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

SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, al) ⊦ sl.head.gstate.set ⊆ sl.last.gstate.set ∧ SetInv(sl.last.gstate, sl.last.lsf, sl.last.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

SetInv(sl.head.gstate, sl.head.lsf, sl.head.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, al) ⊦ sl.head.gstate.set ⊆ sl.last.gstate.set ∧ SetInv(sl.last.gstate, sl.last.lsf, sl.last.pcf)

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: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: sl, s, sl0

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

sl = s + sl0, sl ≠ [], SetInv(s.gstate, s.lsf, s.pcf), frag*(IOAut(setInit, setStep, setExt))(sl, al) ⊦ s.gstate.set ⊆ sl.last.gstate.set ∧ SetInv(sl.last.gstate, sl.last.lsf, sl.last.pcf)

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)

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

SetInv(s.gstate, s.lsf, s.pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl0, al) ⊦ s.gstate.set ⊆ (s ' + sl0).last.gstate.set ∧ SetInv((s ' + sl0).last.gstate, (s ' + sl0).last.lsf, (s ' + sl0).last.pcf)

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 5 from proof frag-rely-inv (previous step) (back to tree)

s = mkstate(gstate, lsf, pcf), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl0, al) ⊦ gstate.set ⊆ (s ' + sl0).last.gstate.set ∧ SetInv((s ' + sl0).last.gstate, (s ' + sl0).last.lsf, (s ' + sl0).last.pcf)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al) ⊦ gstate.set ⊆ (mkstate(gstate, lsf, pcf) ' + sl0).last.gstate.set ∧ SetInv ((mkstate(gstate, lsf, pcf) ' + sl0).last.gstate, (mkstate(gstate, lsf, pcf) ' + sl0).last.lsf, (mkstate(gstate, lsf, pcf) ' + sl0).last.pcf)

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(gstate, lsf, pcf) ' + sl0).last, 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: (mkstate(gstate, lsf, pcf) ' + sl0).last, gstate0, lsf0, pcf0

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

(mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

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 [], +;; ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s) ⊦ (a ' + x).head = a; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ mkstate(gstate, lsf, pcf) = mkstate(gstate0, lsf0, pcf0) ↔ gstate = gstate0 ∧ lsf = lsf0 ∧ pcf = pcf0; (ws) ⊦ # s = # s0 → (s ⊆ s0 ↔ s = s0); (s)

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

(mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, a + al0), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

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

(mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, a ' + al0), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

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, mkstate(gstate, lsf, pcf), setExt, sl0, 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, mkstate(gstate, lsf, pcf), setExt, sl0, a, setInit, setStep

Proof step 10 from proof frag-rely-inv (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 11 from proof frag-rely-inv (previous step) (back to tree)

(mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(sl0, al0), setStep(mkstate(gstate, lsf, pcf), a, sl0.head), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: sl0

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ mkstate(gstate, lsf, pcf) = mkstate(gstate0, lsf0, pcf0) ↔ gstate = gstate0 ∧ lsf = lsf0 ∧ pcf = pcf0; (ws) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

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

(mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0), sl0 = s + sl, SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(sl0, al0), setStep(mkstate(gstate, lsf, pcf), a, sl0.head), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

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 ' + x).head = a; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s)

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

(s ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, s), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: s

Additional information

Used simplifier rules: SetState freely generated by mkstate;;

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

(s ' + sl).last = mkstate(gstate0, lsf0, pcf0), s = mkstate(gstate1, lsf1, pcf1), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(s ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, s), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

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 15 from proof frag-rely-inv (previous step) (back to tree)

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), SetInv(gstate1, lsf1, pcf1), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)), ∀ gstate, lsf, pcf, sl0, gstate0, lsf0, pcf0. (mkstate(gstate, lsf, pcf) ' + sl0).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate, lsf, pcf) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate, lsf, pcf) ' + sl0, al0) → gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: gstate1, lsf1, pcf1, sl, gstate0, lsf0, pcf0 abort: F discard: T computed: T

Additional information

Used terms: gstate1, lsf1, pcf1, sl, gstate0, lsf0, pcf0

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

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate1, lsf1, pcf1), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)), (mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0) ∧ SetInv(gstate1, lsf1, pcf1) ∧ frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0) → gstate1.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0) ⊦ gstate.set ⊆ gstate0.set ∧ SetInv(gstate0, lsf0, pcf0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic

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

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), SetInv(gstate1, lsf1, pcf1), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)), gstate1.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate1.set ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: Set-proofs, name:SetInv-thm, seq:⊦ SetInv(gs, lsf, pcf) ∧ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → SetInv(gs0, lsf0, pcf0), subst:[lsf, pcf, pcf0, lsf0, gs0, gs, a] → [lsf, pcf, pcf1, lsf1, gstate1, gstate, a]

Additional information

Used terms: lsf, pcf, pcf1, lsf1, gstate1, gstate, a

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

⊦ SetInv(gs, lsf, pcf) ∧ setStep(mkstate(gs, lsf, pcf), a, mkstate(gs0, lsf0, pcf0)) → SetInv(gs0, lsf0, pcf0)

A spec-lemma

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

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate0, lsf0, pcf0), SetInv(gstate1, lsf1, pcf1), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)), gstate1.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate1.set, SetInv(gstate, lsf, pcf) ∧ setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)) → SetInv(gstate1, lsf1, pcf1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate, lsf, pcf), SetInv(gstate1, lsf1, pcf1), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), setStep(mkstate(gstate, lsf, pcf), a, mkstate(gstate1, lsf1, pcf1)), gstate1.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate1.set ⊦

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, pcf1, lsf1, gstate1, gstate, a], <all paths>

Additional information

Used terms: lsf, pcf, pcf1, lsf1, gstate1, gstate, a

Proof step 21 from proof frag-rely-inv (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 22 from proof frag-rely-inv (previous step) (back to tree)

(mkstate(gstate1, lsf1, pcf1) ' + sl).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate0, lsf0, pcf0), SetInv(gstate1, lsf1, pcf1), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt))(mkstate(gstate1, lsf1, pcf1) ' + sl, al0), gstate1.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate1.set, ∃ tid, ls, pc. lstep(gstate, lsf(tid), pcf(tid), a, gstate1, ls, pc) ∧ lsf1 = lsf(tid := ls) ∧ pcf1 = pcf(tid := pc) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ 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 23 from proof frag-rely-inv (previous step) (back to tree)

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate0, lsf0, pcf0), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), 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 := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: lsf(tid).tid = tid

Additional information

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

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

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), lsf(tid).tid = tid, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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;

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

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), SetInv(gstate0, lsf0, pcf0), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), 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 := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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; SetPC freely generated by idle, invInsert, invMember, retInsert, retMember;; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ 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 26 from proof frag-rely-inv (previous step) (back to tree)

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = invInsert, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s)

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

( mkstate (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = invInsert, SetInv(gstate0, lsf0, pcf0), SetInv (gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))), 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 := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl, al0), gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)).set ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:gstepf-invInsert-DOINSERT, seq:⊦ ls.tid = tid → gstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ gs.set:= (gs.set ++ ls.le);gs), 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 28 from proof frag-rely-inv (previous step) (back to tree)

⊦ ls.tid = tid → gstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ gs.set:= (gs.set ++ ls.le);gs)

A spec-lemma

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

( mkstate ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), lsf(tid).tid = tid, pcf(tid) = invInsert, SetInv(gstate, lsf, pcf), SetInv ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl, al0), (b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ (b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate).set ⊦

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; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic

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

( mkstate ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = invInsert, SetInv(gstate0, lsf0, pcf0), SetInv ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate ((b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate), lsf(tid := lstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b))), pcf(tid := pcstepf(gstate, lsf(tid), invInsert, ADoInsert(tid, b)))) ' + sl, al0), (b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ (b ⊃ gstate.set:= (gstate.set ++ lsf(tid).le);gstate).set ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 1

Additional information

Used simplifier rules: ⊦ (gs.set:= set).set = set; (s) ⊦ s ++ a ⊆ s0 ↔ s ⊆ s0 ∧ a ∈ s0; (s)

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

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = invMember, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s)

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

( mkstate (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := pcstepf(gstate, lsf(tid), invMember, ADoMember(tid)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = invMember, SetInv(gstate0, lsf0, pcf0), SetInv (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := pcstepf(gstate, lsf(tid), invMember, ADoMember(tid)))), SetInv(gstate, lsf, pcf), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), invMember, ADoMember(tid)), lsf(tid := lstepf(gstate, lsf(tid), invMember, ADoMember(tid))), pcf(tid := pcstepf(gstate, lsf(tid), invMember, ADoMember(tid)))) ' + sl, al0), gstepf(gstate, lsf(tid), invMember, ADoMember(tid)).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), invMember, ADoMember(tid)).set ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:gstepf-invMember-DOMEMBER, seq: ⊦ ls.tid = tid → gstepf(gs, ls, invMember, ADoMember(tid)) = gs, 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; Used terms: gstate, lsf(tid), tid

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

⊦ ls.tid = tid → gstepf(gs, ls, invMember, ADoMember(tid)) = gs

A spec-lemma

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

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = retInsert, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ pre(gs, ls, retInsert, a) ↔ (∃ b0. a = ARetInsert(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s)

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

( 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 := pcstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = retInsert, SetInv(gstate0, lsf0, pcf0), 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), 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 := pcstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)))) ' + sl, al0), gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), retInsert, ARetInsert(tid, lsf(tid).lb)).set ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:gstepf-retInsert-RETINSERT, seq: ⊦ ls.tid = tid → gstepf(gs, ls, retInsert, ARetInsert(tid, b0)) = gs, 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 36 from proof frag-rely-inv (previous step) (back to tree)

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

A spec-lemma

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

( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = retMember, SetInv(gstate, lsf, pcf), SetInv(gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))), SetInv(gstate0, lsf0, pcf0), frag*(IOAut(setInit, setStep, setExt)) ( mkstate (gstepf(gstate, lsf(tid), pcf(tid), a), lsf(tid := lstepf(gstate, lsf(tid), pcf(tid), a)), pcf(tid := pcstepf(gstate, lsf(tid), pcf(tid), a))) ' + sl, al0), pre(gstate, lsf(tid), pcf(tid), a), gstepf(gstate, lsf(tid), pcf(tid), a).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), pcf(tid), a).set ⊦

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; sub-ntrans-01: ⊦ s0 ⊆ s1 ∧ ¬ s ⊆ s1 → ¬ s ⊆ s0; (forward) from specification set-basic ⊦ pre(gs, ls, retMember, a) ↔ (∃ b0. a = ARetMember(ls.tid, b0) ∧ (b0 ↔ ls.lb)); (s)

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

( 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 := pcstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)))) ' + sl ).last = mkstate(gstate0, lsf0, pcf0), pcf(tid) = retMember, SetInv(gstate0, lsf0, pcf0), 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 := pcstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)))), SetInv(gstate, lsf, pcf), 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 := pcstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)))) ' + sl, al0), gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)).set ⊆ gstate0.set, ¬ gstate.set ⊆ gstate0.set, ¬ gstate.set ⊆ gstepf(gstate, lsf(tid), retMember, ARetMember(tid, lsf(tid).lb)).set ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: Set, name:gstepf-retMember-RETMEMBER, seq: ⊦ ls.tid = tid → gstepf(gs, ls, retMember, ARetMember(tid, b0)) = gs, 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 39 from proof frag-rely-inv (previous step) (back to tree)

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

A spec-lemma

Proof Tree for frag-rely-inv (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