### Nuprl Lemma : groupoid-nerve-filler0_wf

`∀[G:Groupoid]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].`
`∀[box:open_box(cubical-nerve(cat(G));I;J;x;i)].`
`  groupoid-nerve-filler0(I;x;box) ∈ cubical-nerve(cat(G))(I) supposing ↑null(J)`

Proof

Definitions occuring in Statement :  groupoid-nerve-filler0: `groupoid-nerve-filler0(I;x;box)` cubical-nerve: `cubical-nerve(X)` open_box: `open_box(X;I;J;x;i)` I-cube: `X(I)` nameset: `nameset(L)` coordinate_name: `Cname` groupoid-cat: `cat(G)` groupoid: `Groupoid` null: `null(as)` list: `T List` int_seg: `{i..j-}` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` top: `Top` bfalse: `ff` false: `False` prop: `ℙ` subtype_rel: `A ⊆r B` nameset: `nameset(L)` l_subset: `l_subset(T;as;bs)` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` coordinate_name: `Cname` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` guard: `{T}` sq_stable: `SqStable(P)` squash: `↓T` groupoid-nerve-filler0: `groupoid-nerve-filler0(I;x;box)` groupoid: `Groupoid` groupoid-cat: `cat(G)` pi1: `fst(t)` open_box: `open_box(X;I;J;x;i)` l_exists: `(∃x∈L. P[x])` exists: `∃x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` I-face: `I-face(X;I)` pi2: `snd(t)` l_all: `(∀x∈L.P[x])` le: `A ≤ B` less_than': `less_than'(a;b)` nat_plus: `ℕ+` less_than: `a < b` true: `True` decidable: `Dec(P)` uiff: `uiff(P;Q)` select: `L[n]`
Lemmas referenced :  nameset_wf list-cases null_nil_lemma product_subtype_list null_cons_lemma assert_wf null_wf3 subtype_rel_list top_wf open_box_wf cubical-nerve_wf groupoid-cat_wf coordinate_name_wf int_seg_wf list_wf groupoid_wf cons_member list-diff_wf cname_deq_wf cons_wf nil_wf member-list-diff l_member_wf not_wf or_wf equal_wf subtype_base_sq set_subtype_base int_subtype_base iota_wf name-morph_subtype nameset_subtype cubical-nerve-I-cube sq_stable__l_member decidable__equal-coordinate_name functor-comp_wf poset-cat_wf poset-functor_wf I-face_wf length_of_nil_lemma int_seg_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf reduce_hd_cons_lemma le_wf length_of_cons_lemma false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff intformnot_wf itermAdd_wf intformeq_wf int_formula_prop_not_lemma int_term_value_add_lemma int_formula_prop_eq_lemma lelt_wf length_wf member_singleton
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis dependent_functionElimination unionElimination sqequalRule promote_hyp hypothesis_subsumption productElimination isect_memberEquality voidElimination voidEquality axiomEquality equalityTransitivity equalitySymmetry applyEquality independent_isectElimination lambdaEquality because_Cache setElimination rename natural_numberEquality lambdaFormation independent_functionElimination addLevel orFunctionality productEquality instantiate imageMemberEquality baseClosed imageElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll cumulativity dependent_set_memberEquality applyLambdaEquality pointwiseFunctionality baseApply closedConclusion addEquality equalityElimination

Latex:
\mforall{}[G:Groupoid].  \mforall{}[I:Cname  List].  \mforall{}[J:nameset(I)  List].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(cat(G));I;J;x;i)].
groupoid-nerve-filler0(I;x;box)  \mmember{}  cubical-nerve(cat(G))(I)  supposing  \muparrow{}null(J)

Date html generated: 2017_10_05-PM-03_43_38
Last ObjectModification: 2017_07_28-AM-11_27_03

Theory : cubical!sets

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