### Nuprl Lemma : nerve_box_edge_wf2

`∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].`
`∀[box:open_box(cubical-nerve(C);I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].`
`  nerve_box_edge(box;c;y) ∈ cat-arrow(C) nerve_box_label(box;c) nerve_box_label(box;flip(c;y)) `
`  supposing (∃j∈J. ¬(j = y ∈ Cname)) ∨ (((c x) = i ∈ ℕ2) ∧ (¬↑null(J)))`

Proof

Definitions occuring in Statement :  nerve_box_edge: `nerve_box_edge(box;c;y)` nerve_box_label: `nerve_box_label(box;L)` cubical-nerve: `cubical-nerve(X)` open_box: `open_box(X;I;J;x;i)` name-morph-flip: `flip(f;y)` name-morph: `name-morph(I;J)` nameset: `nameset(L)` coordinate_name: `Cname` l_exists: `(∃x∈L. P[x])` null: `null(as)` nil: `[]` list: `T List` int_seg: `{i..j-}` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` or: `P ∨ Q` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` natural_number: `\$n` equal: `s = t ∈ T` cat-arrow: `cat-arrow(C)` small-category: `SmallCategory`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nerve_box_edge: `nerve_box_edge(box;c;y)` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than: `a < b` squash: `↓T` name-morph: `name-morph(I;J)` so_apply: `x[s]` cat-ob: `cat-ob(C)` pi1: `fst(t)` poset-cat: `poset-cat(J)` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` l_exists: `(∃x∈L. P[x])` exists: `∃x:A. B[x]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` guard: `{T}` nameset: `nameset(L)` false: `False` coordinate_name: `Cname` int_upper: `{i...}` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` prop: `ℙ` cons: `[a / b]` bfalse: `ff` respects-equality: `respects-equality(S;T)` cand: `A c∧ B` uiff: `uiff(P;Q)` name-morph-flip: `flip(f;y)` bool: `𝔹` unit: `Unit` sq_type: `SQType(T)` sq_stable: `SqStable(P)` decidable: `Dec(P)` subtract: `n - m` bnot: `¬bb` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` istype: `istype(T)`
Lemmas referenced :  nerve_box_label_same nerve-box-common-face_wf2 subtype_rel_set name-morph_wf nil_wf coordinate_name_wf cat-ob_wf poset-cat_wf equal-wf-T-base int_seg_wf subtype_rel_self nameset_wf list-cases stuck-spread istype-base istype-void length_of_nil_lemma null_nil_lemma int_seg_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf istype-int 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 product_subtype_list null_cons_lemma name-morph-flip_wf subtype_rel-equal cat-arrow_wf l_exists_wf not_wf equal_wf l_member_wf extd-nameset-nil istype-assert null_wf3 subtype_rel_list top_wf extd-nameset-respects-equality open_box_wf cubical-nerve_wf list_wf small-category_wf name-morph_subtype list-diff_wf cname_deq_wf cons_wf face-dimension_wf nameset_subtype list-diff-subset member-poset-cat-arrow poset-cat-arrow-iff eq-cname_wf eqtt_to_assert assert-eq-cname subtype_base_sq set_subtype_base le_wf int_subtype_base lelt_wf sq_stable__le sq_stable__l_member decidable__equal-coordinate_name decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf extd-nameset_wf isname_wf subtype_rel_dep_function istype-le face-cube_wf cubical-nerve-I-cube functor-arrow_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality sqequalRule lambdaEquality_alt setElimination rename productElimination imageElimination because_Cache baseClosed universeIsType independent_isectElimination inhabitedIsType lambdaFormation_alt unionElimination dependent_functionElimination isect_memberEquality_alt voidElimination natural_numberEquality equalityTransitivity equalitySymmetry applyLambdaEquality approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality Error :memTop,  independent_pairFormation promote_hyp hypothesis_subsumption inrFormation_alt equalityIstype dependent_set_memberEquality_alt productIsType axiomEquality unionIsType setIsType functionIsType isectIsTypeImplies sqequalBase equalityElimination instantiate cumulativity intEquality imageMemberEquality functionEquality

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[I:Cname  List].  \mforall{}[J:nameset(I)  List].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(C);I;J;x;i)].  \mforall{}[y:nameset(I)].  \mforall{}[c:\{c:name-morph(I;[])|  (c  y)  =  0\}  ].
nerve\_box\_edge(box;c;y)  \mmember{}  cat-arrow(C)  nerve\_box\_label(box;c)  nerve\_box\_label(box;flip(c;y))
supposing  (\mexists{}j\mmember{}J.  \mneg{}(j  =  y))  \mvee{}  (((c  x)  =  i)  \mwedge{}  (\mneg{}\muparrow{}null(J)))

Date html generated: 2020_05_21-AM-10_54_43
Last ObjectModification: 2019_12_08-PM-07_05_35

Theory : cubical!sets

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