### Nuprl Lemma : open_box-nil

`∀[X:CubicalSet]. ∀[I:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2].`
`  open_box(X;I;[];x;i) ≡ {L:I-face(X;I) List| (||L|| = 1 ∈ ℤ) ∧ (face-name(hd(L)) = <x, i> ∈ (nameset(I) × ℕ2))} `

Proof

Definitions occuring in Statement :  open_box: `open_box(X;I;J;x;i)` face-name: `face-name(f)` I-face: `I-face(X;I)` cubical-set: `CubicalSet` nameset: `nameset(L)` coordinate_name: `Cname` length: `||as||` hd: `hd(l)` nil: `[]` list: `T List` int_seg: `{i..j-}` ext-eq: `A ≡ B` uall: `∀[x:A]. B[x]` and: `P ∧ Q` set: `{x:A| B[x]} ` pair: `<a, b>` product: `x:A × B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` ext-eq: `A ≡ B` and: `P ∧ Q` subtype_rel: `A ⊆r B` open_box: `open_box(X;I;J;x;i)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` ge: `i ≥ j ` guard: `{T}` int_seg: `{i..j-}` nameset: `nameset(L)` lelt: `i ≤ j < k` all: `∀x:A. B[x]` implies: `P `` Q` sq_stable: `SqStable(P)` squash: `↓T` coordinate_name: `Cname` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` pi1: `fst(t)` I-face: `I-face(X;I)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` cons: `[a / b]` l_exists: `(∃x∈L. P[x])` sq_type: `SQType(T)` select: `L[n]` l_all: `(∀x∈L.P[x])` le: `A ≤ B` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` subtract: `n - m` pairwise: `(∀x,y∈L.  P[x; y])` less_than: `a < b` true: `True` face-name: `face-name(f)` pi2: `snd(t)` uiff: `uiff(P;Q)` rev_implies: `P `` Q` adjacent-compatible: `adjacent-compatible(X;I;L)`
Lemmas referenced :  length_wf_nat set_subtype_base le_wf int_subtype_base face-name_wf hd_wf int_seg_properties sq_stable__l_member coordinate_name_wf decidable__equal-coordinate_name sq_stable__le decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf open_box_wf nil_wf adjacent-compatible_wf l_member_wf l_subset_wf nameset_wf l_exists_wf I-face_wf equal_wf int_seg_wf l_all_wf2 not_wf subtract_wf itermSubtract_wf intformless_wf int_term_value_subtract_lemma int_formula_prop_less_lemma decidable__lt istype-le istype-less_than cons_wf pairwise_wf2 list_wf cubical-set_wf list-cases product_subtype_list length_of_nil_lemma length_of_cons_lemma reduce_hd_cons_lemma subtype_base_sq lelt_wf decidable__equal_int istype-false non_neg_length length_wf itermAdd_wf int_term_value_add_lemma member_singleton pi1_wf_top decidable__equal_int_seg le_antisymmetry_iff null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse l_subset_nil_left bool_wf ppcc-problem unit_wf2 iff_weakening_equal nameset_subtype l_all_cons cons_member pairwise-singleton l_all_single l_all_nil select_wf product_subtype_base nameset_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut independent_pairFormation lambdaEquality_alt sqequalHypSubstitution setElimination thin rename dependent_set_memberEquality_alt hypothesisEquality sqequalRule productIsType equalityIsType4 because_Cache productElimination extract_by_obid isectElimination hypothesis applyEquality intEquality closedConclusion natural_numberEquality independent_isectElimination equalityIsType1 dependent_functionElimination independent_functionElimination lambdaFormation_alt inhabitedIsType imageMemberEquality baseClosed imageElimination equalityTransitivity equalitySymmetry unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination universeIsType independent_pairEquality functionIsType productEquality setIsType instantiate cumulativity axiomEquality isectIsTypeImplies promote_hyp hypothesis_subsumption addEquality applyLambdaEquality unionEquality inlEquality_alt inlFormation_alt

Latex:
\mforall{}[X:CubicalSet].  \mforall{}[I:Cname  List].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
open\_box(X;I;[];x;i)  \mequiv{}  \{L:I-face(X;I)  List|  (||L||  =  1)  \mwedge{}  (face-name(hd(L))  =  <x,  i>)\}

Date html generated: 2019_11_05-PM-00_28_11
Last ObjectModification: 2018_11_08-PM-01_16_01

Theory : cubical!sets

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