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

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

noPending(t, bq, pq), rely(t, pq, bq, pq0, bq0), bq(t) ⊒w bq0(t), ¬ noPending(t, bq0, pq0) ⊦

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); noPending: ⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = []); FOWs-base: ⊦ [].FOWs(t, l) = []; ⊦ [] ⊒ x ↔ x = []; (s) ⊦ [].filter(p) = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s)

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

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

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: l

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

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: l abort: F discard: F computed: T

Additional information

Used terms: l

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

bq(t) = [], pq(l) | t = [], pq(l) | t ⊒ (pq0(l) | t) + bq0(t).FOWs(t, l), ∀ l. pq(l) | t = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l0. bq0(t).filter(λ be. be = SF ∨ be.l = l0) = [] ⊦ pq0(l) | t = [] ∧ bq0(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) ⊦ [] ⊒ x ↔ x = []; (s)

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

bq0(t).FOWs(t, l) = [], bq(t) = [], pq(l) | t = [], pq0(l) | t = [], bq0(t) ≠ [], ∀ l. pq(l) | t = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ 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:elim-first, seq: ⊦ x = [] ↔ ¬(∃ a, y. x = a ' + y), subst:[x] → [bq0(t)]; ['a] → [Bevent], <all paths>

Additional information

Used terms: bq0(t)

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

⊦ x = [] ↔ ¬(∃ a, y. x = a ' + y)

A spec-lemma

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

bq0(t).FOWs(t, l) = [], bq(t) = [], pq0(l) | t = [], pq(l) | t = [], ∀ l0. bq0(t).filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ l. pq(l) | t = [], ∃ a, y. bq0(t) = a ' + y ⊦

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

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

bq0(t).FOWs(t, l) = [], bq(t) = [], bq0(t) = be ' + bel, pq(l) | t = [], pq0(l) | t = [], be ∈ bq0(t), ∀ l. pq(l) | t = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + bq0(t).FOWs(t, l0), ∀ 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 10 from proof rely-noPending-stable-strong (previous step) (back to tree)

(be ' + bel).FOWs(t, l) = [], bq(t) = [], bq0(t) = be ' + bel, pq0(l) | t = [], pq(l) | t = [], be ∈ be ' + bel, ∀ l0. (be ' + bel).filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (be ' + bel).FOWs(t, l0), ∀ l. pq(l) | t = [] ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: be

Additional information

Used simplifier rules: FOWs-SF: ⊦ (SF ' + bel).FOWs(t, l) = bel.FOWs(t, l); Bevent freely generated by SF, W, FO;; ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 11 from proof rely-noPending-stable-strong (previous step) (back to tree)

(be ' + bel).FOWs(t, l) = [], be = W(l1, v), bq(t) = [], bq0(t) = be ' + bel, pq(l) | t = [], pq0(l) | t = [], be ∈ be ' + bel, ∀ l. pq(l) | t = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (be ' + bel).FOWs(t, l0), ∀ l0. (be ' + bel).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-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 in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ W(l, v).l = l; (s) ⊦ W(l, v) ≠ SF; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s)

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = W(l1, v) ' + bel, pq0(l) | t = [], pq(l) | t = [], l ≠ l1, W(l1, v) ∈ bq0(t), ∀ l. pq(l) | t = [], ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ W(v, t) ';[]) + bel.FOWs(t, l0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: l1 abort: F discard: F computed: F

Additional information

Used terms: l1

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = W(l1, v) ' + bel, pq(l) | t = [], pq0(l) | t = [], l ≠ l1, W(l1, v) ∈ bq0(t), pq(l1) | t ⊒ (pq0(l1) | t) + (l1 = l1 ⊃ W(v, t) ';[]) + bel.FOWs(t, l1), ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ W(v, t) ';[]) + bel.FOWs(t, l0), ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l. pq(l) | 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

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = W(l1, v) ' + bel, pq0(l) | t = [], pq(l) | t = [], l ≠ l1, W(l1, v) ∈ bq0(t), pq(l1) | t ⊒ (pq0(l1) | t) + W(v, t) ' + bel.FOWs(t, l1), ∀ l. pq(l) | t = [], ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ W(v, t) ';[]) + bel.FOWs(t, l0) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: l1 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ [] ⊒ x ↔ x = []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: l1

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

(be ' + bel).FOWs(t, l) = [], be = FO(l1), bq(t) = [], bq0(t) = be ' + bel, pq(l) | t = [], pq0(l) | t = [], be ∈ be ' + bel, ∀ l. pq(l) | t = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (be ' + bel).FOWs(t, l0), ∀ l0. (be ' + bel).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 in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ FO(l).l = l; (s) ⊦ FO(l) ≠ SF; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s)

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = FO(l1) ' + bel, pq0(l) | t = [], pq(l) | t = [], l ≠ l1, FO(l1) ∈ bq0(t), ∀ l. pq(l) | t = [], ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ FO(t) ';[]) + bel.FOWs(t, l0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: l1 abort: F discard: F computed: F

Additional information

Used terms: l1

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = FO(l1) ' + bel, pq(l) | t = [], pq0(l) | t = [], l ≠ l1, FO(l1) ∈ bq0(t), pq(l1) | t ⊒ (pq0(l1) | t) + (l1 = l1 ⊃ FO(t) ';[]) + bel.FOWs(t, l1), ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ FO(t) ';[]) + bel.FOWs(t, l0), ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l. pq(l) | 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

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

bel.FOWs(t, l) = [], bq(t) = [], bq0(t) = FO(l1) ' + bel, pq0(l) | t = [], pq(l) | t = [], l ≠ l1, FO(l1) ∈ bq0(t), pq(l1) | t ⊒ (pq0(l1) | t) + FO(t) ' + bel.FOWs(t, l1), ∀ l. pq(l) | t = [], ∀ l0. l1 ≠ l0 ∧ bel.filter(λ be. be = SF ∨ be.l = l0) = [], ∀ l0. pq(l0) | t ⊒ (pq0(l0) | t) + (l0 = l1 ⊃ FO(t) ';[]) + bel.FOWs(t, l0) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: l1 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ [] ⊒ x ↔ x = []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: l1

Proof Tree for rely-noPending-stable-strong (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