Nuprl Lemma : assert-graph-rcvset

`∀a:Id ─→ Id ─→ Id. ∀b:Id. ∀S:Id List. ∀G:Graph(S). ∀k:Knd.`
`  (↑graph-rcvset(a;b;S;G;k) `⇐⇒` ∃i,j:Id. ((i ∈ S) ∧ (j ∈ S) ∧ (i─→j)∈G ∧ (k = rcv((link(a i j) from i to j),b) ∈ Knd)))`

Proof

Definitions occuring in Statement :  graph-rcvset: `graph-rcvset(a;b;S;G;k)` rcv: `rcv(l,tg)` Knd: `Knd` mk_lnk: `(link(n) from i to j)` id-graph-edge: `(i─→j)∈G` id-graph: `Graph(S)` Id: `Id` l_member: `(x ∈ l)` list: `T List` assert: `↑b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` apply: `f a` function: `x:A ─→ B[x]` equal: `s = t ∈ T`
Lemmas :  assert_wf graph-rcvset_wf exists_wf Id_wf l_member_wf id-graph-edge_wf Knd_wf rcv_wf mk_lnk_wf id-graph_wf list_wf deq-member_wf id-deq_wf bool_wf eqtt_to_assert assert-deq-member eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot eq_id_wf assert-eq-id subtype_rel_list bfalse_wf false_wf iff_transitivity iff_weakening_uiff assert_of_band and_wf atom2_subtype_base id-graph-edge-implies-member union_subtype_base IdLnk_wf product_subtype_base isrcv_rcv_lemma tag_rcv_lemma lnk_rcv_lemma lname_mk_lnk_lemma lsrc_mk_lnk_lemma ldst_mk_lnk_lemma member_wf
\mforall{}a:Id  {}\mrightarrow{}  Id  {}\mrightarrow{}  Id.  \mforall{}b:Id.  \mforall{}S:Id  List.  \mforall{}G:Graph(S).  \mforall{}k:Knd.
(\muparrow{}graph-rcvset(a;b;S;G;k)
\mLeftarrow{}{}\mRightarrow{}  \mexists{}i,j:Id.  ((i  \mmember{}  S)  \mwedge{}  (j  \mmember{}  S)  \mwedge{}  (i{}\mrightarrow{}j)\mmember{}G  \mwedge{}  (k  =  rcv((link(a  i  j)  from  i  to  j),b))))

Date html generated: 2015_07_17-AM-09_13_29
Last ObjectModification: 2015_01_28-AM-07_59_49

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