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Proof step 1 from proof pendingOk-prop-FO (conclusion) (back to tree)

⊦ pendingOk(bq, pq) ∧ isPropFO(n, l, bq(t)) → pendingOk(bq(t := bq(t) -p n), pq(l := pq(l) + FO(t) '))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pendingOk-prop-FO (previous step) (back to tree)

isPropFO(n, l, bq(t)), pendingOk(bq, pq), ¬ pendingOk(bq(t := bq(t) -p n), pq(l := pq(l) + FO(t) ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: pendingOk: ⊦ pendingOk(bq, p) ↔ (∀ l, t. #(p(l) |W | t) + #(bq(t) |WL l) ≤ 1); 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 + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 3 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) ≠ 0, isPropFO(n, l, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) = 1, (t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: t = t0

Additional information

Used formulas: t = t0

Proof step 4 from proof pendingOk-prop-FO (previous step) (back to tree)

t = t0, #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) ≠ 0, isPropFO(n, l, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1, (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: isPropFO-restrict-delpos: ⊦ isPropFO(n, l, bel) → bel -p n |WL l0 = bel |WL l0;

Proof step 5 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, isPropFO(n, l, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) = 1

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: l = l0

Additional information

Used formulas: l = l0

Proof step 6 from proof pendingOk-prop-FO (previous step) (back to tree)

l = l0, #((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, isPropFO(n, l, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) = 1, (l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ FO(t) ' |W = []; (s) ⊦ (fowl + fowl0)|W = fowl |W + fowl0 |W; (s)

Proof step 7 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≠ 0, isPropFO(n, l0, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: l0, t0 abort: F discard: F computed: T

Additional information

Used terms: l0, t0

Proof step 8 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≠ 0, isPropFO(n, l0, bq(t0)), #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1, pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s)

Proof step 9 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) = 1, isPropFO(n, l0, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1, pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-data, name:le-one, seq: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1, subst:[x] → [bq(t0) |WL l0]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 0); (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (ws) ⊦ # x = 1 ↔ x = x.head + []; (as) ⊦ a + x = a ' + x; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≠ n → (m ≤ n ↔ m < n); (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ a ' ≠ []; (s) Used terms: bq(t0) |WL l0

Proof step 10 from proof pendingOk-prop-FO (previous step) (back to tree)

⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1

A spec-lemma

Proof step 11 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, l ≠ l0, isPropFO(n, l, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0) = 1, (l = l0 ⊃ pq(l0) + FO(t0) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≠ 0, l ≠ l0, isPropFO(n, l, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used terms: l0, t0

Proof step 13 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≠ 0, l ≠ l0, isPropFO(n, l, bq(t0)), #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1, pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s)

Proof step 14 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) = 1, l ≠ l0, isPropFO(n, l, bq(t0)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1, pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-data, name:le-one, seq: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1, subst:[x] → [bq(t0) |WL l0]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 0); (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (ws) ⊦ # x = 1 ↔ x = x.head + []; (as) ⊦ a + x = a ' + x; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≠ n → (m ≤ n ↔ m < n); (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ a ' ≠ []; (s) Used terms: bq(t0) |WL l0

Proof step 15 from proof pendingOk-prop-FO (previous step) (back to tree)

⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1

A spec-lemma

Proof step 16 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) ≠ 0, t ≠ t0, isPropFO(n, l, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1, (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #((t = t0 ⊃ bq(t0) -p n;bq(t0)) |WL l0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, t ≠ t0, isPropFO(n, l, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used terms: l0, t0

Proof step 18 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, t ≠ t0, isPropFO(n, l, bq(t)), #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1, (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: l = l0

Additional information

Used formulas: l = l0

Proof step 19 from proof pendingOk-prop-FO (previous step) (back to tree)

l = l0, #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, t ≠ t0, isPropFO(n, l, bq(t)), #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ FO(t) ' |W = []; (s) ⊦ (fowl + fowl0)|W = fowl |W + fowl0 |W; (s) ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s)

Proof step 20 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) = 1, t ≠ t0, isPropFO(n, l0, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-data, name:le-one, seq: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1, subst:[x] → [bq(t0) |WL l0]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 0); (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (ws) ⊦ # x = 1 ↔ x = x.head + []; (as) ⊦ a + x = a ' + x; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≠ n → (m ≤ n ↔ m < n); (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ a ' ≠ []; (s) Used terms: bq(t0) |WL l0

Proof step 21 from proof pendingOk-prop-FO (previous step) (back to tree)

⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1

A spec-lemma

Proof step 22 from proof pendingOk-prop-FO (previous step) (back to tree)

#((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) + #(bq(t0) |WL l0) ≠ 0, l ≠ l0, t ≠ t0, isPropFO(n, l, bq(t)), #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ (l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #((l = l0 ⊃ pq(l0) + FO(t) ';pq(l0)) |W | t0) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s)

Proof step 23 from proof pendingOk-prop-FO (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) = 1, l ≠ l0, t ≠ t0, isPropFO(n, l, bq(t)), ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦ pq(l0) |W | t0 = [] ∧ #(bq(t0) |WL l0) = 1, bq(t0) |WL l0 = [] ∧ #(pq(l0) |W | t0) = 1

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-data, name:le-one, seq: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1, subst:[x] → [bq(t0) |WL l0]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 0); (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (ws) ⊦ # x = 1 ↔ x = x.head + []; (as) ⊦ a + x = a ' + x; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≠ n → (m ≤ n ↔ m < n); (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ a ' ≠ []; (s) Used terms: bq(t0) |WL l0

Proof step 24 from proof pendingOk-prop-FO (previous step) (back to tree)

⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1

A spec-lemma

Proof Tree for pendingOk-prop-FO (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