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



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof abs-is-fwsim (conclusion) (back to tree)

⊦ fwsim(HashsetIOA, SetIOA, λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof abs-is-fwsim (previous step) (back to tree)

¬ fwsim(HashsetIOA, SetIOA, λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: HashsetIOA-def: ⊦ HashsetIOA = IOAut(hashsetInit, hashsetStep, hashsetExt);

Proof step 3 from proof abs-is-fwsim (previous step) (back to tree)

¬ fwsim (IOAut(hashsetInit, hashsetStep, hashsetExt), SetIOA, λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set-Test-IOAut, name:SetIOA-def, seq: ⊦ SetIOA = IOAut(setInit, setStep, setExt), subst:, <all paths>

Additional information

Used terms:

Proof step 4 from proof abs-is-fwsim (previous step) (back to tree)

⊦ SetIOA = IOAut(setInit, setStep, setExt)

A spec-lemma

Proof step 5 from proof abs-is-fwsim (previous step) (back to tree)

¬ fwsim (IOAut(hashsetInit, hashsetStep, hashsetExt), IOAut(setInit, setStep, setExt), λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOforward, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, r, astart, cext, cstart, astep, aext] → [hashsetStep, λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf), setInit, hashsetExt, hashsetInit, setStep, setExt]; ['as, 'ca, 'e, 'aa, 'cs] → [SetState, HashsetAction, EAction, SetAction, HashsetState], <all paths>

Additional information

Used terms: hashsetStep, λ as, cs. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf), setInit, hashsetExt, hashsetInit, setStep, setExt

Proof step 6 from proof abs-is-fwsim (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

A spec-lemma

Proof step 7 from proof abs-is-fwsim (previous step) (back to tree)

(∃ cs. ¬(hashsetInit(cs) → (∃ as. (abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf)) ∧ setInit(as)))) ∨ (∃ cs, ca, cs0, as. ¬ ( (abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf)) ∧ hashsetStep(cs, ca, cs0) → (∃ as0. move(IOAut(setInit, setStep, setExt))(as, hashsetExt !(ca '), as0) ∧ abs(cs0, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs0.gstate, cs0.lsf, cs0.pcf)))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = τ ⊃ [];ext(a).e '); (s)

Proof step 8 from proof abs-is-fwsim (previous step) (back to tree)

¬(hashsetInit(cs) → (∃ as. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf) ∧ setInit(as))) ∨ ¬ ( abs(cs0, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs0.gstate, cs0.lsf, cs0.pcf) ∧ hashsetStep(cs0, ca, cs1) → (∃ as0. move(IOAut(setInit, setStep, setExt))(as, (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf))) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 9 from proof abs-is-fwsim (previous step) (back to tree)

hashsetInit(cs) ⊦ ∃ as. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf) ∧ setInit(as)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cs

Additional information

Used simplifier rules: HashsetState freely generated by mkstate;;

Proof step 10 from proof abs-is-fwsim (previous step) (back to tree)

cs = mkstate(gs, lsf, pcf), hashsetInit(cs) ⊦ ∃ as. abs(cs, as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(cs.gstate, cs.lsf, cs.pcf) ∧ setInit(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s) ⊦ mkstate(gstate, lsf, pcf).gstate = gstate; (s) ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ m > n ↔ n < m; (s) ⊦ hashsetInit(mkstate(gs, lsf, pcf)) ↔ (∃ n. n > 0 ∧ gs.ar = mkar(n, ⊥) ∧ pcf = (λ tid. idle)) ∧ (∀ tid. lsf(tid).tid = tid); (s)

Proof step 11 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Hashset-proofs, name:init-state-inv, seq: (∃ n. n > 0 ∧ gs.ar = mkar(n, ⊥) ∧ pcf = (λ tid. idle)) ∧ (∀ tid. tid = lsf(tid).tid) ⊦ HashsetInv(gs, lsf, pcf), subst:[gs, pcf, lsf] → [gs, λ tid. idle, lsf]

Additional information

Used terms: gs, λ tid. idle, lsf

Proof step 12 from proof abs-is-fwsim (previous step) (back to tree)

(∃ n. n > 0 ∧ gs.ar = mkar(n, ⊥) ∧ pcf = (λ tid. idle)) ∧ (∀ tid. tid = lsf(tid).tid) ⊦ HashsetInv(gs, lsf, pcf)

A spec-lemma

Proof step 13 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as), (∃ n. n > 0 ∧ gs.ar = mkar(n, ⊥) ∧ (λ tid. idle) = (λ tid. idle)) ∧ (∀ tid. tid = lsf(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) ⊦ m > n ↔ n < m; (s)

Proof step 14 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as), (∃ n0. n0 ≠ 0 ∧ mkar(n, ⊥) = mkar(n0, ⊥)) ∧ tid = lsf(tid).tid

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 2

Additional information

Proof step 15 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as), ∃ n0. n0 ≠ 0 ∧ mkar(n, ⊥) = mkar(n0, ⊥)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: n

Additional information

Used terms: n

Proof step 16 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, tid ≠ lsf(tid).tid, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: tid

Additional information

Used terms: tid

Proof step 17 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ HashsetInv(gs, lsf, λ tid. idle) ∧ setInit(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ setInit(as)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Hashset-refines-Set, name:abs-init, seq:hashsetInit(mkstate(gs, lsf, pcf)), HashsetInv(gs, lsf, pcf) ⊦ ∃ ags, alsf, apcf. setInit(mkstate(ags, alsf, apcf)) ∧ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)), subst:[gs, lsf, pcf] → [gs, lsf, λ tid. idle]

Additional information

Used terms: gs, lsf, λ tid. idle

Proof step 19 from proof abs-is-fwsim (previous step) (back to tree)

hashsetInit(mkstate(gs, lsf, pcf)), HashsetInv(gs, lsf, pcf) ⊦ ∃ ags, alsf, apcf. setInit(mkstate(ags, alsf, apcf)) ∧ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf))

A spec-lemma

Proof step 20 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), ¬ hashsetInit(mkstate(gs, lsf, λ tid. idle)), ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ setInit(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ m > n ↔ n < m; (s) ⊦ hashsetInit(mkstate(gs, lsf, pcf)) ↔ (∃ n. n > 0 ∧ gs.ar = mkar(n, ⊥) ∧ pcf = (λ tid. idle)) ∧ (∀ tid. lsf(tid).tid = tid); (s)

Proof step 21 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ setInit(as), ∃ n0. n0 ≠ 0 ∧ mkar(n, ⊥) = mkar(n0, ⊥)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: n

Additional information

Used terms: n

Proof step 22 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid, ∃ ags, alsf, apcf. setInit(mkstate(ags, alsf, apcf)) ∧ abs(mkstate(gs, lsf, λ tid. idle), mkstate(ags, alsf, apcf)) ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ setInit(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s) ⊦ setInit(mkstate(gs, lsf, pcf)) ↔ (gs.set = ∅ ∧ pcf = (λ tid. idle)) ∧ (∀ tid. lsf(tid).tid = tid); (s)

Proof step 23 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), ags.set = ∅, n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), absset(mkar(n, ⊥), ∅), ∀ tid. labs(gs, lsf(tid), idle, ags, alsf(tid), idle), ∀ tid. alsf(tid).tid = tid, ∀ tid. lsf(tid).tid = tid ⊦ ∃ as. abs(mkstate(gs, lsf, λ tid. idle), as) ∧ SetInv(as.gstate, as.lsf, as.pcf) ∧ setInit(as)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: mkstate(ags, alsf, λ tid. idle) abort: F discard: T computed: F

Additional information

Used terms: mkstate(ags, alsf, λ tid. idle)

Proof step 24 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), ags.set = ∅, n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), absset(mkar(n, ⊥), ∅), ∀ tid. lsf(tid).tid = tid, ∀ tid. alsf(tid).tid = tid, ∀ tid. labs(gs, lsf(tid), idle, ags, alsf(tid), idle) ⊦ abs(mkstate(gs, lsf, λ tid. idle), mkstate(ags, alsf, λ tid. idle)) ∧ SetInv(mkstate(ags, alsf, λ tid. idle).gstate, mkstate(ags, alsf, λ tid. idle).lsf, mkstate(ags, alsf, λ tid. idle).pcf) ∧ setInit(mkstate(ags, alsf, λ tid. idle))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ setInit(mkstate(gs, lsf, pcf)) ↔ (gs.set = ∅ ∧ pcf = (λ tid. idle)) ∧ (∀ tid. lsf(tid).tid = tid); (s) ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s) ⊦ mkstate(gstate, lsf, pcf).gstate = gstate; (s) ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s)

Proof step 25 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), ags.set = ∅, n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), absset(mkar(n, ⊥), ∅), ¬ SetInv(ags, alsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid, ∀ tid. alsf(tid).tid = tid, ∀ tid. labs(gs, lsf(tid), idle, ags, alsf(tid), idle) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Set-proofs, name:init-state-inv, seq:(gs.set = ∅ ∧ pcf = (λ tid. idle)) ∧ (∀ tid. tid = lsf(tid).tid) ⊦ SetInv(gs, lsf, pcf), subst:[gs, pcf, lsf] → [ags, λ tid. idle, alsf]

Additional information

Used terms: ags, λ tid. idle, alsf

Proof step 26 from proof abs-is-fwsim (previous step) (back to tree)

(gs.set = ∅ ∧ pcf = (λ tid. idle)) ∧ (∀ tid. tid = lsf(tid).tid) ⊦ SetInv(gs, lsf, pcf)

A spec-lemma

Proof step 27 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), ags.set = ∅, n ≠ 0, HashsetInv(gs, lsf, λ tid. idle), absset(mkar(n, ⊥), ∅), ¬ SetInv(ags, alsf, λ tid. idle), ∀ tid. labs(gs, lsf(tid), idle, ags, alsf(tid), idle), ∀ tid. alsf(tid).tid = tid, ∀ tid. lsf(tid).tid = tid ⊦ (λ tid. idle) = (λ tid. idle) ∧ (∀ tid. tid = alsf(tid).tid)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof abs-is-fwsim (previous step) (back to tree)

gs.ar = mkar(n, ⊥), ags.set = ∅, n ≠ 0, tid ≠ alsf(tid).tid, HashsetInv(gs, lsf, λ tid. idle), absset(mkar(n, ⊥), ∅), ¬ SetInv(ags, alsf, λ tid. idle), ∀ tid. lsf(tid).tid = tid, ∀ tid. alsf(tid).tid = tid, ∀ tid. labs(gs, lsf(tid), idle, ags, alsf(tid), idle) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: tid

Additional information

Used terms: tid

Proof step 29 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(cs0.gstate, cs0.lsf, cs0.pcf), SetInv(as.gstate, as.lsf, as.pcf), abs(cs0, as), hashsetStep(cs0, ca, cs1) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(as, (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cs0

Additional information

Used simplifier rules: HashsetState freely generated by mkstate;;

Proof step 30 from proof abs-is-fwsim (previous step) (back to tree)

cs0 = mkstate(gs, lsf, pcf), HashsetInv(cs0.gstate, cs0.lsf, cs0.pcf), SetInv(as.gstate, as.lsf, as.pcf), abs(cs0, as), hashsetStep(cs0, ca, cs1) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(as, (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 31 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), SetInv(as.gstate, as.lsf, as.pcf), abs(mkstate(gs, lsf, pcf), as), hashsetStep(mkstate(gs, lsf, pcf), ca, cs1) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(as, (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: as

Additional information

Used simplifier rules: SetState freely generated by mkstate;;

Proof step 32 from proof abs-is-fwsim (previous step) (back to tree)

as = mkstate(ags, alsf, apcf), HashsetInv(gs, lsf, pcf), SetInv(as.gstate, as.lsf, as.pcf), abs(mkstate(gs, lsf, pcf), as), hashsetStep(mkstate(gs, lsf, pcf), ca, cs1) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(as, (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s) ⊦ mkstate(gstate, lsf, pcf).gstate = gstate; (s) ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s)

Proof step 33 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, cs1), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cs1

Additional information

Used simplifier rules: HashsetState freely generated by mkstate;;

Proof step 34 from proof abs-is-fwsim (previous step) (back to tree)

cs1 = mkstate(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, cs1), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(cs1, as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(cs1.gstate, cs1.lsf, cs1.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 35 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(gs0, lsf0, pcf0)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Hashset-proofs, name:HashsetInv-thm, seq:⊦ HashsetInv(gs, lsf, pcf) ∧ hashsetStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → HashsetInv(gs0, lsf0, pcf0), subst:[lsf, gs, gs0, lsf0, pcf0, a, pcf] → [lsf, gs, gs0, lsf0, pcf0, ca, pcf]

Additional information

Used terms: lsf, gs, gs0, lsf0, pcf0, ca, pcf

Proof step 36 from proof abs-is-fwsim (previous step) (back to tree)

⊦ HashsetInv(gs, lsf, pcf) ∧ hashsetStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → HashsetInv(gs0, lsf0, pcf0)

A spec-lemma

Proof step 37 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), HashsetInv(gs, lsf, pcf) ∧ hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)) → HashsetInv(gs0, lsf0, pcf0), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf) ∧ HashsetInv(gs0, lsf0, pcf0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 38 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Hashset-refines-Set, name:abs-theorem, seq:HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), hashsetStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)), abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)), ren(a, ags, alsf) = aa ⊦ (aa = τ ⊃ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags, alsf, apcf)) ;∃ ags0, alsf0, apcf0. setStep(mkstate(ags, alsf, apcf), aa, mkstate(ags0, alsf0, apcf0)) ∧ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags0, alsf0, apcf0)) ), subst:[lsf, gs, alsf, aa, gs0, lsf0, apcf, pcf0, a, pcf, ags] → [lsf, gs, alsf, ren(ca, ags, alsf), gs0, lsf0, apcf, pcf0, ca, pcf, ags]

Additional information

Used simplifier rules: ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s) ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) Used terms: lsf, gs, alsf, ren(ca, ags, alsf), gs0, lsf0, apcf, pcf0, ca, pcf, ags

Proof step 39 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), hashsetStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)), abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)), ren(a, ags, alsf) = aa ⊦ (aa = τ ⊃ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags, alsf, apcf)) ;∃ ags0, alsf0, apcf0. setStep(mkstate(ags, alsf, apcf), aa, mkstate(ags0, alsf0, apcf0)) ∧ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags0, alsf0, apcf0)) )

A spec-lemma

Proof step 40 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), (ren(ca, ags, alsf) = τ ⊃ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags, alsf, apcf)) ;∃ ags0, alsf0, apcf0. setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)) ∧ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags0, alsf0, apcf0)) ), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s)

Proof step 41 from proof abs-is-fwsim (previous step) (back to tree)

HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), (ren(ca, ags, alsf) = τ ⊃ absset(gs0.ar, ags.set) ∧ (∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid))) ; setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)) ∧ absset(gs0.ar, ags0.set) ∧ (∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid))) ), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Proof step 42 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) = τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: mkstate(ags, alsf, apcf) abort: F discard: T computed: T

Additional information

Used terms: mkstate(ags, alsf, apcf)

Proof step 43 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) = τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), absset(gs0.ar, ags.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags, alsf, apcf)) ∧ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags, alsf, apcf)) ∧ SetInv(mkstate(ags, alsf, apcf).gstate, mkstate(ags, alsf, apcf).lsf, mkstate(ags, alsf, apcf).pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s) ⊦ mkstate(gstate, lsf, pcf).gstate = gstate; (s) ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s)

Proof step 44 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) = τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ¬ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags, alsf, apcf)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

Used simplifier rules: ⊦ move(IOAut(start, step, ext))(s, [], s); (s)

Proof step 45 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) = τ, hashsetExt(ca) ≠ τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), absset(gs0.ar, ags.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ¬ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), hashsetExt(ca).e ', mkstate(ags, alsf, apcf)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: opt, name:elim-rqui, seq: ⊦ oe ≠ τ → (e = oe.e ↔ oe = ⌜e⌝ ), subst:[oe, e] → [hashsetExt(ca), _e]; ['e] → [EAction]

Additional information

Used simplifier rules: ⊦ oe ≠ τ → (e = oe.e ↔ oe = ⌜e⌝ ); (elim) Used terms: hashsetExt(ca), _e

Proof step 46 from proof abs-is-fwsim (previous step) (back to tree)

hashsetExt(ca) = ⌜_e⌝ , ren(ca, ags, alsf) = τ, hashsetExt(ca) ≠ τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ¬ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), _e ', mkstate(ags, alsf, apcf)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜e⌝ ≠ τ; (s)

Proof step 47 from proof abs-is-fwsim (previous step) (back to tree)

hashsetExt(ca) = ⌜_e⌝ , ren(ca, ags, alsf) = τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), ¬ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), _e ', mkstate(ags, alsf, apcf)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags, alsf(tid), apcf(tid)) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: ca

Additional information

Used simplifier rules: ext-tau: ⊦ hashsetExt(τ) = τ; HashsetAction freely generated by τ, doINSERT, doMEMBER, invINSERT, invMEMBER, retINSERT, retMEMBER ;; ⊦ ren(retMEMBER(tid, b), ags, alsf) = ARetMember(tid, b); (s) ⊦ ARetMember(tid, b) ≠ τ; (s) ⊦ ren(retINSERT(tid, b), ags, alsf) = ARetInsert(tid, b); (s) ⊦ ARetInsert(tid, b) ≠ τ; (s) ⊦ ren(invMEMBER(tid, e), ags, alsf) = AInvMember(tid, e); (s) ⊦ AInvMember(tid, e) ≠ τ; (s) ⊦ ren(invINSERT(tid, e), ags, alsf) = AInvInsert(tid, e); (s) ⊦ AInvInsert(tid, e) ≠ τ; (s) ⊦ ren(doMEMBER(tid), ags, alsf) = ADoMember(tid); (s) ⊦ ADoMember(tid) ≠ τ; (s) ⊦ ren(doINSERT(tid, b), ags, alsf) = ADoInsert(tid, b); (s) ⊦ ADoInsert(tid, b) ≠ τ; (s) ⊦ ren(τ, ags, alsf) = τ; (s) ⊦ ⌜e⌝ ≠ τ; (s)

Proof step 48 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags0.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

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 49 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), SetInv(ags0, alsf0, apcf0), absset(gs0.ar, ags0.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦ ∃ as0. move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), as0) ∧ abs(mkstate(gs0, lsf0, pcf0), as0) ∧ SetInv(as0.gstate, as0.lsf, as0.pcf)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: mkstate(ags0, alsf0, apcf0) abort: F discard: T computed: F

Additional information

Used terms: mkstate(ags0, alsf0, apcf0)

Proof step 50 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags0, alsf0, apcf0), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags0.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)) ⊦ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags0, alsf0, apcf0)) ∧ abs(mkstate(gs0, lsf0, pcf0), mkstate(ags0, alsf0, apcf0)) ∧ SetInv(mkstate(ags0, alsf0, apcf0).gstate, mkstate(ags0, alsf0, apcf0).lsf, mkstate(ags0, alsf0, apcf0).pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mkstate(gstate, lsf, pcf).pcf = pcf; (s) ⊦ mkstate(gstate, lsf, pcf).lsf = lsf; (s) ⊦ mkstate(gstate, lsf, pcf).gstate = gstate; (s) ⊦ gabs(gs, lsf, pcf, ags, alsf, apcf) ↔ absset(gs.ar, ags.set); (s) ⊦ abs(mkstate(gs, lsf, pcf), mkstate(ags, alsf, apcf)) ↔ gabs(gs, lsf, pcf, ags, alsf, apcf) ∧ (∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid))); (s)

Proof step 51 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), SetInv(ags0, alsf0, apcf0), absset(gs0.ar, ags0.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ¬ move(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(setInit, setStep, setExt), mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags0, alsf0, apcf0)]; ['s, 'e, 'a] → [SetState, EAction, SetAction], <all paths>

Additional information

Used terms: IOAut(setInit, setStep, setExt), mkstate(ags, alsf, apcf), (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), mkstate(ags0, alsf0, apcf0)

Proof step 52 from proof abs-is-fwsim (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

A spec-lemma

Proof step 53 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags0, alsf0, apcf0), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags0.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ∀ al, sl. ¬ ( frag*(IOAut(setInit, setStep, setExt))(sl, al) ∧ mkstate(ags, alsf, apcf) = sl[0] ∧ mkstate(ags0, alsf0, apcf0) = sl.last ∧ IOAut(setInit, setStep, setExt).ext !(al) = (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e ')) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 54 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), SetInv(ags0, alsf0, apcf0), absset(gs0.ar, ags0.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ al, sl. ¬ frag*(IOAut(setInit, setStep, setExt))(sl, al) ∨ mkstate(ags, alsf, apcf) ≠ sl[0] ∨ mkstate(ags0, alsf0, apcf0) ≠ sl.last ∨ setExt !(al) ≠ (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e ') ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: ren(ca, ags, alsf) ', mkstate(ags, alsf, apcf) ' + mkstate(ags0, alsf0, apcf0) ' abort: F discard: T computed: F

Additional information

Used terms: ren(ca, ags, alsf) ', mkstate(ags, alsf, apcf) ' + mkstate(ags0, alsf0, apcf0) '

Proof step 55 from proof abs-is-fwsim (previous step) (back to tree)

ren(ca, ags, alsf) ≠ τ, HashsetInv(gs0, lsf0, pcf0), HashsetInv(gs, lsf, pcf), SetInv(ags0, alsf0, apcf0), SetInv(ags, alsf, apcf), absset(gs.ar, ags.set), absset(gs0.ar, ags0.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ¬ frag*(IOAut(setInit, setStep, setExt))(mkstate(ags, alsf, apcf) ' + mkstate(ags0, alsf0, apcf0) ', ren(ca, ags, alsf) ') ∨ mkstate(ags, alsf, apcf) ≠ (mkstate(ags, alsf, apcf) ' + mkstate(ags0, alsf0, apcf0) ')[0] ∨ mkstate(ags0, alsf0, apcf0) ≠ (mkstate(ags, alsf, apcf) ' + mkstate(ags0, alsf0, apcf0) ').last ∨ setExt !(ren(ca, ags, alsf) ') ≠ (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e ') ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = τ ⊃ [];ext(a).e '); (s) ⊦ mkstate(gstate, lsf, pcf) = mkstate(gstate0, lsf0, pcf0) ↔ gstate = gstate0 ∧ lsf = lsf0 ∧ pcf = pcf0; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + a ').last = a; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ frag*(IOAut(start, step, ext))(s ' + s0 ', a ') ↔ step(s, a, s0); (s)

Proof step 56 from proof abs-is-fwsim (previous step) (back to tree)

(setExt(ren(ca, ags, alsf)) = τ ⊃ [];setExt(ren(ca, ags, alsf)).e ') ≠ (hashsetExt(ca) = τ ⊃ [];hashsetExt(ca).e '), ren(ca, ags, alsf) ≠ τ, HashsetInv(gs, lsf, pcf), HashsetInv(gs0, lsf0, pcf0), SetInv(ags, alsf, apcf), SetInv(ags0, alsf0, apcf0), absset(gs0.ar, ags0.set), absset(gs.ar, ags.set), hashsetStep(mkstate(gs, lsf, pcf), ca, mkstate(gs0, lsf0, pcf0)), setStep(mkstate(ags, alsf, apcf), ren(ca, ags, alsf), mkstate(ags0, alsf0, apcf0)), ∀ tid. labs(gs0, lsf0(tid), pcf0(tid), ags0, alsf0(tid), apcf0(tid)), ∀ tid. labs(gs, lsf(tid), pcf(tid), ags, alsf(tid), apcf(tid)) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: ca

Additional information

Used simplifier rules: ext-tau: ⊦ hashsetExt(τ) = τ; ext-DoInsert: ⊦ hashsetExt(doINSERT(tid, b)) = τ; ext-DoMember: ⊦ hashsetExt(doMEMBER(tid)) = τ; ext-InvInsert: ⊦ hashsetExt(invINSERT(tid, e)) = ⌜InvInsert(tid, e)⌝ ; ext-InvMember: ⊦ hashsetExt(invMEMBER(tid, e)) = ⌜InvMember(tid, e)⌝ ; ext-RetInsert: ⊦ hashsetExt(retINSERT(tid, b)) = ⌜RetInsert(tid, b)⌝ ; ext-RetMember: ⊦ hashsetExt(retMEMBER(tid, b)) = ⌜RetMember(tid, b)⌝ ; HashsetAction freely generated by τ, doINSERT, doMEMBER, invINSERT, invMEMBER, retINSERT, retMEMBER ;; ⊦ ren(retMEMBER(tid, b), ags, alsf) = ARetMember(tid, b); (s) ⊦ setExt(ARetMember(tid, b)) = ⌜RetMember(tid, b)⌝ ; (s) ⊦ ⌜e⌝ ≠ τ; (s) ⊦ ⌜e⌝ .e = e; (s) ⊦ a ' = a0 ' ↔ a = a0; (s) ⊦ ren(retINSERT(tid, b), ags, alsf) = ARetInsert(tid, b); (s) ⊦ setExt(ARetInsert(tid, b)) = ⌜RetInsert(tid, b)⌝ ; (s) ⊦ ren(invMEMBER(tid, e), ags, alsf) = AInvMember(tid, e); (s) ⊦ setExt(AInvMember(tid, e)) = ⌜InvMember(tid, e)⌝ ; (s) ⊦ ren(invINSERT(tid, e), ags, alsf) = AInvInsert(tid, e); (s) ⊦ setExt(AInvInsert(tid, e)) = ⌜InvInsert(tid, e)⌝ ; (s) ⊦ ren(doMEMBER(tid), ags, alsf) = ADoMember(tid); (s) ⊦ setExt(ADoMember(tid)) = τ; (s) ⊦ ren(doINSERT(tid, b), ags, alsf) = ADoInsert(tid, b); (s) ⊦ setExt(ADoInsert(tid, b)) = τ; (s) ⊦ a ' ≠ []; (s) ⊦ ren(τ, ags, alsf) = τ; (s)

Proof Tree for abs-is-fwsim (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