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Proof step 1 from proof frag-retIns (conclusion) (back to tree)

# sl = # al + 2, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, al + ARetInsert(tid, b) ') ⊦ (∃ n. n < # al ∧ al[n] = ADoInsert(tid, b) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al → al[m] ≠ τ ∧ al[m].tid ≠ tid)) ∨ (b ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (∀ m. m < # al → al[m] ≠ τ ∧ al[m].tid ≠ tid)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof frag-retIns (previous step) (back to tree)

# sl = # al + 2, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, al + ARetInsert(tid, b) ') ⊦ ∃ n. n < # al ∧ al[n] = ADoInsert(tid, b) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al → al[m] ≠ τ ∧ al[m].tid ≠ tid), (b ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (∀ m. m < # al → al[m] ≠ τ ∧ al[m].tid ≠ tid)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: al

Additional information

Used simplifier rules: List('a) freely generated by [], +;;

Proof step 3 from proof frag-retIns (previous step) (back to tree)

# sl = # [] + 2, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, [] + ARetInsert(tid, b) ') ⊦ ∃ n. n < # [] ∧ [][n] = ADoInsert(tid, b) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # [] → [][m] ≠ τ ∧ [][m].tid ≠ tid), (b ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (∀ m. m < # [] → [][m] ≠ τ ∧ [][m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 4 from proof frag-retIns (previous step) (back to tree)

# sl = 2, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, ARetInsert(tid, b) ') ⊦ (sl.last.lsf(tid).lb ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl, s, sl0]; ['a] → [SetState]

Additional information

Used simplifier rules: ⊦ # [] = 0; (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl, s, sl0

Proof step 5 from proof frag-retIns (previous step) (back to tree)

# sl = 2, sl = s + sl0, sl ≠ [], SetInv(s.gstate, s.lsf, s.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, ARetInsert(tid, b) ') ⊦ (sl.last.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ s.pcf(tid) = retInsert

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.head ' → x.last = x.head; (s) ⊦ x ≠ [] → (a ' + x).last = x.last; (s) ⊦ a + x = a ' + x; (s) ⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ') ↔ step(s, a, sl.head) ∧ sl ≠ [] ∧ sl.tail = []; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ x + [] = x; (s) ⊦ # x = 1 ↔ x = x.head + []; (as) n0 < n ⊦ (* n) = m0 + (* n0) ↔ n - n0 = m0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 6 from proof frag-retIns (previous step) (back to tree)

sl0.tail = [], sl0 = sl0.head ', sl0 ≠ [], SetInv(s.gstate, s.lsf, s.pcf), setStep(s, ARetInsert(tid, b), sl0.head) ⊦ (sl0.head.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ sl0.head.lsf(tid).lb) ∧ s.pcf(tid) = retInsert

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] → [sl0, s0, sl]; ['a] → [SetState]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: sl0, s0, sl

Proof step 7 from proof frag-retIns (previous step) (back to tree)

sl = [], sl0 = s0 ', sl0 = s0 + sl, sl0 ≠ [], SetInv(s.gstate, s.lsf, s.pcf), setStep(s, ARetInsert(tid, b), s0) ⊦ (s0.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ s0.lsf(tid).lb) ∧ s.pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' = a0 ' ↔ a = a0; (s)

Proof step 8 from proof frag-retIns (previous step) (back to tree)

SetInv(s.gstate, s.lsf, s.pcf), setStep(s, ARetInsert(tid, b), s0) ⊦ (s0.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ s0.lsf(tid).lb) ∧ s.pcf(tid) = retInsert

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Set, name:elim-mkstate, seq: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf), subst:[s, gstate, lsf, pcf] → [s, gstate, lsf, pcf]

Additional information

Used simplifier rules: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf); (elim) Used terms: s, gstate, lsf, pcf

Proof step 9 from proof frag-retIns (previous step) (back to tree)

s = mkstate(gstate, lsf, pcf), SetInv(gstate, lsf, pcf), setStep(s, ARetInsert(tid, b), s0) ⊦ (s0.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ s0.lsf(tid).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof frag-retIns (previous step) (back to tree)

SetInv(gstate, lsf, pcf), setStep(mkstate(gstate, lsf, pcf), ARetInsert(tid, b), s0) ⊦ (s0.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ s0.lsf(tid).lb) ∧ pcf(tid) = retInsert

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: s0

Additional information

Used simplifier rules: SetState freely generated by mkstate;;

Proof step 11 from proof frag-retIns (previous step) (back to tree)

s0 = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), setStep(mkstate(gstate, lsf, pcf), ARetInsert(tid, b), s0) ⊦ (s0.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ s0.lsf(tid).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: SetInv-thm: ⊦ SetInv(gs, lsf, pcf) ∧ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → SetInv(gs0, lsf0, pcf0); (forward) from specification Set-proofs ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s)

Proof step 12 from proof frag-retIns (previous step) (back to tree)

SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), setStep(mkstate(gstate, lsf, pcf), ARetInsert(tid, b), mkstate(gstate0, lsf0, pcf0)) ⊦ (lsf0(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf0(tid).lb) ∧ pcf(tid) = retInsert

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:setStep, seq:⊦ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) ↔ (∃ tid, ls, pc. lstep(gs, lsf(tid), pcf(tid), a, gs0, ls, pc) ∧ lsf0 = lsf(tid := ls) ∧ pcf0 = pcf(tid := pc)), subst:[lsf, pcf, pcf0, lsf0, gs0, gs, a] → [lsf, pcf, pcf0, lsf0, gstate0, gstate, ARetInsert(tid, b)], <all paths>

Additional information

Used terms: lsf, pcf, pcf0, lsf0, gstate0, gstate, ARetInsert(tid, b)

Proof step 13 from proof frag-retIns (previous step) (back to tree)

⊦ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) ↔ (∃ tid, ls, pc. lstep(gs, lsf(tid), pcf(tid), a, gs0, ls, pc) ∧ lsf0 = lsf(tid := ls) ∧ pcf0 = pcf(tid := pc))

A spec-lemma

Proof step 14 from proof frag-retIns (previous step) (back to tree)

SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), ∃ tid0, ls, pc. lstep(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b), gstate0, ls, pc) ∧ lsf0 = lsf(tid0 := ls) ∧ pcf0 = pcf(tid0 := pc) ⊦ (lsf0(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf0(tid).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ lstep(gs, ls, pc, a, gs0, ls0, pc0) ↔ pre(gs, ls, pc, a) ∧ ls0 = lstepf(gs, ls, pc, a) ∧ gs0 = gstepf(gs, ls, pc, a) ∧ pc0 = pcstepf(gs, ls, pc, a); (s)

Proof step 15 from proof frag-retIns (previous step) (back to tree)

SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b))), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)))), SetInv(gstate, lsf, pcf), pre(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)) ⊦ ((tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb ↔ lsf(tid).lb) ∧ (b ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb) ∧ pcf(tid) = retInsert

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: tid0 = tid

Additional information

Used formulas: tid0 = tid

Proof step 16 from proof frag-retIns (previous step) (back to tree)

tid0 = tid, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b))), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)))), pre(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)) ⊦ ((tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb ↔ lsf(tid).lb) ∧ (b ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof frag-retIns (previous step) (back to tree)

SetInv (gstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)))), SetInv(gstate, lsf, pcf), pre(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)) ⊦ (lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)).lb ↔ lsf(tid).lb) ∧ (b ↔ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)).lb) ∧ pcf(tid) = retInsert

User applied rule constructor cut

And obtained the following new goals

Rule argument

term: pcf(tid)

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; SetPC freely generated by idle, invInsert, invMember, retInsert, retMember;; ⊦ pre(gs, ls, retMember, a) ↔ (∃ b0. a = ARetMember(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s) ⊦ ARetInsert(tid, b) ≠ ARetMember(tid0, b0); (s) ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s) ⊦ ADoMember(tid) ≠ ARetInsert(tid0, b); (s) ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s) ⊦ ADoInsert(tid, b) ≠ ARetInsert(tid0, b0); (s) ⊦ gstepf(gs, ls, idle, a) = gs; (s) ⊦ pre(gs, ls, idle, a) ↔ (∃ e. a = AInvInsert(ls.tid, e) ∧ e ≠ ⊥) ∨ (∃ e. a = AInvMember(ls.tid, e) ∧ e ≠ ⊥); (s) ⊦ AInvInsert(tid, e) ≠ ARetInsert(tid0, b); (s) ⊦ AInvMember(tid, e) ≠ ARetInsert(tid0, b); (s)

Proof step 18 from proof frag-retIns (previous step) (back to tree)

pcf(tid) = retInsert, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)))), pre(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)) ⊦ (lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)).lb ↔ lsf(tid).lb) ∧ (b ↔ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b)).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ARetInsert(tid, b) = ARetInsert(tid0, b0) ↔ tid = tid0 ∧ (b ↔ b0); (ws) ⊦ pre(gs, ls, retInsert, a) ↔ (∃ b0. a = ARetInsert(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s)

Proof step 19 from proof frag-retIns (previous step) (back to tree)

pcf(tid) = retInsert, ¬(lsf(tid).lb ↔ lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)).lb), SetInv (gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb))), pcf(tid := pcstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)))), SetInv(gstate, lsf, pcf) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-retInsert-RETINSERT, seq: ⊦ ls.tid = tid → lstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = ls, subst:[gs, ls, tid, b0] → [gstate, lsf(tid), tid, lsf(tid).lb], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; Used terms: gstate, lsf(tid), tid, lsf(tid).lb

Proof step 20 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = ls

A spec-lemma

Proof step 21 from proof frag-retIns (previous step) (back to tree)

tid0 ≠ tid, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b))), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)))), pre(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)) ⊦ ((tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb ↔ lsf(tid).lb) ∧ (b ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), ARetInsert(tid, b));lsf(tid)).lb) ∧ pcf(tid) = retInsert

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof frag-retIns (previous step) (back to tree)

tid0 ≠ tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b))), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)))), SetInv(gstate, lsf, pcf), pre(gstate, lsf(tid0), pcf(tid0), ARetInsert(tid, b)) ⊦ (b ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert

User applied rule constructor cut

And closed the goal!

Rule argument

term: pcf(tid0)

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; SetPC freely generated by idle, invInsert, invMember, retInsert, retMember;; ⊦ pre(gs, ls, retMember, a) ↔ (∃ b0. a = ARetMember(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s) ⊦ ARetInsert(tid, b) ≠ ARetMember(tid0, b0); (s) ⊦ pre(gs, ls, retInsert, a) ↔ (∃ b0. a = ARetInsert(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s) ⊦ ARetInsert(tid, b) = ARetInsert(tid0, b0) ↔ tid = tid0 ∧ (b ↔ b0); (ws) ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s) ⊦ ADoMember(tid) ≠ ARetInsert(tid0, b); (s) ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s) ⊦ ADoInsert(tid, b) ≠ ARetInsert(tid0, b0); (s) ⊦ gstepf(gs, ls, idle, a) = gs; (s) ⊦ pre(gs, ls, idle, a) ↔ (∃ e. a = AInvInsert(ls.tid, e) ∧ e ≠ ⊥) ∨ (∃ e. a = AInvMember(ls.tid, e) ∧ e ≠ ⊥); (s) ⊦ AInvInsert(tid, e) ≠ ARetInsert(tid0, b); (s) ⊦ AInvMember(tid, e) ≠ ARetInsert(tid0, b); (s)

Proof step 23 from proof frag-retIns (previous step) (back to tree)

# sl = #(a + al0) + 2, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, (a + al0) + ARetInsert(tid, b) '), ∀ sl, tid, b. # sl = # al0 + 2 ∧ SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl, al0 + ARetInsert(tid, b) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid, b) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid)) ∨ (b ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. n < #(a + al0) ∧ (a + al0)[n] = ADoInsert(tid, b) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < #(a + al0) → (a + al0)[m] ≠ τ ∧ (a + al0)[m].tid ≠ tid), (b ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (∀ m. m < #(a + al0) → (a + al0)[m] ≠ τ ∧ (a + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 24 from proof frag-retIns (previous step) (back to tree)

# sl = 3 + # al0, SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, (a ' + al0) + ARetInsert(tid, b) '), ∀ sl0, tid0, b0. # sl0 = # al0 + 2 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, sl.last.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (sl.last.lsf(tid).lb ↔ sl.head.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl.head.pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl, s, sl1]; ['a] → [SetState]

Additional information

Used simplifier rules: ⊦ # [] = 0; (s) ⊦ 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 x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl, s, sl1

Proof step 25 from proof frag-retIns (previous step) (back to tree)

# sl = 3 + # al0, sl = s + sl1, sl ≠ [], SetInv(s.gstate, s.lsf, s.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, (a ' + al0) + ARetInsert(tid, b) '), ∀ sl0, tid0, b0. # sl0 = # al0 + 2 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, sl.last.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ sl[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (sl.last.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ sl.last.lsf(tid).lb) ∧ s.pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 26 from proof frag-retIns (previous step) (back to tree)

# sl1 = # al0 + 2, SetInv(s.gstate, s.lsf, s.pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl1, (a ' + al0) + ARetInsert(tid, b) '), ∀ sl0, tid0, b0. # sl0 = # sl1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl1).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl1).last.lsf(tid).lb) ∧ (s ' + sl1)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl1).last.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ (s ' + sl1).last.lsf(tid).lb) ∧ s.pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Set, name:elim-mkstate, seq: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf), subst:[s, gstate, lsf, pcf] → [s, gstate, lsf, pcf]

Additional information

Used simplifier rules: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf); (elim) Used terms: s, gstate, lsf, pcf

Proof step 27 from proof frag-retIns (previous step) (back to tree)

# sl1 = # al0 + 2, s = mkstate(gstate, lsf, pcf), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl1, (a ' + al0) + ARetInsert(tid, b) '), ∀ sl0, tid0, b0. # sl0 = # sl1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl1).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl1).last.lsf(tid).lb) ∧ (s ' + sl1)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl1).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (s ' + sl1).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof frag-retIns (previous step) (back to tree)

# sl1 = # al0 + 2, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl1, (a ' + al0) + ARetInsert(tid, b) '), ∀ sl0, tid0, b0. # sl0 = # sl1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + sl1)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[al, s, ext, sl, a, start, step] → [al0 + ARetInsert(tid, b) ', mkstate(gstate, lsf, pcf), setExt, sl1, a, setInit, setStep]; ['s, 'a, 'e] → [SetState, SetAction, EAction], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: al0 + ARetInsert(tid, b) ', mkstate(gstate, lsf, pcf), setExt, sl1, a, setInit, setStep

Proof step 29 from proof frag-retIns (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 30 from proof frag-retIns (previous step) (back to tree)

# sl1 = # al0 + 2, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(sl1, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, sl1.head), ∀ sl0, tid0, b0. # sl0 = # sl1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + sl1)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [sl1, s, sl]; ['a] → [SetState]

Additional information

Used simplifier rules: ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 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 x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl1, s, sl

Proof step 31 from proof frag-retIns (previous step) (back to tree)

# sl1 = # al0 + 2, sl1 = s + sl, sl1 ≠ [], SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(sl1, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, s), ∀ sl0, tid0, b0. # sl0 = # sl1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + sl1)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (mkstate(gstate, lsf, pcf) ' + sl1).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

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 ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s)

Proof step 32 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, s), ∀ sl0, tid0, b0. # sl0 = # sl + 1 ∧ SetInv(sl0.head.gstate, sl0.head.lsf, sl0.head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(sl0, al0 + ARetInsert(tid0, b0) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid0, sl0.last.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0[n].pcf(tid0) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0)) ∨ (sl0.last.lsf(tid0).lb ↔ sl0.head.lsf(tid0).lb) ∧ (b0 ↔ sl0.last.lsf(tid0).lb) ∧ sl0.head.pcf(tid0) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid0) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: s ' + sl, tid, b abort: F discard: T computed: T

Additional information

Used terms: s ' + sl, tid, b

Proof step 33 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, s), #(s ' + sl) = # sl + 1 ∧ SetInv((s ' + sl).head.gstate, (s ' + sl).head.lsf, (s ' + sl).head.pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0 + ARetInsert(tid, b) ') → (∃ n. n < # al0 ∧ al0[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid)) ∨ ((s ' + sl).last.lsf(tid).lb ↔ (s ' + sl).head.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (s ' + sl).head.pcf(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

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 ⊦ (a ' + x).head = a; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 34 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, s), SetInv(s.gstate, s.lsf, s.pcf) → n < # al0 ∧ al0[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ∨ ((s ' + sl).last.lsf(tid).lb ↔ s.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ s.pcf(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Set, name:elim-mkstate, seq: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf), subst:[s, gstate, lsf, pcf] → [s, gstate0, lsf0, pcf0]

Additional information

Used simplifier rules: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf); (elim) Used terms: s, gstate0, lsf0, pcf0

Proof step 35 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, s = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, s), SetInv(gstate0, lsf0, pcf0) → n < # al0 ∧ al0[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ∨ ((s ' + sl).last.lsf(tid).lb ↔ lsf0(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ pcf0(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (s ' + sl).last.lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + s ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((s ' + sl).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (s ' + sl).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: SetInv-thm: ⊦ SetInv(gs, lsf, pcf) ∧ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → SetInv(gs0, lsf0, pcf0); (forward) from specification Set-proofs

Proof step 36 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), n < # al0 ∧ al0[n] = ADoInsert(tid, (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ (b ↔ (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ∨ ((mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb ↔ lsf0(tid).lb) ∧ (b ↔ (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ pcf0(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ (b ↔ (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), ((mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ (mkstate(gstate0, lsf0, pcf0) ' + sl).last.lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Set, name:elim-mkstate, seq: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf), subst:[s, gstate, lsf, pcf] → [(mkstate(gstate0, lsf0, pcf0) ' + sl).last, gstate1, lsf1, pcf1]

Additional information

Used simplifier rules: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf); (elim) Used terms: (mkstate(gstate0, lsf0, pcf0) ' + sl).last, gstate1, lsf1, pcf1

Proof step 37 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), n < # al0 ∧ al0[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ (mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ∨ (lsf1(tid).lb ↔ lsf0(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf0(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Set, name:elim-mkstate, seq: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf), subst:[s, gstate, lsf, pcf] → [(mkstate(gstate0, lsf0, pcf0) ' + sl)[n], gstate2, lsf2, pcf2]

Additional information

Used simplifier rules: ⊦ (gstate = s.gstate ∧ lsf = s.lsf) ∧ pcf = s.pcf ↔ s = mkstate(gstate, lsf, pcf); (elim) Used terms: (mkstate(gstate0, lsf0, pcf0) ' + sl)[n], gstate2, lsf2, pcf2

Proof step 38 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), n < # al0 ∧ al0[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf2(tid) = invInsert ∧ (∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ∨ (lsf1(tid).lb ↔ lsf0(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf0(tid) = retInsert ∧ (∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid) ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

Proof step 39 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), b ↔ lsf1(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), al0[n] = ADoInsert(tid, lsf1(tid).lb), pcf2(tid) = invInsert, n < # al0, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 40 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), al0[n] = ADoInsert(tid, lsf1(tid).lb), pcf2(tid) = invInsert, al0 ≠ [], n < # al0, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: n + 1 abort: F discard: F computed: F

Additional information

Used terms: n + 1

Proof step 41 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), al0[n] = ADoInsert(tid, lsf1(tid).lb), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf2(tid) = invInsert, al0 ≠ [], n < # al0, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), ¬ # al0 < n + 1 ∧ (a ' + al0)[n + 1] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n + 1].pcf(tid) = invInsert ∧ (∀ m0. n + 1 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 42 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), al0[n] = ADoInsert(tid, lsf1(tid).lb), pcf2(tid) = invInsert, al0 ≠ [], n + 1 < # al0, n + 1 < m0, n < # al0, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), m0 ≤ # al0, ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid, (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[n, x, y] → [m0, a ', al0]; ['a] → [SetAction], <all paths>

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ m < n → n ≠ 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: m0, a ', al0

Proof step 43 from proof frag-retIns (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 44 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), al0[n] = ADoInsert(tid, lsf1(tid).lb), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf2(tid) = invInsert, al0 ≠ [], n < # al0, n + 1 < m0, n + 1 < # al0, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), m0 ≤ # al0, ¬ m0 < # a ', ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), al0[m0 - # a '] ≠ τ ∧ al0[m0 - # a '].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ m < n → n ≠ 0; (s) ⊦ m < 1 ↔ m = 0; (s)

Proof step 45 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), al0[n] = ADoInsert(tid, lsf1(tid).lb), pcf2(tid) = invInsert, al0 ≠ [], n + 1 < # al0, n + 1 < m0, n < # al0, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), m0 ≤ # al0, ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), al0[m0 - 1] ≠ τ ∧ al0[m0 - 1].tid ≠ tid

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [m0, m1]

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s) n0 < n ⊦ ¬ m + (* n) < (* n0); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m0, m1

Proof step 46 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), al0[n] = ADoInsert(tid, lsf1(tid).lb), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), m0 = m1 + 1, pcf2(tid) = invInsert, al0 ≠ [], m0 ≠ 0, n < # al0, n + 1 < m0, n + 1 < # al0, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), m0 ≤ # al0, ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), al0[m1] ≠ τ ∧ al0[m1].tid ≠ tid, (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 47 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), al0[n] = ADoInsert(tid, lsf1(tid).lb), (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf2(tid) = invInsert, al0 ≠ [], n < m1, m1 < # al0, n < # al0, n + 1 < # al0, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. n < m ∧ m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), al0[m1] ≠ τ ∧ al0[m1].tid ≠ tid, (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule all left

And closed the goal!

Rule argument

pos: left 15 terms: m1 abort: F discard: T computed: T

Additional information

Used terms: m1

Proof step 48 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf0(tid).lb, b ↔ lsf1(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf0(tid) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, b) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n. ¬ # al0 < n ∧ (a ' + al0)[n] = ADoInsert(tid, lsf1(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n].pcf(tid) = invInsert ∧ (∀ m0. n < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ (b ↔ lsf1(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 49 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf0(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf0(tid) = retInsert, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf0(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ ∃ n0. ¬ # al0 < n0 ∧ (a ' + al0)[n0] = ADoInsert(tid, lsf0(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[n0].pcf(tid) = invInsert ∧ (∀ m0. n0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid), (lsf0(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: 0 abort: F discard: T computed: T

Additional information

Used terms: 0

Proof step 50 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf0(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf0(tid) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf0(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf0(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), ¬ # al0 < 0 ∧ (a ' + al0)[0] = ADoInsert(tid, lsf0(tid).lb) ∧ (mkstate(gstate, lsf, pcf) ' + mkstate(gstate0, lsf0, pcf0) ' + sl)[0].pcf(tid) = invInsert ∧ (∀ m0. 0 < m0 ∧ ¬ # al0 < m0 → (a ' + al0)[m0] ≠ τ ∧ (a ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s) ⊦ ¬ n < 0; (s)

Proof step 51 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf0(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf0(tid) = retInsert, SetInv(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf0(tid).lb) '), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate0, lsf0, pcf0)), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf0(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf0(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf0(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf0(tid).lb) ' + al0)[m0].tid ≠ tid)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:setStep, seq:⊦ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) ↔ (∃ tid, ls, pc. lstep(gs, lsf(tid), pcf(tid), a, gs0, ls, pc) ∧ lsf0 = lsf(tid := ls) ∧ pcf0 = pcf(tid := pc)), subst:[lsf, pcf, pcf0, lsf0, gs0, gs, a] → [lsf, pcf, pcf0, lsf0, gstate0, gstate, a], <all paths>

Additional information

Used terms: lsf, pcf, pcf0, lsf0, gstate0, gstate, a

Proof step 52 from proof frag-retIns (previous step) (back to tree)

⊦ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) ↔ (∃ tid, ls, pc. lstep(gs, lsf(tid), pcf(tid), a, gs0, ls, pc) ∧ lsf0 = lsf(tid := ls) ∧ pcf0 = pcf(tid := pc))

A spec-lemma

Proof step 53 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate0, lsf0, pcf0) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf0(tid).lb, (mkstate(gstate0, lsf0, pcf0) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf0(tid) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate0, lsf0, pcf0) ' + sl, al0 + ARetInsert(tid, lsf0(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid, ∃ tid, ls, pc. lstep(gstate, lsf(tid), pcf(tid), a, gstate0, ls, pc) ∧ lsf0 = lsf(tid := ls) ∧ pcf0 = pcf(tid := pc) ⊦ a = ADoInsert(tid, lsf0(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf0(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf0(tid).lb) ' + al0)[m0].tid ≠ tid), (lsf0(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ lstep(gs, ls, pc, a, gs0, ls0, pc0) ↔ pre(gs, ls, pc, a) ∧ ls0 = lstepf(gs, ls, pc, a) ∧ gs0 = gstepf(gs, ls, pc, a) ∧ pc0 = pcstepf(gs, ls, pc, a); (s)

Proof step 54 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), (tid0 = tid ⊃ pcstepf(gstate, lsf(tid), pcf(tid), a);pcf(tid)) = retInsert, lsf1(tid).lb ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), a);lsf(tid)).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: tid0 = tid

Additional information

Used formulas: tid0 = tid

Proof step 55 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), (tid0 = tid ⊃ pcstepf(gstate, lsf(tid), pcf(tid), a);pcf(tid)) = retInsert, lsf1(tid).lb ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), a);lsf(tid)).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), tid0 = tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 56 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid), (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid)

User applied rule constructor cut

And obtained the following new goals

Rule argument

term: pcf(tid)

Additional information

Used simplifier rules: SetPC freely generated by idle, invInsert, invMember, retInsert, retMember;;

Proof step 57 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = idle, pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ idle ≠ invInsert; (s) ⊦ idle ≠ retInsert; (s) ⊦ pre(gs, ls, idle, a) ↔ (∃ e. a = AInvInsert(ls.tid, e) ∧ e ≠ ⊥) ∨ (∃ e. a = AInvMember(ls.tid, e) ∧ e ≠ ⊥); (s) ⊦ gstepf(gs, ls, idle, a) = gs; (s)

Proof step 58 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), idle, a).lb, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = idle, pcstepf(gstate, lsf(tid), idle, a) = retInsert, SetInv(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid, a = AInvInsert(tid, e) ∧ e ≠ ⊥ ∨ a = AInvMember(tid, e0) ∧ e0 ≠ ⊥ ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 11

Additional information

Proof step 59 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), idle, a).lb, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), a = AInvInsert(tid, e), pcf(tid) = idle, pcstepf(gstate, lsf(tid), idle, a) = retInsert, e ≠ ⊥, SetInv(gstate, lsf, pcf), SetInv(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 60 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvInsert(tid, e))), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), idle, AInvInsert(tid, e)).lb, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvInsert(tid, e))), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = idle, pcstepf(gstate, lsf(tid), idle, AInvInsert(tid, e)) = retInsert, e ≠ ⊥, SetInv(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvInsert(tid, e))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvInsert(tid, e))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-idle-INVINSERT, seq: ⊦ ls.tid = tid → lstepf(gs, ls, idle, AInvInsert(tid, e)) = ls.le:= e, subst:[gs, ls, tid, e] → [gstate, lsf(tid), tid, e], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → pcstepf(gs, ls, idle, AInvInsert(tid, e)) = invInsert; (s) ⊦ invInsert ≠ retInsert; (s) Used terms: gstate, lsf(tid), tid, e

Proof step 61 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, idle, AInvInsert(tid, e)) = ls.le:= e

A spec-lemma

Proof step 62 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), idle, a).lb, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), a = AInvMember(tid, e0), pcf(tid) = idle, pcstepf(gstate, lsf(tid), idle, a) = retInsert, e0 ≠ ⊥, SetInv(gstate, lsf, pcf), SetInv(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 63 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvMember(tid, e0))), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), idle, AInvMember(tid, e0)).lb, (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvMember(tid, e0))), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = idle, pcstepf(gstate, lsf(tid), idle, AInvMember(tid, e0)) = retInsert, e0 ≠ ⊥, SetInv(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvMember(tid, e0))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid := lstepf(gstate, lsf(tid), idle, AInvMember(tid, e0))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-idle-INVMEMBER, seq: ⊦ ls.tid = tid → lstepf(gs, ls, idle, AInvMember(tid, e)) = ls.le:= e, subst:[gs, ls, tid, e] → [gstate, lsf(tid), tid, e0], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → pcstepf(gs, ls, idle, AInvMember(tid, e)) = invMember; (s) ⊦ invMember ≠ retInsert; (s) Used terms: gstate, lsf(tid), tid, e0

Proof step 64 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, idle, AInvMember(tid, e)) = ls.le:= e

A spec-lemma

Proof step 65 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ADoInsert(tid, b) = ADoInsert(tid0, b0) ↔ tid = tid0 ∧ (b ↔ b0); (ws) ⊦ invInsert ≠ retInsert; (s) ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s)

Proof step 66 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (b ↔ lsf1(tid).lb) ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 67 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, ¬(b ↔ lsf1(tid).lb), SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-invInsert-DOINSERT, seq:⊦ ls.tid = tid → lstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ ls.lb:= true;ls.lb:= false), subst:[gs, ls, tid, b] → [gstate, lsf(tid), tid, b], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; Used terms: gstate, lsf(tid), tid, b

Proof step 68 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ ls.lb:= true;ls.lb:= false)

A spec-lemma

Proof step 69 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf(tid).tid = tid, lsf1(tid).lb ↔ (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false).lb, ( mkstate(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, ¬(b ↔ lsf1(tid).lb), SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → gstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ gs.set:= (gs.set ++ ls.le);gs); (s) ⊦ ls.tid = tid → pcstepf(gs, ls, invInsert, ADoInsert(tid, b)) = retInsert; (s)

Proof step 70 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false).lb, ( mkstate((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, ¬(b ↔ lsf1(tid).lb), SetInv(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate(gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := (b ⊃ lsf(tid).lb:= true;lsf(tid).lb:= false)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 2

Additional information

Used simplifier rules: ⊦ (ls.lb:= lb).lb ↔ lb; (s)

Proof step 71 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, m0 ≠ 0, SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ¬ # al0 < m0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 72 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, m0 ≠ 0, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), m0 ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[n, x, y] → [m0, ADoInsert(tid, lsf1(tid).lb) ', al0]; ['a] → [SetAction], <all paths>

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ m < 1 ↔ m = 0; (s) Used terms: m0, ADoInsert(tid, lsf1(tid).lb) ', al0

Proof step 73 from proof frag-retIns (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 74 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, m0 ≠ 0, SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), m0 ≤ # al0, ¬ m0 < # ADoInsert(tid, lsf1(tid).lb) ', ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m0 - # ADoInsert(tid, lsf1(tid).lb) '] ≠ τ ∧ al0[m0 - # ADoInsert(tid, lsf1(tid).lb) '].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ m < 1 ↔ m = 0; (s)

Proof step 75 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, m0 ≠ 0, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), m0 ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m0 - 1] ≠ τ ∧ al0[m0 - 1].tid ≠ tid

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [m0, m1]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m0, m1

Proof step 76 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), m0 = m1 + 1, pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, m0 ≠ 0, SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), m0 ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m1] ≠ τ ∧ al0[m1].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 77 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).lb, ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invInsert, pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)) = retInsert, al0 ≠ [], m1 < # al0, SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m1] ≠ τ ∧ al0[m1].tid ≠ tid

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: m1 abort: F discard: T computed: T

Additional information

Used terms: m1

Proof step 78 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invMember, pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ invInsert ≠ invMember; (s) ⊦ invMember ≠ retInsert; (s) ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s)

Proof step 79 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), invMember, ADoMember(tid)).lb, ( mkstate (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = invMember, pcstepf(gstate, lsf(tid), invMember, ADoMember(tid)) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-invMember-DOMEMBER, seq: ⊦ ls.tid = tid → lstepf(gs, ls, invMember, ADoMember(tid)) = ls.lb:= (ls.le ∈ gs.set), subst:[gs, ls, tid] → [gstate, lsf(tid), tid], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → pcstepf(gs, ls, invMember, ADoMember(tid)) = retMember; (s) ⊦ retInsert ≠ retMember; (s) Used terms: gstate, lsf(tid), tid

Proof step 80 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, invMember, ADoMember(tid)) = ls.lb:= (ls.le ∈ gs.set)

A spec-lemma

Proof step 81 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ invInsert ≠ retInsert; (s) ⊦ pre(gs, ls, retInsert, a) ↔ (∃ b0. a = ARetInsert(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s)

Proof step 82 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)).lb, ( mkstate (gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, pcstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)) = retInsert, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ (¬ # al0 < m → (ARetInsert(tid, lsf(tid).lb) ' + al0)[m] ≠ τ ∧ (ARetInsert(tid, lsf(tid).lb) ' + al0)[m].tid ≠ tid)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-retInsert-RETINSERT, seq: ⊦ ls.tid = tid → lstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = ls, subst:[gs, ls, tid, b0] → [gstate, lsf(tid), tid, lsf(tid).lb], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → pcstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = idle; (s) ⊦ idle ≠ retInsert; (s) Used terms: gstate, lsf(tid), tid, lsf(tid).lb

Proof step 83 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = ls

A spec-lemma

Proof step 84 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), pcf(tid), a).lb, (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retMember, pcstepf(gstate, lsf(tid), pcf(tid), a) = retInsert, SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid), pcf(tid), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ invInsert ≠ retMember; (s) ⊦ retInsert ≠ retMember; (s) ⊦ pre(gs, ls, retMember, a) ↔ (∃ b0. a = ARetMember(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s)

Proof step 85 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)).lb, ( mkstate (gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retMember, pcstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)) = retInsert, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb))), pcf(tid := retInsert)), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)), lsf(tid := lstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb))), pcf(tid := retInsert)) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:lstepf-retMember-RETMEMBER, seq: ⊦ ls.tid = tid → lstepf(gs, ls, retMember, ARetMember(tid, b0)) = ls, subst:[gs, ls, tid, b0] → [gstate, lsf(tid), tid, lsf(tid).lb], <all paths>

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ ls.tid = tid → pcstepf(gs, ls, retMember, ARetMember(tid, b0)) = idle; (s) ⊦ idle ≠ retInsert; (s) Used terms: gstate, lsf(tid), tid, lsf(tid).lb

Proof step 86 from proof frag-retIns (previous step) (back to tree)

⊦ ls.tid = tid → lstepf(gs, ls, retMember, ARetMember(tid, b0)) = ls

A spec-lemma

Proof step 87 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), (tid0 = tid ⊃ pcstepf(gstate, lsf(tid), pcf(tid), a);pcf(tid)) = retInsert, lsf1(tid).lb ↔ (tid0 = tid ⊃ lstepf(gstate, lsf(tid), pcf(tid), a);lsf(tid)).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), tid0 ≠ tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf1(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (lsf1(tid).lb ↔ lsf(tid).lb) ∧ pcf(tid) = retInsert ∧ (¬ # al0 < m → (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid), a = ADoInsert(tid, lsf1(tid).lb) ∧ pcf(tid) = invInsert ∧ (m0 ≠ 0 ∧ ¬ # al0 < m0 → (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0] ≠ τ ∧ (ADoInsert(tid, lsf1(tid).lb) ' + al0)[m0].tid ≠ tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ invInsert ≠ retInsert; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 88 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, tid0 ≠ tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), m ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[n, x, y] → [m, a ', al0]; ['a] → [SetAction], <all paths>

Additional information

Used terms: m, a ', al0

Proof step 89 from proof frag-retIns (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 90 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, tid0 ≠ tid, m < # a ', SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), m ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ (a ' + al0)[m] ≠ τ ∧ (a ' + al0)[m].tid ≠ tid

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 ⊦ (a ' + x)[0] = a; (s) ⊦ 0 ≤ m; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ # a ' = 0 +1; (s)

Proof step 91 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, tid0 ≠ tid, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ a ≠ τ ∧ a.tid ≠ tid

User applied rule constructor cut

And obtained the following new goals

Rule argument

term: pcf(tid0)

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; SetPC freely generated by idle, invInsert, invMember, retInsert, retMember;; ⊦ pre(gs, ls, retMember, a) ↔ (∃ b0. a = ARetMember(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s) ⊦ ARetMember(tid, b) ≠ τ; (s) ⊦ ARetMember(tid, b).tid = tid; (s) ⊦ pre(gs, ls, retInsert, a) ↔ (∃ b0. a = ARetInsert(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s) ⊦ ARetInsert(tid, b) ≠ τ; (s) ⊦ ARetInsert(tid, b).tid = tid; (s) ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s) ⊦ ADoMember(tid) ≠ τ; (s) ⊦ ADoMember(tid).tid = tid; (s) ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s) ⊦ ADoInsert(tid, b) ≠ τ; (s) ⊦ ADoInsert(tid, b).tid = tid; (s)

Proof step 92 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, pcf(tid0) = idle, tid0 ≠ tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ a ≠ τ ∧ a.tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: inv-tid: ⊦ SetInv(gstate, lsf, pcf) → lsf(tid).tid = tid; ⊦ pre(gs, ls, idle, a) ↔ (∃ e. a = AInvInsert(ls.tid, e) ∧ e ≠ ⊥) ∨ (∃ e. a = AInvMember(ls.tid, e) ∧ e ≠ ⊥); (s) ⊦ gstepf(gs, ls, idle, a) = gs; (s)

Proof step 93 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, (mkstate(gstate, lsf(tid0 := lstepf(gstate, lsf(tid0), idle, a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), idle, a))) ' + sl).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, (mkstate(gstate, lsf(tid0 := lstepf(gstate, lsf(tid0), idle, a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), idle, a))) ' + sl)[n] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, pcf(tid0) = idle, tid0 ≠ tid, SetInv(gstate, lsf(tid0 := lstepf(gstate, lsf(tid0), idle, a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), idle, a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) (mkstate(gstate, lsf(tid0 := lstepf(gstate, lsf(tid0), idle, a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), idle, a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid, a = AInvInsert(tid0, e) ∧ e ≠ ⊥ ∨ a = AInvMember(tid0, e0) ∧ e0 ≠ ⊥ ⊦ a ≠ τ ∧ a.tid ≠ tid

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 12

Additional information

Used simplifier rules: ⊦ AInvMember(tid, e) ≠ τ; (s) ⊦ AInvMember(tid, e).tid = tid; (s) ⊦ AInvInsert(tid, e) ≠ τ; (s) ⊦ AInvInsert(tid, e).tid = tid; (s)

Proof step 94 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, tid0 ≠ tid, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), m ≤ # al0, ¬ m < # a ', ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m - # a '] ≠ τ ∧ al0[m - # a '].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ m < 1 ↔ m = 0; (s)

Proof step 95 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, m ≠ 0, tid0 ≠ tid, SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), m ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m - 1] ≠ τ ∧ al0[m - 1].tid ≠ tid

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [m, m0]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m, m0

Proof step 96 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), m = m0 + 1, pcf(tid) = retInsert, m ≠ 0, tid0 ≠ tid, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), m ≤ # al0, ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m0] ≠ τ ∧ al0[m0].tid ≠ tid

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 97 from proof frag-retIns (previous step) (back to tree)

# sl = # al0 + 1, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl ).last = mkstate(gstate1, lsf1, pcf1), lsf1(tid).lb ↔ lsf(tid).lb, ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl )[n ] = mkstate(gstate2, lsf2, pcf2), pcf(tid) = retInsert, al0 ≠ [], tid0 ≠ tid, m0 < # al0, SetInv(gstate, lsf, pcf), SetInv (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid0), pcf(tid0), a), lsf(tid0 := lstepf(gstate, lsf(tid0), pcf(tid0), a)), pcf(tid0 := pcstepf(gstate, lsf(tid0), pcf(tid0), a))) ' + sl, al0 + ARetInsert(tid, lsf(tid).lb) '), pre(gstate, lsf(tid0), pcf(tid0), a), ∀ m. m < # al0 → al0[m] ≠ τ ∧ al0[m].tid ≠ tid ⊦ al0[m0] ≠ τ ∧ al0[m0].tid ≠ tid

User applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: m0 abort: F discard: T computed: T

Additional information

Used terms: m0

Proof Tree for frag-retIns (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