Nuprl Lemma : A-open-box-image_wf

[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I,J,K:Cname List]. ∀[alpha:X(I)]. ∀[f:name-morph(I;K)]. ∀[x:nameset(I)]. ∀[i:ℕ2].
  ∀[bx:A-open-box(X;A;I;alpha;J;x;i)]
    (A-open-box-image(X;A;I;K;f;alpha;bx) ∈ A-open-box(X;A;K;f(alpha);map(f;J);f x;i)) 
  supposing nameset([x J]) ⊆name-morph-domain(f;I)


Proof




Definitions occuring in Statement :  A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx) A-open-box: A-open-box(X;A;I;alpha;J;x;i) cubical-type: {X ⊢ _} cube-set-restriction: f(s) I-cube: X(I) cubical-set: CubicalSet name-morph-domain: name-morph-domain(f;I) name-morph: name-morph(I;J) nameset: nameset(L) coordinate_name: Cname map: map(f;as) cons: [a b] list: List int_seg: {i..j-} uimplies: supposing a subtype_rel: A ⊆B uall: [x:A]. B[x] member: t ∈ T apply: a natural_number: $n
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] subtype_rel: A ⊆B implies:  Q name-morph-domain: name-morph-domain(f;I) nameset: nameset(L) name-morph: name-morph(I;J) prop: iff: ⇐⇒ Q and: P ∧ Q guard: {T} so_lambda: λ2x.t[x] so_apply: x[s] uiff: uiff(P;Q) A-open-box: A-open-box(X;A;I;alpha;J;x;i) rev_implies:  Q or: P ∨ Q l_member: (x ∈ l) exists: x:A. B[x] cand: c∧ B coordinate_name: Cname int_upper: {i...} sq_type: SQType(T) nat: squash: T sq_stable: SqStable(P) ge: i ≥  decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) false: False not: ¬A top: Top l_all: (∀x∈L.P[x]) int_seg: {i..j-} lelt: i ≤ j < k le: A ≤ B A-face: A-face(X;A;I;alpha) pi1: fst(t) isname: isname(z) le_int: i ≤j bnot: ¬bb ifthenelse: if then else fi  lt_int: i <j true: True A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx) A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] pairwise: (∀x,y∈L.  P[x; y]) less_than: a < b rev_uimplies: rev_uimplies(P;Q) l_subset: l_subset(T;as;bs) l_exists: (∃x∈L. P[x]) pi2: snd(t) A-face-name: A-face-name(f) A-face-image: A-face-image(X;A;I;K;f;alpha;face) spreadn: spread3
Lemmas referenced :  name-morph-domain_wf member_filter_2 coordinate_name_wf isname_wf l_member_wf assert-isname subtype_rel_sets filter_wf5 equal_wf nameset_wf cons_wf nameset_subtype l_subset_right_cons_trivial list-subtype list_wf map_wf cons_member subtype_base_sq set_subtype_base le_wf int_subtype_base select_wf sq_stable__le nat_properties 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 subtype_rel_list A-open-box_wf subtype_rel_wf int_seg_wf name-morph_wf I-cube_wf cubical-type_wf cubical-set_wf lelt_wf length_wf A-face_wf not_wf A-face-name_wf subtract_wf int_seg_properties sq_stable__l_member decidable__equal-coordinate_name itermSubtract_wf intformless_wf int_term_value_subtract_lemma int_formula_prop_less_lemma decidable__lt sq_stable__assert squash_wf true_wf pi1_wf_top top_wf subtype_rel_product cubical-type-at_wf list-diff_wf cname_deq_wf nil_wf cube-set-restriction_wf face-map_wf2 iff_weakening_equal A-face-image_wf pairwise-map A-face-compatible_wf A-face-compatible-image select_member member-map assert_of_le_int nameset_subtype_extd-nameset nameset_subtype_base l_member_subtype assert_wf eq-cname_wf decidable__l_exists_better-extract decidable__assert assert-eq-cname length-map select-map A-face-name-image product_subtype_base pi2_wf and_wf isname-nameset assert_of_tt equal_functionality_wrt_subtype_rel2 extd-nameset_wf map_cons_lemma length_of_cons_lemma length-map-sq A-adjacent-compatible_wf l_subset_wf all_wf l_exists_wf l_all_wf2 pairwise_wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation hypothesisEquality applyEquality hypothesis sqequalHypSubstitution sqequalRule thin extract_by_obid isectElimination setElimination rename dependent_functionElimination lambdaEquality setEquality productElimination independent_functionElimination because_Cache equalityTransitivity equalitySymmetry independent_isectElimination promote_hyp functionExtensionality voidElimination voidEquality inlFormation dependent_set_memberEquality instantiate cumulativity intEquality natural_numberEquality applyLambdaEquality imageMemberEquality baseClosed imageElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality independent_pairFormation computeAll axiomEquality hyp_replacement productEquality independent_pairEquality universeEquality equalityElimination addEquality comment

Latex:
\mforall{}[X:CubicalSet].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[I,J,K:Cname  List].  \mforall{}[alpha:X(I)].  \mforall{}[f:name-morph(I;K)].
\mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
    \mforall{}[bx:A-open-box(X;A;I;alpha;J;x;i)]
        (A-open-box-image(X;A;I;K;f;alpha;bx)  \mmember{}  A-open-box(X;A;K;f(alpha);map(f;J);f  x;i)) 
    supposing  nameset([x  /  J])  \msubseteq{}r  name-morph-domain(f;I)



Date html generated: 2017_10_05-AM-10_23_11
Last ObjectModification: 2017_07_28-AM-11_21_46

Theory : cubical!sets


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