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Proof step 1 from proof B-postfix-weak-FOWs-postfix (conclusion) (back to tree)

⊦ bel ⊒w bel0 → bel.FOWs(t, l) ⊒ bel0.FOWs(t, l)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel ⊒w bel0, ¬ bel.FOWs(t, l) ⊒ bel0.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: B-postfix-weak: ⊦ bel0 ⊒w bel1 ↔ (∀ l. bel0 |SF l ⊒ bel1 |SF l); B-restrict-SF-l: ⊦ bel |SF l = bel.filter(λ be. be = SF ∨ be.l = l);

Proof step 3 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

¬ bel.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0. bel.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: l abort: F discard: T computed: T

Additional information

Used terms: l

Proof step 4 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ bel.FOWs(t, l) ⊒ bel0.FOWs(t, l) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: bel

Additional information

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

Proof step 5 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

[].filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ [].FOWs(t, l) ⊒ bel0.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FOWs-base: ⊦ [].FOWs(t, l) = []; ⊦ [] ⊒ x ↔ x = []; (s) ⊦ [].filter(p) = []; (s)

Proof step 6 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel0.filter(λ be. be = SF ∨ be.l = l) = [], bel0.FOWs(t, l) ≠ [] ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: bel0

Additional information

Used simplifier rules: FOWs-base: ⊦ [].FOWs(t, l) = []; List('a) freely generated by [], +;; ⊦ [].filter(p) = []; (s)

Proof step 7 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

(be + bel).filter(λ be. be = SF ∨ be.l = l) = [], (be + bel).FOWs(t, l) ≠ [], ∀ l, t. ¬(bel.filter(λ be. be = SF ∨ be.l = l) = [] ∧ bel.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 + y) + z = x + y + z; from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 8 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) = [], be.l ≠ l, be ≠ SF, ∀ l0, t0. bel.filter(λ be. be = SF ∨ be.l = l0) ≠ [] ∨ bel.FOWs(t0, l0) = [] ⊦ be '.FOWs(t, l) = [] ∧ bel.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 9 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) = [], be = W(l1, v), be.l ≠ l, be ≠ SF, ∀ l0, t0. bel.filter(λ be. be = SF ∨ be.l = l0) ≠ [] ∨ bel.FOWs(t0, l0) = [] ⊦ be '.FOWs(t, l) = [] ∧ bel.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) ';[]); ⊦ W(l, v) ≠ SF; (s) ⊦ W(l, v).l = l; (s)

Proof step 10 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) = [], bel.FOWs(t, l) ≠ [], l1 ≠ l, ∀ l0, t0. bel.filter(λ be. be = SF ∨ be.l = l0) ≠ [] ∨ bel.FOWs(t0, l0) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l, t

Additional information

Used terms: l, t

Proof step 11 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) = [], be = FO(l1), be.l ≠ l, be ≠ SF, ∀ l0, t0. bel.filter(λ be. be = SF ∨ be.l = l0) ≠ [] ∨ bel.FOWs(t0, l0) = [] ⊦ be '.FOWs(t, l) = [] ∧ bel.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) ';[]); ⊦ FO(l) ≠ SF; (s) ⊦ FO(l).l = l; (s)

Proof step 12 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel.filter(λ be. be = SF ∨ be.l = l) = [], bel.FOWs(t, l) ≠ [], l1 ≠ l, ∀ l0, t0. bel.filter(λ be. be = SF ∨ be.l = l0) ≠ [] ∨ bel.FOWs(t0, l0) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l, t

Additional information

Used terms: l, t

Proof step 13 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

(be + bel1).filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ (be + bel1).FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l, bel0, t. ¬(bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l) ∧ ¬ bel1.FOWs(t, l) ⊒ bel0.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 + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 14 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

(be = SF ∨ be.l = l ⊃ be ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 15 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: l, bel0, t abort: F discard: F computed: F

Additional information

Used terms: l, bel0, t

Proof step 16 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l, ¬ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l) ∨ bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

Proof step 17 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix-rec-right, seq: ⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x, subst:[x, a, y] → [bel0.filter(λ be. be = SF ∨ be.l = l), be, bel1.filter(λ be. be = SF ∨ be.l = l)]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: bel0.filter(λ be. be = SF ∨ be.l = l), be, bel1.filter(λ be. be = SF ∨ be.l = l)

Proof step 18 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x

A spec-lemma

Proof step 19 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

¬ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l, bel0.filter(λ be. be = SF ∨ be.l = l) = be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ∨ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s) a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 20 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel0.filter(λ be. be = SF ∨ be.l = l) = be ' + bel1.filter(λ be. be = SF ∨ be.l = l), be ∈ bel0, ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ¬ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:restrict-SF-eq-FOWs-eq, seq: ⊦ bel0 |SF l = bel1 |SF l → bel0.FOWs(t, l) = bel1.FOWs(t, l), subst:[bel0, l, bel1, t] → [bel0, l, be ' + bel1, t]

Additional information

Used simplifier rules: B-restrict-SF-l: ⊦ bel |SF l = bel.filter(λ be. be = SF ∨ be.l = l); FOWs-append: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l); ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x ⊒ x; (s) Used terms: bel0, l, be ' + bel1, t

Proof step 21 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

⊦ bel0 |SF l = bel1 |SF l → bel0.FOWs(t, l) = bel1.FOWs(t, l)

Lemma restrict-SF-eq-FOWs-eq

Proof step 22 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = l ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: be

Additional information

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

Proof step 23 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be = W(l1, v), bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = 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) ';[]); ⊦ W(l, v).l = l; (s) ⊦ W(l, v) ≠ SF; (s)

Proof step 24 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), W(l, v) ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ W(v, t) ' + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix-rec-right, seq: ⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x, subst:[x, a, y] → [bel0.FOWs(t, l), W(v, t), bel1.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: bel0.FOWs(t, l), W(v, t), bel1.FOWs(t, l)

Proof step 25 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x

A spec-lemma

Proof step 26 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be = FO(l1), bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), be ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), be = SF ∨ be.l = 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) ';[]); ⊦ FO(l).l = l; (s) ⊦ FO(l) ≠ SF; (s)

Proof step 27 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), FO(l) ' + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ FO(t) ' + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix-rec-right, seq: ⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x, subst:[x, a, y] → [bel0.FOWs(t, l), FO(t), bel1.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: bel0.FOWs(t, l), FO(t), bel1.FOWs(t, l)

Proof step 28 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

⊦ a ' + y ⊒ x ↔ x = a ' + y ∨ y ⊒ x

A spec-lemma

Proof step 29 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be.l ≠ l, be ≠ SF, [] + bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, 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 = x; (s)

Proof step 30 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be.l ≠ l, be ≠ SF, bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: l, bel0, t abort: F discard: F computed: T

Additional information

Used terms: l, bel0, t

Proof step 31 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be.l ≠ l, be ≠ SF, bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0), ¬ bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l) ∨ bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 32 from proof B-postfix-weak-FOWs-postfix (previous step) (back to tree)

be.l ≠ l, be ≠ SF, bel1.filter(λ be. be = SF ∨ be.l = l) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l), bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ¬ be '.FOWs(t, l) + bel1.FOWs(t, l) ⊒ bel0.FOWs(t, l), ∀ l0, bel, t0. ¬ bel1.filter(λ be. be = SF ∨ be.l = l0) ⊒ bel.filter(λ be. be = SF ∨ be.l = l0) ∨ bel1.FOWs(t0, l0) ⊒ bel.FOWs(t0, l0) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: be

Additional information

Used simplifier rules: FOWs-one-W: ⊦ W(l0, v) '.FOWs(t, l) = (l = l0 ⊃ W(v, t) ';[]); FOWs-one-FO: ⊦ FO(l0) '.FOWs(t, l) = (l = l0 ⊃ FO(t) ';[]); Bevent freely generated by SF, W, FO;; ⊦ FO(l).l = l; (s) ⊦ FO(l) ≠ SF; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ W(l, v).l = l; (s) ⊦ W(l, v) ≠ SF; (s)

Proof Tree for B-postfix-weak-FOWs-postfix (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