[Imprint][Data Privacy Statement][E-Mail]



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Proof step 1 from proof pendingsok-del-pos (conclusion) (back to tree)

⊦ pendingOk(p) → pendingOk(p(l := p(l) -p n))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof pendingsok-del-pos (previous step) (back to tree)

pendingOk(p), ¬ pendingOk(p(l := p(l) -p n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: pendingOk: ⊦ pendingOk(p) ↔ (∀ l, t. #(p(l) |W | t) ≤ 1); filterwrites-def: ⊦ fowl |W = fowl.filter(λ fow. fow.W?); filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t); ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ # x.filter(p) = x.count(p); (s)

Proof step 3 from proof pendingsok-del-pos (previous step) (back to tree)

(l = l0 ⊃ p(l0) -p n;p(l0)).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (l = l0 ⊃ p(l0) -p n;p(l0)).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, ∀ l0, t. p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 4 from proof pendingsok-del-pos (previous step) (back to tree)

l = l0, (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (l = l0 ⊃ p(l0) -p n;p(l0)).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, ∀ l0, t. p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof pendingsok-del-pos (previous step) (back to tree)

(p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, ∀ l0, t. p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: l0, t

Proof step 6 from proof pendingsok-del-pos (previous step) (back to tree)

(p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ fowl. fowl = p(l0)

Additional information

Used formulas: ∃ fowl. fowl = p(l0)

Proof step 7 from proof pendingsok-del-pos (previous step) (back to tree)

(p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∃ fowl. fowl = p(l0) ⊦

User applied rule exists left

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 8 from proof pendingsok-del-pos (previous step) (back to tree)

fowl = p(l0), (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (p(l0) -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: rotate, discard

Additional information

Proof step 9 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl -p n).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: n

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 10 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl -p 0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl -p 0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: fowl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ (a ' + x) -p 0 = x; (s) ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ (x + y).count(p) = x.count(p) + y.count(p); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ [] -p n = []; (s) ⊦ [].filter(p) = []; (s) ⊦ [].count(p) = 0; (s)

Proof step 11 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl -p n0 +1).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl -p n0 +1).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl, t. ¬ ( (fowl -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0 ∧ (fowl -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1 ∧ fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 12 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl -p (n0 + 1)).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl -p (n0 + 1)).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ [] -p n = []; (s) ⊦ [].filter(p) = []; (s) ⊦ [].count(p) = 0; (s)

Proof step 13 from proof pendingsok-del-pos (previous step) (back to tree)

fowl = fow0 + fowl1, (fowl -p (n0 + 1)).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl -p (n0 + 1)).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fowl.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.count(p) = (p(a) ⊃ 1;0); (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) ⊦ [].count(p) = 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + y).count(p) = x.count(p) + y.count(p); (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a + x = a ' + x; (s) ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws) ⊦ (a ' + x) -p (n + 1) = a + (x -p n); (s)

Proof step 14 from proof pendingsok-del-pos (previous step) (back to tree)

(fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) = 0 ∧ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) = 1, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 15 from proof pendingsok-del-pos (previous step) (back to tree)

(fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) ≠ 0, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s) ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws)

Proof step 16 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = t) = 0, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) = 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: fow0.W?

Additional information

Used simplifier rules: ⊦ [].count(p) = 0; (s) Used formulas: fow0.W?

Proof step 17 from proof pendingsok-del-pos (previous step) (back to tree)

(fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) = 1, fowl1.filter(.W?).count(λ fow. fow.tid = t) = 0, fow0.W?, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.count(p) = (p(a) ⊃ 1;0); (s)

Proof step 18 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) ≠ 0, fow0.W?, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: fowl1, fow0.tid abort: F discard: T computed: T

Additional information

Used terms: fowl1, fow0.tid

Proof step 19 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) ≠ 0, fow0.W?, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∨ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1 ∨ ¬ fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) ≤ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 20 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, fow0.W? ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 3

Additional information

Proof step 21 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1 ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: n0

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 22 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p 0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: fowl1

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ (a ' + x) -p 0 = x; (s) ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ (x + y).count(p) = x.count(p) + y.count(p); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ [].count(p) = 0; (s) ⊦ [] -p n = []; (s) ⊦ [].filter(p) = []; (s)

Proof step 23 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p n +1).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, ∀ fowl1, fow0. ¬(fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ (fowl1 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 24 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl1 -p (n + 1)).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fowl1

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ [] -p n = []; (s) ⊦ [].filter(p) = []; (s) ⊦ [].count(p) = 0; (s)

Proof step 25 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p (n + 1)).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl1.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, fowl1 = fow2 + fowl0, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + y).count(p) = x.count(p) + y.count(p); (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a + x = a ' + x; (s) ⊦ (a ' + x) -p (n + 1) = a + (x -p n); (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)

Proof step 26 from proof pendingsok-del-pos (previous step) (back to tree)

fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fow2.W? ⊃ fow2 ';[]).count(λ fow. fow.tid = fow0.tid) = 0, (fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 27 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, fow2 '.count(λ fow. fow.tid = fow0.tid) = 0, fow2.W?, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.count(p) = (p(a) ⊃ 1;0); (s)

Proof step 28 from proof pendingsok-del-pos (previous step) (back to tree)

fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fow2.tid ≠ fow0.tid, fow2.W?, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: FOW, name:elim-W, seq: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)), subst:[fow, val, tid] → [fow2, v, t]

Additional information

Used simplifier rules: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)); (elim) Used terms: fow2, v, t

Proof step 29 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, fow2 = W(v, t), t ≠ fow0.tid, fow2.W?, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 30 from proof pendingsok-del-pos (previous step) (back to tree)

fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, t ≠ fow0.tid, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: fowl0, fow0 abort: F discard: T computed: F

Additional information

Used terms: fowl0, fow0

Proof step 31 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, [].count(λ fow. fow.tid = fow0.tid) = 0, ¬ fow2.W?, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ [].count(p) = 0; (s)

Proof step 32 from proof pendingsok-del-pos (previous step) (back to tree)

fowl0.filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0, (fowl0 -p n).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 1, ¬ fow2.W?, ∀ fowl, fow. fowl.filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 0 ∨ (fowl -p n).filter(.W?).count(λ fow1. fow1.tid = fow.tid) ≠ 1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: fowl0, fow0 abort: F discard: T computed: F

Additional information

Used terms: fowl0, fow0

Proof step 33 from proof pendingsok-del-pos (previous step) (back to tree)

(fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 34 from proof pendingsok-del-pos (previous step) (back to tree)

fow0 '.count(λ fow. fow.tid = t) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fow0.W?, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.count(p) = (p(a) ⊃ 1;0); (s)

Proof step 35 from proof pendingsok-del-pos (previous step) (back to tree)

(fow0.tid = t ⊃ 1;0) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fow0.W?, (fow0.tid = t ⊃ 1;0) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: FOW, name:elim-W, seq: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)), subst:[fow, val, tid] → [fow0, v, t1]

Additional information

Used simplifier rules: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)); (elim) Used terms: fow0, v, t1

Proof step 36 from proof pendingsok-del-pos (previous step) (back to tree)

fow0 = W(v, t1), (t1 = t ⊃ 1;0) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, fow0.W?, (t1 = t ⊃ 1;0) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t1) = 0 ∧ t1 = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 37 from proof pendingsok-del-pos (previous step) (back to tree)

(t1 = t ⊃ 1;0) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (t1 = t ⊃ 1;0) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) = 0 ∧ t1 = t

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 38 from proof pendingsok-del-pos (previous step) (back to tree)

t1 = t, 1 + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (t1 = t ⊃ 1;0) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) = 0 ∧ t1 = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws) ⊦ m + 1 ≠ 0; (s)

Proof step 39 from proof pendingsok-del-pos (previous step) (back to tree)

fowl1.filter(.W?).count(λ fow. fow.tid = t) = 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: fowl1, t

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) Used terms: fowl1, t

Proof step 40 from proof pendingsok-del-pos (previous step) (back to tree)

0 + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, t1 ≠ t, (t1 = t ⊃ 1;0) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) = 0 ∧ t1 = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 41 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, t1 ≠ t, fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: fowl1, t

Additional information

Used terms: fowl1, t

Proof step 42 from proof pendingsok-del-pos (previous step) (back to tree)

[].count(λ fow. fow.tid = t) + (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (fow0.W? ⊃ fow0 ';[]).count(λ fow. fow.tid = t) + fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ¬ fow0.W?, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦ (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = fow0.tid) = 0 ∧ fow0.tid = t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 43 from proof pendingsok-del-pos (previous step) (back to tree)

(fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, (fowl1 -p n0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, fowl1.filter(.W?).count(λ fow. fow.tid = t) ≤ 1, ¬ fow0.W?, ∀ fowl0, t0. (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 0 ∨ (fowl0 -p n0).filter(.W?).count(λ fow. fow.tid = t0) = 1 ∨ ¬ fowl0.filter(.W?).count(λ fow. fow.tid = t0) ≤ 1 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: fowl1, t

Additional information

Used terms: fowl1, t

Proof step 44 from proof pendingsok-del-pos (previous step) (back to tree)

p(l0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, (l = l0 ⊃ p(l0) -p n;p(l0)).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, l ≠ l0, ∀ l0, t. p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 45 from proof pendingsok-del-pos (previous step) (back to tree)

p(l0).filter(.W?).count(λ fow. fow.tid = t) ≠ 0, p(l0).filter(.W?).count(λ fow. fow.tid = t) ≠ 1, l ≠ l0, ∀ l0, t. p(l0).filter(.W?).count(λ fow. fow.tid = t) ≤ 1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t abort: F discard: T computed: F

Additional information

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

Proof Tree for pendingsok-del-pos (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