### Nuprl Lemma : cubical-interval-non-trivial-box

`∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].`
`  ∀bx:open_box(cubical-interval();I;J;x;i). ∀h:name-morph(I;[]).`
`    ((¬(J = [] ∈ (nameset(I) List)))`
`    `` (¬(filter(λf.(h (fst(f)) =z fst(snd(f)));bx)`
`       = []`
`       ∈ ({x:{f:I-face(cubical-interval();I)| (f ∈ bx)} | ↑(h (fst(x)) =z fst(snd(x)))}  List))))`

Proof

Definitions occuring in Statement :  open_box: `open_box(X;I;J;x;i)` I-face: `I-face(X;I)` cubical-interval: `cubical-interval()` name-morph: `name-morph(I;J)` nameset: `nameset(L)` coordinate_name: `Cname` l_member: `(x ∈ l)` filter: `filter(P;l)` nil: `[]` list: `T List` int_seg: `{i..j-}` assert: `↑b` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` lambda: `λx.A[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` not: `¬A` false: `False` prop: `ℙ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` nameset: `nameset(L)` open_box: `open_box(X;I;J;x;i)` and: `P ∧ Q` name-morph: `name-morph(I;J)` I-face: `I-face(X;I)` pi1: `fst(t)` pi2: `snd(t)` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` top: `Top` or: `P ∨ Q` cons: `[a / b]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` l_exists: `(∃x∈L. P[x])` exists: `∃x:A. B[x]` l_all: `(∀x∈L.P[x])` guard: `{T}` lelt: `i ≤ j < k` sq_stable: `SqStable(P)` squash: `↓T` coordinate_name: `Cname` int_upper: `{i...}` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` less_than: `a < b` rev_uimplies: `rev_uimplies(P;Q)` face-name: `face-name(f)` true: `True`
Lemmas referenced :  equal-wf-T-base list_wf filter_type list-subtype not_wf nameset_wf name-morph_wf nil_wf coordinate_name_wf open_box_wf cubical-interval_wf subtype_rel_list int_seg_wf I-face_wf l_member_wf assert_wf eq_int_wf l_all_wf2 null-filter2 null_wf3 filter_wf5 top_wf assert_of_null list-cases product_subtype_list cons_member cons_wf extd-nameset-nil assert_of_eq_int select_wf int_seg_properties length_wf sq_stable__l_member decidable__equal-coordinate_name sq_stable__le decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma pi1_wf_top equal_wf squash_wf true_wf extd-nameset_subtype_int iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin hypothesis sqequalHypSubstitution independent_functionElimination voidElimination extract_by_obid isectElimination because_Cache equalityTransitivity equalitySymmetry sqequalRule baseClosed hypothesisEquality applyEquality independent_isectElimination lambdaEquality setElimination rename dependent_functionElimination natural_numberEquality isect_memberEquality productElimination setEquality addLevel impliesFunctionality voidEquality independent_pairFormation unionElimination promote_hyp hypothesis_subsumption inlFormation dependent_set_memberEquality functionExtensionality applyLambdaEquality imageMemberEquality imageElimination dependent_pairFormation int_eqEquality intEquality computeAll independent_pairEquality universeEquality

Latex:
\mforall{}[I:Cname  List].  \mforall{}[J:nameset(I)  List].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
\mforall{}bx:open\_box(cubical-interval();I;J;x;i).  \mforall{}h:name-morph(I;[]).
((\mneg{}(J  =  []))  {}\mRightarrow{}  (\mneg{}(filter(\mlambda{}f.(h  (fst(f))  =\msubz{}  fst(snd(f)));bx)  =  [])))

Date html generated: 2017_10_05-AM-10_26_36
Last ObjectModification: 2017_07_28-AM-11_22_57

Theory : cubical!sets

Home Index