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



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Proof step 1 from proof rem-stays-reachable-aux1 (conclusion) (back to tree)

r0 ∈ h, (h(r0)).nxt ≠ null, (h(r0)).nxt ∈ h, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h) ↔ (∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ∈ h, r0 ∈ h ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h) ↔ (∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, r0 ∈ h, (h(r0)).nxt ∈ h ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h) ↔ (∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 4 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ reachable((h((h(r0)).nxt)).nxt, r0, h) ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ reachable((h((h(r0)).nxt)).nxt, r0, h) ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-nodups-def, seq:⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x)), subst:[h, r0, r] → [h, (h(r0)).nxt, (h((h(r0)).nxt)).nxt], <all paths>

Additional information

Used terms: h, (h(r0)).nxt, (h((h(r0)).nxt)).nxt

Proof step 6 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

Lemma reachable-nodups-def

Proof step 7 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ∈ h, r0 ∈ h, ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ∃ x. path((h((h(r0)).nxt)).nxt ' + x, h) ∧ ((h((h(r0)).nxt)).nxt ' + x).last = (h(r0)).nxt ∧ ¬ dups((h((h(r0)).nxt)).nxt ' + x) ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

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[reflist] ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 8 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

((h((h(r0)).nxt)).nxt ' + x).last = (h(r0)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x, h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h((h(r0)).nxt)).nxt ∈ x ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name: lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[y, x] → [x, (h((h(r0)).nxt)).nxt '], <all paths>

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used terms: x, (h((h(r0)).nxt)).nxt '

Proof step 9 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 10 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x.last = (h(r0)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, x ≠ [], path((h((h(r0)).nxt)).nxt ' + x, h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h((h(r0)).nxt)).nxt ∈ x ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x.last = (h(r0)).nxt, x.last ≠ null, x.last ≠ (h(x.last)).nxt, x ≠ [], path((h(x.last)).nxt ' + x, h), r0 ∈ h, x.last ∈ h, ¬ dups(x), ¬ reachable((h(x.last)).nxt, r0, h), ¬ (h(x.last)).nxt ∈ x ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [x, r, x0]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, r, x0

Proof step 12 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

r = (h(r0)).nxt, x = x0 + r ', r ≠ null, r ≠ (h(r)).nxt, x ≠ [], path((h(r)).nxt ' + x, h), r ∈ h, r0 ∈ h, ¬ dups(x), ¬ reachable((h(r)).nxt, r0, h), ¬ (h(r)).nxt ∈ x ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

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[reflist] ⊦ a ∈ b ' ↔ a = b; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 13 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: (h((h(r0)).nxt)).nxt ' + x0 abort: F discard: F computed: F

Additional information

Used terms: (h((h(r0)).nxt)).nxt ' + x0

Proof step 14 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x, path((h(r0)).nxt ' + ((h((h(r0)).nxt)).nxt ' + x0) + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ (h((h(r0)).nxt)).nxt ' + x0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s)

Proof step 15 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x, r0 ≠ (h((h(r0)).nxt)).nxt ∧ ¬ r0 ∈ x0

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 2

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; path-first-in-H: ⊦ path(r ' + x, h) → (r ∈ h ↔ true); null-not-start-of-path: ⊦ ¬ path(null ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 16 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ x0, (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, r0 ∈ x0, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list[reflist], name: in-def, seq: ⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z)), subst:[x, a] → [x0, r0], <all paths>

Additional information

Used terms: x0, r0

Proof step 18 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

A spec-lemma

Proof step 19 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x0), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ (h(r0)).nxt ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x0, ∃ y, z. x0 = y + (r0 + z) ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

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[reflist] ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a + x = a ' + x; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws)

Proof step 20 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, path((h((h(r0)).nxt)).nxt ' + x + (r0 ' + x1) + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x + x1), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ (h(r0)).nxt ∈ x, ¬ (h(r0)).nxt ∈ x1, ¬ (h((h(r0)).nxt)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x1 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: path((r0 ' + x1) + (h(r0)).nxt ', h)

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used formulas: path((r0 ' + x1) + (h(r0)).nxt ', h)

Proof step 21 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, path((h((h(r0)).nxt)).nxt ' + x + (r0 ' + x1) + (h(r0)).nxt ', h), path((r0 ' + x1) + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x + x1), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ (h(r0)).nxt ∈ x, ¬ (h(r0)).nxt ∈ x1, ¬ (h((h(r0)).nxt)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x1 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x1

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); Reflist freely generated by [], +; ⊦ a + x = a ' + x; (s) ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 22 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x1 = [], (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, path((r0 ' + x1) + (h(r0)).nxt ', h), path((h((h(r0)).nxt)).nxt ' + x + (r0 ' + x1) + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x + x1), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ (h(r0)).nxt ∈ x, ¬ (h(r0)).nxt ∈ x1, ¬ (h((h(r0)).nxt)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x1 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ (h(r)).nxt = r0; ⊦ ¬ a ∈ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + [] = x; (s) ⊦ [] + x = x; (s)

Proof step 23 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, r0 ≠ null, path((h((h(r0)).nxt)).nxt ' + x + r0 ' + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ reachable((h((h(r0)).nxt)).nxt, r0, h), ¬ r0 ∈ x, ¬ (h(r0)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[r, r0, h] → [(h((h(r0)).nxt)).nxt, r0, h], <all paths>

Additional information

Used terms: (h((h(r0)).nxt)).nxt, r0, h

Proof step 24 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 25 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, r0 ≠ null, path((h((h(r0)).nxt)).nxt ' + x + r0 ' + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x), ¬ r0 ∈ x, ¬ (h(r0)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x, ∀ x. ¬(path((h((h(r0)).nxt)).nxt ' + x, h) ∧ ((h((h(r0)).nxt)).nxt ' + x).last = r0) ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 26 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r0, (h((h(r0)).nxt)).nxt ≠ r0, r0 ≠ null, path((h((h(r0)).nxt)).nxt ' + x + r0 ' + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ r0 ∈ x, ¬ (h(r0)).nxt ∈ x, ¬ (h((h(r0)).nxt)).nxt ∈ x, ∀ x0. ¬ path((h((h(r0)).nxt)).nxt ' + x0, h) ∨ ((h((h(r0)).nxt)).nxt ' + x0).last ≠ r0 ⊦ ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x

User applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: x + r0 ' abort: F discard: T computed: F

Additional information

Used simplifier rules: path-butlast: ⊦ path(r ' + x + r0 ', h) → (path(r ' + x, h) ↔ true); ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x + r0 '

Proof step 27 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ∈ h, r0 ∈ h, ∃ x. path((h(r0)).nxt ' + x + (h(r0)).nxt ', h) ∧ ¬ r0 ∈ x ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h(r0)).nxt ' + x + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ r0 ∈ x ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ (h(r)).nxt = r0; Reflist freely generated by [], +; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 29 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x = r + x0, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, path((h(r0)).nxt ' + x + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s)

Proof step 30 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0 ⊦ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ∧ ¬ reachable((h((h(r0)).nxt)).nxt, r0, h)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 31 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h), ¬ r0 ∈ x0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[h, r0, r] → [h, (h(r0)).nxt, (h((h(r0)).nxt)).nxt], <all paths>

Additional information

Used terms: h, (h(r0)).nxt, (h((h(r0)).nxt)).nxt

Proof step 32 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 33 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0, ∀ x. ¬(path((h((h(r0)).nxt)).nxt ' + x, h) ∧ ((h((h(r0)).nxt)).nxt ' + x).last = (h(r0)).nxt) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ r0 ∈ x0, ∀ x. ¬ path((h((h(r0)).nxt)).nxt ' + x, h) ∨ ((h((h(r0)).nxt)).nxt ' + x).last ≠ (h(r0)).nxt ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: x0 + (h(r0)).nxt '

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x0 + (h(r0)).nxt '

Proof step 35 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), reachable((h((h(r0)).nxt)).nxt, r0, h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ r0 ∈ x0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 36 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), reachable((h((h(r0)).nxt)).nxt, r0, h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-nodups-def, seq:⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x)), subst:[h, r0, r] → [h, r0, (h((h(r0)).nxt)).nxt], <all paths>

Additional information

Used terms: h, r0, (h((h(r0)).nxt)).nxt

Proof step 37 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

Lemma reachable-nodups-def

Proof step 38 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ r0 ∈ x0, ∃ x. path((h((h(r0)).nxt)).nxt ' + x, h) ∧ ((h((h(r0)).nxt)).nxt ' + x).last = r0 ∧ ¬ dups((h((h(r0)).nxt)).nxt ' + x) ⊦

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[reflist] ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 39 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

((h((h(r0)).nxt)).nxt ' + x).last = r0, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, path((h((h(r0)).nxt)).nxt ' + x, h), path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ r0 ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name: lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[y, x] → [x, (h((h(r0)).nxt)).nxt '], <all paths>

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used terms: x, (h((h(r0)).nxt)).nxt '

Proof step 40 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 41 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x.last = r0, (h(r0)).nxt ≠ (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r0 ≠ (h((h(r0)).nxt)).nxt, x ≠ [], path((h((h(r0)).nxt)).nxt ' + x0 + (h(r0)).nxt ', h), path((h((h(r0)).nxt)).nxt ' + x, h), r0 ∈ h, (h(r0)).nxt ∈ h, ¬ dups(x), ¬ r0 ∈ x0, ¬ (h((h(r0)).nxt)).nxt ∈ x ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 42 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x.last ≠ (h((h(x.last)).nxt)).nxt, (h(x.last)).nxt ≠ (h((h(x.last)).nxt)).nxt, (h(x.last)).nxt ≠ null, x ≠ [], path((h((h(x.last)).nxt)).nxt ' + x, h), path((h((h(x.last)).nxt)).nxt ' + x0 + (h(x.last)).nxt ', h), (h(x.last)).nxt ∈ h, x.last ∈ h, ¬ dups(x), ¬ x.last ∈ x0, ¬ (h((h(x.last)).nxt)).nxt ∈ x ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [x, r, x1]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, r, x1

Proof step 43 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x = x1 + r ', (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, x ≠ [], path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), path((h((h(r)).nxt)).nxt ' + x, h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(x), ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x ⊦

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[reflist] ⊦ a ∈ b ' ↔ a = b; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 44 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x1 + r ', h), path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1 ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:linearity-path, seq:path(r ' + x, h), path(r ' + y, h) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0), subst:[r, x, h, y] → [(h((h(r)).nxt)).nxt, x0 + (h(r)).nxt ', h, x1 + r ']

Additional information

Used terms: (h((h(r)).nxt)).nxt, x0 + (h(r)).nxt ', h, x1 + r '

Proof step 45 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

path(r ' + x, h), path(r ' + y, h) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0)

Lemma linearity-path

Proof step 46 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), path((h((h(r)).nxt)).nxt ' + x1 + r ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1, x0 + (h(r)).nxt ' = x1 + r ' ∨ (∃ r0, y0. x0 + (h(r)).nxt ' = (x1 + r ') + r0 ' + y0) ∨ (∃ r0, x2. x1 + r ' = (x0 + (h(r)).nxt ') + r0 ' + x2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 47 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x1 + r ', h), path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1, x0 + (h(r)).nxt ' = x1 + r ' ∨ x0 + (h(r)).nxt ' = (x1 + r ') + r0 ' + x ∨ x1 + r ' = (x0 + (h(r)).nxt ') + r1 ' + x2 ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

Used simplifier rules: ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ ¬ c ∈ y ∧ c ≠ b → y + b ' ≠ y0 + c ' + z; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list[reflist] a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + a ' = y + b ' ↔ x = y ∧ a = b; (ws)

Proof step 48 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x1 + r ' = (x0 + (h(r)).nxt ') + r1 ' + x2, (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), path((h((h(r)).nxt)).nxt ' + x1 + r ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1 ⊦

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[reflist] ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list[reflist]

Proof step 49 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x1 + r ' = (x0 + (h(r)).nxt ') + r1 ' + x2, (h(r)).nxt ≠ null, (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), path((h((h(r)).nxt)).nxt ' + x1 + r ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(x1), ¬ (h((h(r)).nxt)).nxt ∈ x1, ¬ r ∈ x0, ¬ r ∈ x1, r = (h(r)).nxt ∨ r = r1 ∨ r ∈ x2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name: splitlasteq, seq:y ≠ [] ⊦ x + a ' = y ↔ x = y.butlast ∧ a = y.last, subst:[y, a, x] → [(x0 + (h(r)).nxt ') + r1 ' + x2, r, x1], <all paths>

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (as) ⊦ a ' + x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: (x0 + (h(r)).nxt ') + r1 ' + x2, r, x1

Proof step 50 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

y ≠ [] ⊦ x + a ' = y ↔ x = y.butlast ∧ a = y.last

A spec-lemma

Proof step 51 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

r = ((x0 + (h(r)).nxt ') + r1 ' + x2).last, x1 = ((x0 + (h(r)).nxt ') + r1 ' + x2).butlast, (x0 + (h(r)).nxt ') + r1 ' + x2 ≠ [], (h(r)).nxt ≠ null, (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, r ≠ (h((h(r)).nxt)).nxt, path((h((h(r)).nxt)).nxt ' + x1 + r ', h), path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ (h((h(r)).nxt)).nxt ∈ x1, ¬ r ∈ x0, ¬ r ∈ x1, r = (h(r)).nxt ∨ r = r1 ∨ r ∈ x2 ⊦

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[reflist] x ≠ [] ⊦ x.butlast + x.last ' = x; (s) ⊦ a ' + x ≠ []; (s) ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ (a ' + b ' + x).butlast = a ' + (b ' + x).butlast; (s) ⊦ (x + a ' + y).butlast = x + (a ' + y).butlast; (s) ⊦ a ≠ b → (a ∈ b ' + x ↔ a ∈ x); (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ x + a ' + y ≠ []; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ (x + a ' + y).last = (a ' + y).last; (s)

Proof step 52 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(r1 ' + x2).last ≠ (h((r1 ' + x2).last)).nxt, (r1 ' + x2).last ≠ (h((h((r1 ' + x2).last)).nxt)).nxt, (h((r1 ' + x2).last)).nxt ≠ (h((h((r1 ' + x2).last)).nxt)).nxt, (h((r1 ' + x2).last)).nxt ≠ null, path((h((h((r1 ' + x2).last)).nxt)).nxt ' + x0 + (h((r1 ' + x2).last)).nxt ', h), path((h((h((r1 ' + x2).last)).nxt)).nxt ' + x0 + (h((r1 ' + x2).last)).nxt ' + r1 ' + x2, h), (h((r1 ' + x2).last)).nxt ∈ h, (r1 ' + x2).last ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ (r1 ' + x2).last ∈ (r1 ' + x2).butlast, ¬ (r1 ' + x2).last ∈ x0, ¬ (h((h((r1 ' + x2).last)).nxt)).nxt ∈ (r1 ' + x2).butlast, ¬ (h((h((r1 ' + x2).last)).nxt)).nxt ∈ x0, ¬ (h((r1 ' + x2).last)).nxt ∈ (r1 ' + x2).butlast, ¬ (h((r1 ' + x2).last)).nxt ∈ x0, (r1 ' + x2).last = r1 ∨ (r1 ' + x2).last ∈ x2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name: lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[y, x] → [x2, r1 '], <all paths>

Additional information

Used terms: x2, r1 '

Proof step 53 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 54 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x2 = [], (r1 ' + x2).last ≠ (h((r1 ' + x2).last)).nxt, (r1 ' + x2).last ≠ (h((h((r1 ' + x2).last)).nxt)).nxt, (h((r1 ' + x2).last)).nxt ≠ (h((h((r1 ' + x2).last)).nxt)).nxt, (h((r1 ' + x2).last)).nxt ≠ null, path((h((h((r1 ' + x2).last)).nxt)).nxt ' + x0 + (h((r1 ' + x2).last)).nxt ' + r1 ' + x2, h), path((h((h((r1 ' + x2).last)).nxt)).nxt ' + x0 + (h((r1 ' + x2).last)).nxt ', h), (r1 ' + x2).last ∈ h, (h((r1 ' + x2).last)).nxt ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ (r1 ' + x2).last ∈ (r1 ' + x2).butlast, ¬ (r1 ' + x2).last ∈ x0, ¬ (h((h((r1 ' + x2).last)).nxt)).nxt ∈ (r1 ' + x2).butlast, ¬ (h((h((r1 ' + x2).last)).nxt)).nxt ∈ x0, ¬ (h((r1 ' + x2).last)).nxt ∈ (r1 ' + x2).butlast, ¬ (h((r1 ' + x2).last)).nxt ∈ x0, (r1 ' + x2).last = r1 ∨ (r1 ' + x2).last ∈ x2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + [] = x; (s) ⊦ ¬ a ∈ []; (s) ⊦ a '.butlast = []; (s)

Proof step 55 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r1)).nxt ≠ (h((h(r1)).nxt)).nxt, (h(r1)).nxt ≠ null, r1 ≠ (h(r1)).nxt, r1 ≠ (h((h(r1)).nxt)).nxt, path((h((h(r1)).nxt)).nxt ' + x0 + (h(r1)).nxt ', h), path((h((h(r1)).nxt)).nxt ' + x0 + (h(r1)).nxt ' + r1 ', h), (h(r1)).nxt ∈ h, r1 ∈ h, ¬ dups(x0), ¬ r1 ∈ x0, ¬ (h((h(r1)).nxt)).nxt ∈ x0, ¬ (h(r1)).nxt ∈ x0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-split-last, seq:⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null, subst:[h, r0, x, r] → [h, r1, x0 + (h(r1)).nxt ', (h((h(r1)).nxt)).nxt], <all paths>

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: h, r1, x0 + (h(r1)).nxt ', (h((h(r1)).nxt)).nxt

Proof step 56 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null

Lemma path-split-last

Proof step 57 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x2.last ≠ (h(x2.last)).nxt, x2.last ≠ (h((h(x2.last)).nxt)).nxt, (h(x2.last)).nxt ≠ (h((h(x2.last)).nxt)).nxt, (h(x2.last)).nxt ≠ null, x2 ≠ [], path((h((h(x2.last)).nxt)).nxt ' + x0 + (h(x2.last)).nxt ' + r1 ' + x2, h), path((h((h(x2.last)).nxt)).nxt ' + x0 + (h(x2.last)).nxt ', h), x2.last ∈ h, (h(x2.last)).nxt ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ x2.last ∈ (r1 ' + x2).butlast, ¬ x2.last ∈ x0, ¬ (h((h(x2.last)).nxt)).nxt ∈ (r1 ' + x2).butlast, ¬ (h((h(x2.last)).nxt)).nxt ∈ x0, ¬ (h(x2.last)).nxt ∈ (r1 ' + x2).butlast, ¬ (h(x2.last)).nxt ∈ x0, x2.last = r1 ∨ x2.last ∈ x2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x.last ∈ x ↔ x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) ⊦ x ≠ [] → (a ' + x).butlast = a ' + x.butlast; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws)

Proof step 58 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x2.last ≠ (h(x2.last)).nxt, x2.last ≠ (h((h(x2.last)).nxt)).nxt, x2.last ≠ r1, (h(x2.last)).nxt ≠ (h((h(x2.last)).nxt)).nxt, (h(x2.last)).nxt ≠ null, (h((h(x2.last)).nxt)).nxt ≠ r1, (h(x2.last)).nxt ≠ r1, x2 ≠ [], path((h((h(x2.last)).nxt)).nxt ' + x0 + (h(x2.last)).nxt ', h), path((h((h(x2.last)).nxt)).nxt ' + x0 + (h(x2.last)).nxt ' + r1 ' + x2, h), (h(x2.last)).nxt ∈ h, x2.last ∈ h, ¬ dups(x0 + x2.butlast), ¬ (h(x2.last)).nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x2.butlast, ¬ x2.last ∈ x2.butlast, ¬ x2.last ∈ x0, ¬ (h((h(x2.last)).nxt)).nxt ∈ x2.butlast, ¬ (h((h(x2.last)).nxt)).nxt ∈ x0, ¬ (h(x2.last)).nxt ∈ x2.butlast ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-last[reflist], name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [x2, r, x1]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x2, r, x1

Proof step 59 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

x2 = x1 + r ', (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, (h((h(r)).nxt)).nxt ≠ r1, (h(r)).nxt ≠ r1, r ≠ (h(r)).nxt, r ≠ (h((h(r)).nxt)).nxt, r ≠ r1, x2 ≠ [], path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ' + r1 ' + x2, h), path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(x0 + x1), ¬ (h(r)).nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x1, ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1, ¬ (h((h(r)).nxt)).nxt ∈ x0, ¬ (h(r)).nxt ∈ x1 ⊦

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[reflist] ⊦ x + a ' ≠ []; (s)

Proof step 60 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

(h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, (h((h(r)).nxt)).nxt ≠ r1, (h(r)).nxt ≠ r1, r ≠ (h(r)).nxt, r ≠ (h((h(r)).nxt)).nxt, r ≠ r1, path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ', h), path((h((h(r)).nxt)).nxt ' + x0 + (h(r)).nxt ' + r1 ' + x1 + r ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(x0 + x1), ¬ (h(r)).nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x1, ¬ r ∈ x1, ¬ r ∈ x0, ¬ (h((h(r)).nxt)).nxt ∈ x1, ¬ (h((h(r)).nxt)).nxt ∈ x0, ¬ (h(r)).nxt ∈ x1 ⊦

User applied rule cut formula

And closed the goal!

Rule argument

fma: path((h(r)).nxt ' + r1 ' + x1 + r ', h)

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used formulas: path((h(r)).nxt ' + r1 ' + x1 + r ', h)

Proof Tree for rem-stays-reachable-aux1 (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