### Nuprl Lemma : get_face-wf

`∀[X:CubicalSet]. ∀[I,J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)].`
`  (get_face(x;i;box) ∈ {f:I-face(X;I)| (f ∈ box) ∧ (face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))} )`

Proof

Definitions occuring in Statement :  get_face: `get_face(y;c;box)` 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` l_member: `(x ∈ l)` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` pair: `<a, b>` product: `x:A × B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` open_box: `open_box(X;I;J;x;i)` and: `P ∧ Q` cand: `A c∧ B` get_face: `get_face(y;c;box)` prop: `ℙ` all: `∀x:A. B[x]` I-face: `I-face(X;I)` pi1: `fst(t)` nameset: `nameset(L)` coordinate_name: `Cname` int_upper: `{i...}` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` band: `p ∧b q` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` uimplies: `b supposing a` pi2: `snd(t)` int_seg: `{i..j-}` bfalse: `ff` subtype_rel: `A ⊆r B` face-name: `face-name(f)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` sq_stable: `SqStable(P)` squash: `↓T` sq_type: `SQType(T)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cons: `[a / b]` l_exists: `(∃x∈L. P[x])` l_all: `(∀x∈L.P[x])` less_than: `a < b`
Lemmas referenced :  filter_type I-face_wf l_member_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int equal_wf list-subtype list_wf assert_wf hd_wf face-name_wf open_box_wf int_seg_wf nameset_wf coordinate_name_wf cubical-set_wf subtype_rel_list band_wf iff_transitivity iff_weakening_uiff assert_of_band int_seg_properties decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_wf sq_stable__l_member decidable__equal-coordinate_name sq_stable__le decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf subtype_base_sq int_subtype_base set_wf list-cases product_subtype_list sqequal-nil nil_wf filter_is_nil_implies select_wf length_wf pi1_wf_top set_subtype_base nameset_subtype_base length_cons_ge_one top_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut sqequalHypSubstitution setElimination thin rename productElimination introduction extract_by_obid isectElimination setEquality hypothesisEquality hypothesis lambdaEquality lambdaFormation sqequalRule unionElimination equalityElimination independent_isectElimination because_Cache equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination productEquality independent_pairEquality natural_numberEquality applyEquality dependent_set_memberEquality independent_pairFormation intEquality dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll imageMemberEquality baseClosed imageElimination instantiate cumulativity promote_hyp hypothesis_subsumption applyLambdaEquality

Latex:
\mforall{}[X:CubicalSet].  \mforall{}[I,J:Cname  List].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].  \mforall{}[box:open\_box(X;I;J;x;i)].
(get\_face(x;i;box)  \mmember{}  \{f:I-face(X;I)|  (f  \mmember{}  box)  \mwedge{}  (face-name(f)  =  <x,  i>)\}  )

Date html generated: 2017_10_05-AM-10_21_03
Last ObjectModification: 2017_07_28-AM-11_21_11

Theory : cubical!sets

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