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Proof step 1 from proof rely-FOFollowsW-stable (conclusion) (back to tree)

⊦ rely(t, pq, bq, pq0, bq0) ∧ bq(t) ⊒w bq0(t) ∧ FOFollowsW(t, bq(t), pq(l), l) → FOFollowsW(t, bq0(t), pq0(l), l)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FOFollowsW(t, bq(t), pq(l), l), rely(t, pq, bq, pq0, bq0), bq(t) ⊒w bq0(t), ¬ FOFollowsW(t, bq0(t), pq0(l), 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); rely-def: ⊦ rely(t, pq, bq, pq0, bq0) ↔ (∀ l. rely(t, l, pq(l), bq(t), pq0(l), bq0(t))); rely-local-def: ⊦ rely(t, l, fowl, bel, fowl0, bel0) ↔ (fowl | t) + bel.FOWs(t, l) ⊒ (fowl0 | t) + bel0.FOWs(t, l);

Proof step 3 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FOFollowsW(t, bq(t), pq(l), l), ¬ FOFollowsW(t, bq0(t), pq0(l), l), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOFollowsW-def, seq:⊦ FOFollowsW(t, bel, fowl, l) ↔ (∀ fowl0, fowl1, v. fowl = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bel) ∧ (∀ bel0, bel1, v. bel = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1), subst:[t, bel, fowl, l] → [t, bq0(t), pq0(l), l], <all paths>

Additional information

Used terms: t, bq0(t), pq0(l), l

Proof step 4 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ FOFollowsW(t, bel, fowl, l) ↔ (∀ fowl0, fowl1, v. fowl = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bel) ∧ (∀ bel0, bel1, v. bel = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1)

Lemma FOFollowsW-def

Proof step 5 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FOFollowsW(t, bq(t), pq(l), l), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), (∃ fowl0, fowl1, v. ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t))) ∨ (∃ bel0, bel1, v. ¬(bq0(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FOFollowsW(t, bq(t), pq(l), l), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOFollowsW-def, seq:⊦ FOFollowsW(t, bel, fowl, l) ↔ (∀ fowl0, fowl1, v. fowl = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bel) ∧ (∀ bel0, bel1, v. bel = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1), subst:[t, bel, fowl, l] → [t, bq(t), pq(l), l], <all paths>

Additional information

Used terms: t, bq(t), pq(l), l

Proof step 7 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ FOFollowsW(t, bel, fowl, l) ↔ (∀ fowl0, fowl1, v. fowl = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bel) ∧ (∀ bel0, bel1, v. bel = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1)

Lemma FOFollowsW-def

Proof step 8 from proof rely-FOFollowsW-stable (previous step) (back to tree)

∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:B-postfix-weak-FOWs-postfix, seq: ⊦ bel ⊒w bel0 → bel.FOWs(t, l) ⊒ bel0.FOWs(t, l), subst:[bel, bel0, t, l] → [bq(t), bq0(t), t, l]

Additional information

Used terms: bq(t), bq0(t), t, l

Proof step 9 from proof rely-FOFollowsW-stable (previous step) (back to tree)

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

Lemma B-postfix-weak-FOWs-postfix

Proof step 10 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t) ⊒w bq0(t) → bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

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 11 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) → bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: l0 abort: F discard: F computed: T

Additional information

Used terms: l0

Proof step 12 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) → bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 13 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ l0. (pq(l0) | t) + bq(t).FOWs(t, l0) ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: l

Proof step 14 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), (pq(l) | t) + bq(t).FOWs(t, l) ⊒ (pq0(l) | t) + bq0(t).FOWs(t, l), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix, seq: ⊦ x ⊒ y ↔ (∃ z. z + y = x), subst:[x, y] → [(pq(l) | t) + bq(t).FOWs(t, l), (pq0(l) | t) + bq0(t).FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: (pq(l) | t) + bq(t).FOWs(t, l), (pq0(l) | t) + bq0(t).FOWs(t, l)

Proof step 15 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ x ⊒ y ↔ (∃ z. z + y = x)

A spec-lemma

Proof step 16 from proof rely-FOFollowsW-stable (previous step) (back to tree)

bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ z. z + (pq0(l) | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 18 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ¬(pq0(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq0(t)) ∨ ¬(bq0(t) = bel0 + W(l, v0) ' + bel1 → FO(l) ∈ bel1) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

Proof step 19 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags:

Additional information

Proof step 20 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 + W(v, t) ' + fowl1 | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

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 ⊦ W(v, tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s)

Proof step 21 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix, seq: ⊦ x ⊒ y ↔ (∃ z. z + y = x), subst:[x, y] → [bq(t).FOWs(t, l), bq0(t).FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: bq(t).FOWs(t, l), bq0(t).FOWs(t, l)

Proof step 22 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ x ⊒ y ↔ (∃ z. z + y = x)

A spec-lemma

Proof step 23 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ z. z + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: rotate

Additional information

Proof step 25 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + fowl2 + bq0(t).FOWs(t, l), fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # fowl + #(fowl0 | t) < #(pq(l) | t)

Additional information

Used formulas: # fowl + #(fowl0 | t) < #(pq(l) | t)

Proof step 26 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + fowl2 + bq0(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 27 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) + W(v, t) ' + (fowl1 | t) = (pq(l) | t) + fowl2, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less, seq: ⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x, x0, x1, x2] → [fowl + (fowl0 | t), W(v, t) ' + (fowl1 | t), pq(l) | t, fowl2]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: fowl + (fowl0 | t), W(v, t) ' + (fowl1 | t), pq(l) | t, fowl2

Proof step 28 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 29 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), #(fowl + (fowl0 | t)) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ y. pq(l) | t = (fowl + (fowl0 | t)) + y ∧ W(v, t) ' + (fowl1 | t) = y + fowl2 ⊦

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 + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 30 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), W(v, t) ' + (fowl1 | t) = fowl3 + fowl2, pq0(l) = fowl0 + W(v, t) ' + fowl1, pq(l) | t = (fowl + (fowl0 | t)) + fowl3, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, W(v, t) ∈ fowl3 ∨ W(v, t) ∈ fowl2 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl3

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ x + [] = x; (s) ⊦ [] + x = x; (s) ⊦ ¬ a ∈ []; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 31 from proof rely-FOFollowsW-stable (previous step) (back to tree)

W(v, t) ' + (fowl1 | t) = fowl3 + fowl2, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl3 = fow + fowl4, pq0(l) = fowl0 + W(v, t) ' + fowl1, pq(l) | t = (fowl + (fowl0 | t)) + fowl3, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), W(v, t) ∈ fowl3 ∨ W(v, t) ∈ fowl2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 32 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = fowl4 + fowl2, pq(l) | t = fowl + (fowl0 | t) + W(v, t) ' + fowl4, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-t-split, seq:⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t), subst:[fowl0, t, fow, fowl, fowl1] → [fowl + (fowl0 | t), t, W(v, t), pq(l), fowl4], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pq(l), fowl + (fowl0 | t), W(v, t), t, fowl4

Proof step 33 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t)

A spec-lemma

Proof step 34 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = fowl4 + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ fowl2, fowl3. pq(l) = fowl2 + W(v, t) ' + fowl3 ∧ fowl + (fowl0 | t) = fowl2 | t ∧ W(v, t).tid = t ∧ fowl4 = fowl3 | t ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ W(v, tid).tid = tid; (s)

Proof step 35 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: fowl3, fowl5, v abort: F discard: F computed: T

Additional information

Used terms: fowl3, fowl5, v

Proof step 36 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, pq(l) = fowl3 + W(v, t) ' + fowl5 → FO(t) ∈ fowl5 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 37 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), FO(t) ∈ fowl5 ∨ FO(l) ∈ bq(t) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 14

Additional information

Proof step 38 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), FO(t) ∈ fowl5, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

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(t), fowl5]; ['a] → [FOW], <all paths>

Additional information

Used terms: FO(t), fowl5

Proof step 39 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

A spec-lemma

Proof step 40 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ y, z. fowl5 = y + (FO(t) + 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) ⊦ FO(tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s)

Proof step 41 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq(l) = fowl3 + W(v, t) ' + fowl4 + FO(t) ' + fowl6, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = ((fowl4 | t) + FO(t) ' + (fowl6 | t)) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-t-split, seq:⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t), subst:[fowl0, t, fow, fowl, fowl1] → [fowl4 | t, t, FO(t), fowl1, (fowl6 | t) + fowl2], <all paths>

Additional information

Used simplifier rules: ⊦ FO(tid).tid = tid; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl1, fowl4 | t, FO(t), t, (fowl6 | t) + fowl2

Proof step 42 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t)

A spec-lemma

Proof step 43 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), FO(l) ∈ bq(t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 44 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), FO(l) ∈ bq(t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

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), bq(t)]; ['a] → [Bevent], <all paths>

Additional information

Used terms: FO(l), bq(t)

Proof step 45 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

A spec-lemma

Proof step 46 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ y, z. bq(t) = 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 47 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

Proof step 48 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = (bel + FO(l) ' + bel0).FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), (bel + FO(l) ' + bel0).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. (bel + FO(l) ' + bel0).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOWs-append, seq: ⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l), subst:[t, l, bel0, bel1] → [t, l, bel, FO(l) ' + bel0], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: t, l, bel, FO(l) ' + bel0

Proof step 49 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ (bel0 + bel1).FOWs(t, l) = bel0.FOWs(t, l) + bel1.FOWs(t, l)

Lemma FOWs-append

Proof step 50 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + (FO(l) ' + bel0).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), (bel + FO(l) ' + bel0).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l0. (bel + FO(l) ' + bel0).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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 ⊦ FO(l).l = l; (s) ⊦ FO(l) ≠ SF; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s)

Proof step 51 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel0.FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # bel.FOWs(t, l) < # fowl2

Additional information

Used formulas: # bel.FOWs(t, l) < # fowl2

Proof step 52 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel0.FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # bel.FOWs(t, l) < # fowl2, bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 53 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel0.FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, fowl2 ≠ [], pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # bel.FOWs(t, l) < # fowl2, bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less, seq: ⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x1, x2, x, x0] → [fowl2, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel0.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl2, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel0.FOWs(t, l)

Proof step 54 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 55 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, fowl2 ≠ [], pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # bel.FOWs(t, l) < # fowl2, bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∃ y. fowl2 = bel.FOWs(t, l) + y ∧ FO(t) ' + bel0.FOWs(t, l) = y + bq0(t).FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FO-in-FOWs: ⊦ FO(t0) ∈ bel.FOWs(t, l) ↔ t = t0 ∧ FO(l) ∈ bel; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ m < m + n ↔ n ≠ 0; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 56 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, FO(t) ' + bel0.FOWs(t, l) = fowl4 + bq0(t).FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + bel.FOWs(t, l) + fowl4, fowl4 ≠ [], pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), FO(t) ∈ fowl4, bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl4

Additional information

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

Proof step 57 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FO(t) ' + bel0.FOWs(t, l) = fowl4 + bq0(t).FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, fowl4 = fow + fowl2, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + bel.FOWs(t, l) + fowl4, fowl4 ≠ [], pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), FO(t) ∈ fowl4, bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 58 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, bel0.FOWs(t, l) = fowl2 + bq0(t).FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + bel.FOWs(t, l) + FO(t) ' + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-t-split, seq:⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t), subst:[fowl0, t, fow, fowl, fowl1] → [(fowl5 | t) + bel.FOWs(t, l), t, FO(t), fowl1, fowl2], <all paths>

Additional information

Used simplifier rules: ⊦ FO(tid).tid = tid; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl1, (fowl5 | t) + bel.FOWs(t, l), FO(t), t, fowl2

Proof step 59 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t)

A spec-lemma

Proof step 60 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel0.FOWs(t, l), fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ # bel.FOWs(t, l) < # fowl2, ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 61 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, fowl2 + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel0.FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # fowl2 ≤ # bel.FOWs(t, l), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less-eq, seq: ⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x, x0, x1, x2] → [fowl2, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel0.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl2, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel0.FOWs(t, l)

Proof step 62 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 63 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # fowl2 ≤ # bel.FOWs(t, l), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ y. bel.FOWs(t, l) = fowl2 + y ∧ bq0(t).FOWs(t, l) = y + FO(t) ' + bel0.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 64 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = fowl3 | t, bel.FOWs(t, l) = fowl2 + fowl4, bq0(t).FOWs(t, l) = fowl4 + FO(t) ' + bel0.FOWs(t, l), bq(t) = bel + FO(l) ' + bel0, pq(l) = fowl3 + W(v, t) ' + fowl5, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = (fowl5 | t) + fowl2, pq(l) | t ≠ [], # fowl + #(fowl0 | t) < #(pq(l) | t), # fowl2 ≤ # bel.FOWs(t, l), bel.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ FO(l0) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ l1. bel.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ FO(l1) ';[]) + bel0.filter(λ be. be = SF ∨ be.l = l1) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOWs-FO-split, seq:⊦ 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)), subst:[fowl0, t, l, bel, fowl1] → [fowl4, t, l, bq0(t), bel0.FOWs(t, l)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ x → x ≠ y + a ' + z; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: l, bq0(t), fowl4, t, bel0.FOWs(t, l)

Proof step 65 from proof rely-FOFollowsW-stable (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))

Lemma FOWs-FO-split

Proof step 66 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + ((fowl0 | t) + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = (pq(l) | t) + fowl2 + bq0(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ # fowl + #(fowl0 | t) < #(pq(l) | t), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 67 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), fowl + (fowl0 | t) + W(v, t) ' + (fowl1 | t) = (pq(l) | t) + fowl2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less-eq, seq: ⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x1, x2, x, x0] → [fowl + (fowl0 | t), W(v, t) ' + (fowl1 | t), pq(l) | t, fowl2]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: fowl + (fowl0 | t), W(v, t) ' + (fowl1 | t), pq(l) | t, fowl2

Proof step 68 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 69 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl2 + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), #(pq(l) | t) ≤ #(fowl + (fowl0 | t)), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ y. fowl + (fowl0 | t) = (pq(l) | t) + y ∧ fowl2 = y + W(v, t) ' + (fowl1 | t) ⊦

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 + y) = # x + # y; (s)

Proof step 70 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + fowl3, (fowl3 + W(v, t) ' + (fowl1 | t)) + bq0(t).FOWs(t, l) = bq(t).FOWs(t, l), pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOWs-W-split, seq:⊦ bel.FOWs(t, l) = fowl0 + W(v, t) ' + fowl1 ↔ (∃ bel0, bel1. bel = bel0 + W(l, v) ' + bel1 ∧ fowl0 = bel0.FOWs(t, l) ∧ fowl1 = bel1.FOWs(t, l)), subst:[fowl0, t, l, v, bel, fowl1] → [fowl3, t, l, v, bq(t), (fowl1 | t) + bq0(t).FOWs(t, l)], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, l, bq(t), fowl3, t, (fowl1 | t) + bq0(t).FOWs(t, l)

Proof step 71 from proof rely-FOFollowsW-stable (previous step) (back to tree)

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

Lemma FOWs-W-split

Proof step 72 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + fowl3, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ bel0, bel1. bq(t) = bel0 + W(l, v) ' + bel1 ∧ fowl3 = bel0.FOWs(t, l) ∧ (fowl1 | t) + bq0(t).FOWs(t, l) = bel1.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 73 from proof rely-FOFollowsW-stable (previous step) (back to tree)

(fowl1 | t) + bq0(t).FOWs(t, l) = bel1.FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel1, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: bel0, bel1, v abort: F discard: T computed: T

Additional information

Used terms: bel0, bel1, v

Proof step 74 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), (fowl1 | t) + bq0(t).FOWs(t, l) = bel1.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel1, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 75 from proof rely-FOFollowsW-stable (previous step) (back to tree)

(fowl1 | t) + bq0(t).FOWs(t, l) = bel1.FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel1, pq0(l) = fowl0 + W(v, t) ' + fowl1, FO(l) ∈ bel1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t) ⊦

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), bel1]; ['a] → [Bevent], <all paths>

Additional information

Used terms: FO(l), bel1

Proof step 76 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

A spec-lemma

Proof step 77 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), (fowl1 | t) + bq0(t).FOWs(t, l) = bel1.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel1, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ y, z. bel1 = y + (FO(l) + z) ⊦

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); ⊦ x = y + a ' + z → a ∈ x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

Proof step 78 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), (fowl1 | t) + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel2.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: #(fowl1 | t) ≤ # bel.FOWs(t, l)

Additional information

Used formulas: #(fowl1 | t) ≤ # bel.FOWs(t, l)

Proof step 79 from proof rely-FOFollowsW-stable (previous step) (back to tree)

(fowl1 | t) + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel2.FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), #(fowl1 | t) ≤ # bel.FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less-eq, seq: ⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x, x0, x1, x2] → [fowl1 | t, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel2.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl1 | t, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel2.FOWs(t, l)

Proof step 80 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x ≤ # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 81 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), #(fowl1 | t) ≤ # bel.FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ y. bel.FOWs(t, l) = (fowl1 | t) + y ∧ bq0(t).FOWs(t, l) = y + FO(t) ' + bel2.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 82 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bel.FOWs(t, l) = (fowl1 | t) + fowl2, bq0(t).FOWs(t, l) = fowl2 + FO(t) ' + bel2.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(fowl1 | t) ≤ # bel.FOWs(t, l), #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOWs-FO-split, seq:⊦ 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)), subst:[fowl0, t, l, bel, fowl1] → [fowl2, t, l, bq0(t), bel2.FOWs(t, l)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ x → x ≠ y + a ' + z; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: l, bq0(t), fowl2, t, bel2.FOWs(t, l)

Proof step 83 from proof rely-FOFollowsW-stable (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))

Lemma FOWs-FO-split

Proof step 84 from proof rely-FOFollowsW-stable (previous step) (back to tree)

(fowl1 | t) + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel2.FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ¬ #(fowl1 | t) ≤ # bel.FOWs(t, l), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 85 from proof rely-FOFollowsW-stable (previous step) (back to tree)

(fowl1 | t) + bq0(t).FOWs(t, l) = bel.FOWs(t, l) + FO(t) ' + bel2.FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t ≠ [], # bel.FOWs(t, l) < #(fowl1 | t), #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:append-size-less, seq: ⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2)), subst:[x1, x2, x, x0] → [fowl1 | t, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel2.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl1 | t, bq0(t).FOWs(t, l), bel.FOWs(t, l), FO(t) ' + bel2.FOWs(t, l)

Proof step 86 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ # x < # x1 → (x + x0 = x1 + x2 ↔ (∃ y. x1 = x + y ∧ x0 = y + x2))

A spec-lemma

Proof step 87 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t ≠ [], # bel.FOWs(t, l) < #(fowl1 | t), #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ∃ y. fowl1 | t = bel.FOWs(t, l) + y ∧ FO(t) ' + bel2.FOWs(t, l) = y + bq0(t).FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: FO-in-FOWs: ⊦ FO(t0) ∈ bel.FOWs(t, l) ↔ t = t0 ∧ FO(l) ∈ bel; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 88 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), FO(t) ' + bel2.FOWs(t, l) = fowl2 + bq0(t).FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = bel.FOWs(t, l) + fowl2, fowl1 | t ≠ [], # bel.FOWs(t, l) < #(fowl1 | t), FO(t) ∈ fowl2, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl2

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ x + [] = x; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ a ∈ []; (s)

Proof step 89 from proof rely-FOFollowsW-stable (previous step) (back to tree)

FO(t) ' + bel2.FOWs(t, l) = fowl2 + bq0(t).FOWs(t, l), fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, fowl2 = fow + fowl3, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = bel.FOWs(t, l) + fowl2, fowl1 | t ≠ [], # bel.FOWs(t, l) < #(fowl1 | t), FO(t) ∈ fowl2, #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ fowl1, ¬ FO(l) ∈ bq0(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = 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 ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 90 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (fowl0 | t) = (pq(l) | t) + bel0.FOWs(t, l), bel2.FOWs(t, l) = fowl3 + bq0(t).FOWs(t, l), bq(t) = bel0 + W(l, v) ' + bel + FO(l) ' + bel2, pq0(l) = fowl0 + W(v, t) ' + fowl1, fowl1 | t = bel.FOWs(t, l) + FO(t) ' + fowl3, fowl1 | t ≠ [], # bel.FOWs(t, l) < #(fowl1 | t), #(pq(l) | t) ≤ # fowl + #(fowl0 | t), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bq0(t), ¬ FO(t) ∈ fowl1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-t-split, seq:⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t), subst:[fowl0, t, fow, fowl, fowl1] → [bel.FOWs(t, l), t, FO(t), fowl1, fowl3], <all paths>

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ FO(tid).tid = tid; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl1, bel.FOWs(t, l), FO(t), t, fowl3

Proof step 91 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ fowl | t = fowl0 + fow ' + fowl1 ↔ (∃ fowl2, fowl3. fowl = fowl2 + fow ' + fowl3 ∧ fowl0 = fowl2 | t ∧ fow.tid = t ∧ fowl1 = fowl3 | t)

A spec-lemma

Proof step 92 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bq0(t).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).FOWs(t, l) ⊒ bq0(t).FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bq0(t).filter(λ be. be = SF ∨ be.l = l0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags:

Additional information

Proof step 93 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + (bel0 + W(l, v0) ' + bel1).FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ (bel0 + W(l, v0) ' + bel1).filter(λ be. be = SF ∨ be.l = l0), bq(t).FOWs(t, l) ⊒ (bel0 + W(l, v0) ' + bel1).FOWs(t, l), ¬ FO(l) ∈ bel1, ∀ l0. bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ (bel0 + W(l, v0) ' + bel1).filter(λ be. be = SF ∨ be.l = l0), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

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-W: ⊦ (W(l0, v) ' + bel).FOWs(t, l) = (l = l0 ⊃ W(v, t) ';[]) + bel.FOWs(t, l); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ W(l, v).l = l; (s) ⊦ W(l, v) ≠ SF; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s)

Proof step 94 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).FOWs(t, l) ⊒ bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l), bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix, seq: ⊦ x ⊒ y ↔ (∃ z. z + y = x), subst:[x, y] → [bq(t).FOWs(t, l), bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: bq(t).FOWs(t, l), bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l)

Proof step 95 from proof rely-FOFollowsW-stable (previous step) (back to tree)

⊦ x ⊒ y ↔ (∃ z. z + y = x)

A spec-lemma

Proof step 96 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ z. z + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = bq(t).FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 97 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl0 + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = bq(t).FOWs(t, l), fowl + (pq0(l) | t) + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = (pq(l) | t) + bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: rotate

Additional information

Proof step 98 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = (pq(l) | t) + fowl0 + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l), fowl0 + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = bq(t).FOWs(t, l), bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User 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 ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 99 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl0 + bel0.FOWs(t, l) + W(v0, t) ' + bel1.FOWs(t, l) = bq(t).FOWs(t, l), fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:FOWs-W-split, seq:⊦ bel.FOWs(t, l) = fowl0 + W(v, t) ' + fowl1 ↔ (∃ bel0, bel1. bel = bel0 + W(l, v) ' + bel1 ∧ fowl0 = bel0.FOWs(t, l) ∧ fowl1 = bel1.FOWs(t, l)), subst:[fowl0, t, l, v, bel, fowl1] → [fowl0 + bel0.FOWs(t, l), t, l, v0, bq(t), bel1.FOWs(t, l)], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v0, l, bq(t), fowl0 + bel0.FOWs(t, l), t, bel1.FOWs(t, l)

Proof step 100 from proof rely-FOFollowsW-stable (previous step) (back to tree)

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

Lemma FOWs-W-split

Proof step 101 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∃ bel2, bel3. bq(t) = bel2 + W(l, v0) ' + bel3 ∧ fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l) ∧ bel1.FOWs(t, l) = bel3.FOWs(t, l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 102 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l), fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, bel1.FOWs(t, l) = bel3.FOWs(t, l), bq(t) = bel2 + W(l, v0) ' + bel3, bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: bel2, bel3, v0 abort: F discard: F computed: T

Additional information

Used terms: bel2, bel3, v0

Proof step 103 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l), bel1.FOWs(t, l) = bel3.FOWs(t, l), bq(t) = bel2 + W(l, v0) ' + bel3, bq0(t) = bel0 + W(l, v0) ' + bel1, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, bq(t) = bel2 + W(l, v0) ' + bel3 → FO(l) ∈ bel3, ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 104 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l), fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, bel1.FOWs(t, l) = bel3.FOWs(t, l), bq(t) = bel2 + W(l, v0) ' + bel3, bq0(t) = bel0 + W(l, v0) ' + bel1, FO(l) ∈ bel3, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: FO(t) ∈ bel1.FOWs(t, l)

Additional information

Used simplifier rules: FO-in-FOWs: ⊦ FO(t0) ∈ bel.FOWs(t, l) ↔ t = t0 ∧ FO(l) ∈ bel; Used formulas: FO(t) ∈ bel1.FOWs(t, l)

Proof step 105 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l), bel1.FOWs(t, l) = bel3.FOWs(t, l), bq(t) = bel2 + W(l, v0) ' + bel3, bq0(t) = bel0 + W(l, v0) ' + bel1, FO(l) ∈ bel3, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ bel1.FOWs(t, l), ¬ FO(l) ∈ bel1, ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1), ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1 ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

Proof step 106 from proof rely-FOFollowsW-stable (previous step) (back to tree)

fowl0 + bel0.FOWs(t, l) = bel2.FOWs(t, l), fowl + (pq0(l) | t) = (pq(l) | t) + fowl0, bel1.FOWs(t, l) = bel3.FOWs(t, l), bq(t) = bel2 + W(l, v0) ' + bel3, bq0(t) = bel0 + W(l, v0) ' + bel1, FO(l) ∈ bel3, bq(t).filter(λ be. be = SF ∨ be.l = l0) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l0) + (l = l0 ⊃ W(l0, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l0), ¬ FO(t) ∈ bel3.FOWs(t, l), ¬ FO(l) ∈ bel1, ∀ bel0, bel1, v. bq(t) = bel0 + W(l, v) ' + bel1 → FO(l) ∈ bel1, ∀ fowl0, fowl1, v. pq(l) = fowl0 + W(v, t) ' + fowl1 → FO(t) ∈ fowl1 ∨ FO(l) ∈ bq(t), ∀ l1. bq(t).filter(λ be. be = SF ∨ be.l = l1) ⊒ bel0.filter(λ be. be = SF ∨ be.l = l1) + (l = l1 ⊃ W(l1, v0) ';[]) + bel1.filter(λ be. be = SF ∨ be.l = l1) ⊦

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: FO-in-FOWs: ⊦ FO(t0) ∈ bel.FOWs(t, l) ↔ t = t0 ∧ FO(l) ∈ bel;

Proof Tree for rely-FOFollowsW-stable (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