previous step) (back to tree)

(l = l0 ⊃ W(v, t) ';[]) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 11 from proof FOWs-FO-split (previous step) (back to tree)

W(v, t) ' + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l = l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, 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 ⊦ FO(tid) ≠ W(v, tid0); (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 12 from proof FOWs-FO-split (previous step) (back to tree)

W(v, t) ' + bel0.FOWs(t, l0) = fowl0 + FO(t) ' + fowl1, W(v, t) ∈ fowl0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl0

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ [] + x = x; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws) ⊦ FO(tid) ≠ W(v, tid0); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 13 from proof FOWs-FO-split (previous step) (back to tree)

W(v, t) ' + bel0.FOWs(t, l0) = fowl0 + FO(t) ' + fowl1, fowl0 = fow + fowl, W(v, t) ∈ fowl0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 14 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ W(v, t) ' + fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: t, l0, fowl, fowl1 abort: F discard: F computed: T

Additional information

Used terms: t, l0, fowl, fowl1

Proof step 15 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1, bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ W(v, t) ' + fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 16 from proof FOWs-FO-split (previous step) (back to tree)

∀ t0, l, fowl0, fowl1. bel1.FOWs(t0, l) + (l = l0 ⊃ FO(t0) ';[]) + bel2.FOWs(t0, l) = fowl0 + FO(t0) ' + fowl1 → (∃ bel, bel3. bel1 + FO(l0) ' + bel2 = bel + FO(l) ' + bel3 ∧ fowl0 = bel.FOWs(t0, l) ∧ fowl1 = bel3.FOWs(t0, l)) ⊦ ∃ bel, bel3. W(l0, v) ' + bel1 + FO(l0) ' + bel2 = bel + FO(l0) ' + bel3 ∧ W(v, t) ' + bel1.FOWs(t, l0) = bel.FOWs(t, l0) ∧ bel2.FOWs(t, l0) = bel3.FOWs(t, l0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: W(l0, v) ' + bel1, bel2

Additional information

Used simplifier rules: FOWs-W: ⊦ (W(l0, v) ' + bel).FOWs(t, l) = (l = l0 ⊃ W(v, t) ';[]) + bel.FOWs(t, l); ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: W(l0, v) ' + bel1, bel2

Proof step 17 from proof FOWs-FO-split (previous step) (back to tree)

[] + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, 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 ⊦ [] + x = x; (s)

Proof step 18 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: t, l, fowl0, fowl1 abort: F discard: F computed: T

Additional information

Used terms: t, l, fowl0, fowl1

Proof step 19 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. W(l0, v) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 20 from proof FOWs-FO-split (previous step) (back to tree)

l ≠ l0, ∀ t0, l1, fowl0, fowl1. bel1.FOWs(t0, l1) + (l1 = l ⊃ FO(t0) ';[]) + bel2.FOWs(t0, l1) = fowl0 + FO(t0) ' + fowl1 → (∃ bel, bel3. bel1 + FO(l) ' + bel2 = bel + FO(l1) ' + bel3 ∧ fowl0 = bel.FOWs(t0, l1) ∧ fowl1 = bel3.FOWs(t0, l1)) ⊦ ∃ bel, bel3. W(l0, v) ' + bel1 + FO(l) ' + bel2 = bel + FO(l) ' + bel3 ∧ bel1.FOWs(t, l) = bel.FOWs(t, l) ∧ bel2.FOWs(t, l) = bel3.FOWs(t, l)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: W(l0, v) ' + bel1, bel2

Additional information

Used simplifier rules: FOWs-W: ⊦ (W(l0, v) ' + bel).FOWs(t, l) = (l = l0 ⊃ W(v, t) ';[]) + bel.FOWs(t, l); ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: W(l0, v) ' + bel1, bel2

Proof step 21 from proof FOWs-FO-split (previous step) (back to tree)

be '.FOWs(t, l) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, be = FO(l0), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. be ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-one-FO: ⊦ FO(l0) '.FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]);

Proof step 22 from proof FOWs-FO-split (previous step) (back to tree)

(l = l0 ⊃ FO(t) ';[]) + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 23 from proof FOWs-FO-split (previous step) (back to tree)

FO(t) ' + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l = l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, 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 ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 24 from proof FOWs-FO-split (previous step) (back to tree)

FO(t) ' + bel0.FOWs(t, l0) = fowl0 + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl0

Additional information

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

Proof step 25 from proof FOWs-FO-split (previous step) (back to tree)

FO(t) ' + bel0.FOWs(t, l0) = fowl0 + FO(t) ' + fowl1, fowl0 = [], ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, 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 ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ [] + x = x; (s)

Proof step 26 from proof FOWs-FO-split (previous step) (back to tree)

∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ [] = bel1.FOWs(t, l0) ∧ bel0.FOWs(t, l0) = bel2.FOWs(t, l0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: [], bel0

Additional information

Used simplifier rules: FOWs-base: ⊦ [].FOWs(t, l) = []; ⊦ [] + x = x; (s) ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: [], bel0

Proof step 27 from proof FOWs-FO-split (previous step) (back to tree)

FO(t) ' + bel0.FOWs(t, l0) = fowl0 + FO(t) ' + fowl1, fowl0 = fow + fowl, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 28 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ FO(t) ' + fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: t, l0, fowl, fowl1 abort: F discard: F computed: T

Additional information

Used terms: t, l0, fowl, fowl1

Proof step 29 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1, bel0.FOWs(t, l0) = fowl + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l0) ' + bel2 ∧ fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l0) ' + bel2 ∧ FO(t) ' + fowl = bel1.FOWs(t, l0) ∧ fowl1 = bel2.FOWs(t, l0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 30 from proof FOWs-FO-split (previous step) (back to tree)

∀ t0, l, fowl0, fowl1. bel1.FOWs(t0, l) + (l = l0 ⊃ FO(t0) ';[]) + bel2.FOWs(t0, l) = fowl0 + FO(t0) ' + fowl1 → (∃ bel, bel3. bel1 + FO(l0) ' + bel2 = bel + FO(l) ' + bel3 ∧ fowl0 = bel.FOWs(t0, l) ∧ fowl1 = bel3.FOWs(t0, l)) ⊦ ∃ bel, bel3. FO(l0) ' + bel1 + FO(l0) ' + bel2 = bel + FO(l0) ' + bel3 ∧ FO(t) ' + bel1.FOWs(t, l0) = bel.FOWs(t, l0) ∧ bel2.FOWs(t, l0) = bel3.FOWs(t, l0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: FO(l0) ' + bel1, bel2

Additional information

Used simplifier rules: FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: FO(l0) ' + bel1, bel2

Proof step 31 from proof FOWs-FO-split (previous step) (back to tree)

[] + bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, 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 ⊦ [] + x = x; (s)

Proof step 32 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: t, l, fowl0, fowl1 abort: F discard: F computed: T

Additional information

Used terms: t, l, fowl0, fowl1

Proof step 33 from proof FOWs-FO-split (previous step) (back to tree)

bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1, l ≠ l0, bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)), ∀ t, l, fowl0, fowl1. bel0.FOWs(t, l) = fowl0 + FO(t) ' + fowl1 → (∃ bel1, bel2. bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)) ⊦ ∃ bel1, bel2. FO(l0) ' + bel0 = bel1 + FO(l) ' + bel2 ∧ fowl0 = bel1.FOWs(t, l) ∧ fowl1 = bel2.FOWs(t, l)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 34 from proof FOWs-FO-split (previous step) (back to tree)

l ≠ l0, ∀ t0, l1, fowl0, fowl1. bel1.FOWs(t0, l1) + (l1 = l ⊃ FO(t0) ';[]) + bel2.FOWs(t0, l1) = fowl0 + FO(t0) ' + fowl1 → (∃ bel, bel3. bel1 + FO(l) ' + bel2 = bel + FO(l1) ' + bel3 ∧ fowl0 = bel.FOWs(t0, l1) ∧ fowl1 = bel3.FOWs(t0, l1)) ⊦ ∃ bel, bel3. FO(l0) ' + bel1 + FO(l) ' + bel2 = bel + FO(l) ' + bel3 ∧ bel1.FOWs(t, l) = bel.FOWs(t, l) ∧ bel2.FOWs(t, l) = bel3.FOWs(t, l)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: FO(l0) ' + bel1, bel2

Additional information

Used simplifier rules: FOWs-FO: ⊦ (FO(l0) ' + bel).FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]) + bel.FOWs(t, l); ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: FO(l0) ' + bel1, bel2

Proof Tree for FOWs-FO-split (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