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Proof step 1 from proof pendingOk-add-pending (conclusion) (back to tree)

⊦ noPending(t, bq, pq) ∧ pendingOk(bq, pq) → pendingOk(bq(t := bq(t) + W(l, v) '), pq)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pendingOk-add-pending (previous step) (back to tree)

noPending(t, bq, pq), pendingOk(bq, pq), ¬ pendingOk(bq(t := bq(t) + W(l, v) '), pq) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: noPending: ⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = []); 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-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 4 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 5 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 6 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-append: ⊦ bel0 + bel1 |WL l = (bel0 |WL l) + (bel1 |WL l); restrict-l-empty: ⊦ [] |WL l = []; a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ #(x + y) = # x + # y; (s)

Proof step 7 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], W(l, v) ' |WL l0 ≠ [], ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used terms: l0, t0

Proof step 8 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], W(l, v) ' |WL l0 ≠ [], #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-empty: ⊦ [] |WL l = []; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ a + x = a ' + x; (s) ⊦ # x = 1 ↔ x = x.head + []; (as) le-one: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1; (forward) from specification list-data c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s)

Proof step 9 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], pq(l0) |W | t0 = (pq(l0) |W | t0).head ', #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], W(l, v) ' |WL l0 ≠ [], #(pq(l0) |W | t0) ≤ 1, ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [pq(l0) |W | t0, fow, fowl]; ['a] → [FOW]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: pq(l0) |W | t0, fow, fowl

Proof step 10 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], pq(l0) |W | t0 = fow ', pq(l0) |W | t0 = fow + fowl, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], W(l, v) ' |WL l0 ≠ [], #(pq(l0) |W | t0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) le-one: ⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1; (forward) from specification list-data in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 11 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], pq(l0) |W | t0 = fow ', W(l, v) ' |WL l0 ≠ [], ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:restrict-l-W-other, seq: ⊦ l0 ≠ l → W(l0, v) ' |WL l = [], subst:[l0, v, l] → [l, v, l0], <all paths>

Additional information

Used terms: l, v, l0

Proof step 12 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ l0 ≠ l → W(l0, v) ' |WL l = []

Lemma restrict-l-W-other

Proof step 13 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], l = l0, pq(l0) |W | t0 = fow ', W(l, v) ' |WL l0 ≠ [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-W-same: ⊦ W(l, v) ' |WL l = W(l, v) '; ⊦ a ' ≠ []; (s)

Proof step 14 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], pq(l0) |W | t0 = fow ', ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

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 15 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], pq(l0) |W | t0 = fow ', pq(l0) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-W-t-swap, seq: ⊦ fowl |W | t = (fowl | t)|W, subst:[fowl, t] → [pq(l0), t0], <all paths>

Additional information

Used simplifier rules: ⊦ [] |W = []; (s) ⊦ a ' ≠ []; (s) Used terms: pq(l0), t0

Proof step 16 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ fowl |W | t = (fowl | t)|W

A spec-lemma

Proof step 17 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used terms: l0, t0

Proof step 19 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t) = [], #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≠ 0, t ≠ t0, pq(l0) |W | t0 ≠ [], bq(t0) |WL l0 ≠ [], #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l. pq(l) | t = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 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 20 from proof pendingOk-add-pending (previous step) (back to tree)

#(pq(l0) |W | t0) + #(bq(t0) |WL l0) = 1, bq(t) = [], t ≠ t0, pq(l0) |W | t0 ≠ [], bq(t0) |WL l0 ≠ [], ∀ l. pq(l) | t = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 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) Used terms: bq(t0) |WL l0

Proof step 21 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ # x ≤ 1 ∧ x ≠ [] → # x = 1

A spec-lemma

Proof step 22 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 23 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t) = [], t = t0, #(pq(l0) |W | t0) ≠ 1, #(pq(l0) |W | t0) + #(bq(t0) + W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], ∀ l. pq(l) | t = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-append: ⊦ bel0 + bel1 |WL l = (bel0 |WL l) + (bel1 |WL l); restrict-l-empty: ⊦ [] |WL l = []; a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 24 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(pq(l0) |W | t0) ≠ 1, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, pq(l0) |W | t0 ≠ [], ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used simplifier rules: restrict-l-empty: ⊦ [] |WL l = []; ⊦ # [] = 0; (s) ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m ≠ n → (m ≤ n ↔ m < n); (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ # x = 0 ↔ x = []; (s) Used terms: l0, t0

Proof step 25 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (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 Used terms: l0, t0

Proof step 26 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 27 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 28 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-append: ⊦ bel0 + bel1 |WL l = (bel0 |WL l) + (bel1 |WL l); restrict-l-empty: ⊦ [] |WL l = []; a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws)

Proof step 29 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(W(l, v) ' |WL l0) ≠ 1, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, W(l, v) ' |WL l0 ≠ [], ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used terms: l0, t0

Proof step 30 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(W(l, v) ' |WL l0) ≠ 1, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, W(l, v) ' |WL l0 ≠ [], #(pq(l0) |W | t0) + #(bq(t0) |WL l0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-empty: ⊦ [] |WL l = []; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s)

Proof step 31 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(W(l, v) ' |WL l0) ≠ 1, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, W(l, v) ' |WL l0 ≠ [], #(pq(l0) |W | t0) ≤ 1, ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1, ∀ l. pq(l) | t0 = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:restrict-l-W-other, seq: ⊦ l0 ≠ l → W(l0, v) ' |WL l = [], subst:[l0, v, l] → [l, v, l0], <all paths>

Additional information

Used simplifier rules: restrict-l-W-same: ⊦ W(l, v) ' |WL l = W(l, v) '; ⊦ # a ' = 0 +1; (s) Used terms: l, v, l0

Proof step 32 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ l0 ≠ l → W(l0, v) ' |WL l = []

Lemma restrict-l-W-other

Proof step 33 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (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 Used terms: l0, t0

Proof step 35 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 36 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-l-append: ⊦ bel0 + bel1 |WL l = (bel0 |WL l) + (bel1 |WL l); restrict-l-empty: ⊦ [] |WL l = []; ⊦ # [] = 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ n + 0 = n; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws)

Proof step 37 from proof pendingOk-add-pending (previous step) (back to tree)

bq(t0) = [], #(pq(l0) |W | t0) ≠ 1, #(W(l, v) ' |WL l0) ≠ 1, #(pq(l0) |W | t0) + #(W(l, v) ' |WL l0) ≠ 0, ∀ l. pq(l) | t0 = [], ∀ l, t. #(pq(l) |W | t) + #(bq(t) |WL l) ≤ 1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:restrict-l-W-other, seq: ⊦ l0 ≠ l → W(l0, v) ' |WL l = [], subst:[l0, v, l] → [l, v, l0], <all paths>

Additional information

Used simplifier rules: restrict-l-W-same: ⊦ W(l, v) ' |WL l = W(l, v) '; ⊦ # a ' = 0 +1; (s) Used terms: l, v, l0

Proof step 38 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ l0 ≠ l → W(l0, v) ' |WL l = []

Lemma restrict-l-W-other

Proof step 39 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s)

Proof step 40 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: l0 abort: F discard: F computed: F

Additional information

Used terms: l0

Proof step 41 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:filter-W-t-swap, seq: ⊦ fowl |W | t = (fowl | t)|W, subst:[fowl, t] → [pq(l0), t0], <all paths>

Additional information

Used simplifier rules: ⊦ [] |W = []; (s) Used terms: pq(l0), t0

Proof step 42 from proof pendingOk-add-pending (previous step) (back to tree)

⊦ fowl |W | t = (fowl | t)|W

A spec-lemma

Proof step 43 from proof pendingOk-add-pending (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 44 from proof pendingOk-add-pending (previous step) (back to tree)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: l0, t0 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s) ⊦ m ≠ 1 → (m + n = 1 ↔ m = 0 ∧ n = 1); (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 Used terms: l0, t0

Proof Tree for pendingOk-add-pending (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