Nuprl Lemma : cubical-path-0-ap-morph

[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
[u:{I+i,s(phi) ⊢ _:(A)<rho> iota}]. ∀[a:cubical-path-0(Gamma;A;I;i;rho;phi;u)]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I].
[j:{j:ℕ| ¬j ∈ J} ].
  ((a (i0)(rho) g) ∈ cubical-path-0(Gamma;A;J;j;g,i=j(rho);g(phi);(u)subset-trans(I+i;J+j;g,i=j;s(phi))))


Proof




Definitions occuring in Statement :  cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u) csm-ap-term: (t)s cubical-term: {X ⊢ _:A} csm-ap-type: (AF)s cubical-type-ap-morph: (u f) cubical-type: {X ⊢ _} subset-trans: subset-trans(I;J;f;x) subset-iota: iota cubical-subset: I,psi face-presheaf: 𝔽 csm-comp: F context-map: <rho> formal-cube: formal-cube(I) cube-set-restriction: f(s) I_cube: A(I) cubical_set: CubicalSet nc-e': g,i=j nc-0: (i0) nc-s: s add-name: I+i names-hom: I ⟶ J fset-member: a ∈ s fset: fset(T) int-deq: IntDeq nat: uall: [x:A]. B[x] not: ¬A member: t ∈ T set: {x:A| B[x]} 
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u) not: ¬A implies:  Q subtype_rel: A ⊆B uimplies: supposing a nat: so_lambda: λ2x.t[x] so_apply: x[s] prop: false: False all: x:A. B[x] ge: i ≥  decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] and: P ∧ Q cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0) csm-ap-term: (t)s cubical-term-at: u(a) subset-trans: subset-trans(I;J;f;x) csm-ap: (s)x name-morph-satisfies: (psi f) 1 squash: T bdd-distributive-lattice: BoundedDistributiveLattice bounded-lattice-hom: Hom(l1;l2) lattice-hom: Hom(l1;l2) I_cube: A(I) functor-ob: ob(F) pi1: fst(t) face-presheaf: 𝔽 lattice-point: Point(l) record-select: r.x face_lattice: face_lattice(I) face-lattice: face-lattice(T;eq) free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]) constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P) mk-bounded-distributive-lattice: mk-bounded-distributive-lattice mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o) record-update: r[x := v] ifthenelse: if then else fi  eq_atom: =a y bfalse: ff btrue: tt true: True guard: {T} iff: ⇐⇒ Q rev_implies:  Q fl-morph: <f> fl-lift: fl-lift(T;eq;L;eqL;f0;f1) face-lattice-property free-dist-lattice-with-constraints-property lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac) lattice-extend: lattice-extend(L;eq;eqL;f;ac) lattice-fset-join: \/(s) reduce: reduce(f;k;as) list_ind: list_ind fset-image: f"(s) f-union: f-union(domeq;rngeq;s;x.g[x]) list_accum: list_accum cube-set-restriction: f(s) pi2: snd(t) uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q) context-map: <rho> subset-iota: iota csm-comp: F compose: g functor-arrow: arrow(F) respects-equality: respects-equality(S;T)
Lemmas referenced :  istype-nat fset-member_wf nat_wf int-deq_wf strong-subtype-deq-subtype strong-subtype-set3 le_wf istype-int strong-subtype-self istype-void names-hom_wf cubical-path-0_wf cubical-term_wf cubical-subset_wf add-name_wf cube-set-restriction_wf face-presheaf_wf2 nc-s_wf f-subset-add-name csm-ap-type_wf cubical_set_cumulativity-i-j cubical-type-cumulativity csm-comp_wf formal-cube_wf1 subset-iota_wf context-map_wf I_cube_wf nat_properties decidable__le full-omega-unsat 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 istype-le fset_wf cubical-type_wf cubical_set_wf cubical-type-ap-morph_wf nc-0_wf subtype_rel-equal cubical-type-at_wf nc-e'_wf cubical-subset-I_cube-member member-cubical-subset-I_cube nh-comp_wf equal_wf squash_wf true_wf istype-universe lattice-point_wf face_lattice_wf subtype_rel_set bounded-lattice-structure_wf lattice-structure_wf lattice-axioms_wf bounded-lattice-structure-subtype bounded-lattice-axioms_wf lattice-meet_wf lattice-join_wf fl-morph_wf subtype_rel_self fl-morph-restriction iff_weakening_equal cube-set-restriction-comp nh-comp-assoc nc-e'-lemma2 subtype_rel_weakening ext-eq_weakening cubical-type-ap-morph-comp-eq-general cubical-type-cumulativity2 cubical-term-at_wf cubical-subset-I_cube name-morph-satisfies_wf name-morph-satisfies-comp nh-id_wf nh-id-right uiff_transitivity2 s-comp-nc-0 csm-ap-type-at istype-cubical-type-at subset-trans_wf csm-ap-term_wf subtype-respects-equality face-lattice-property free-dist-lattice-with-constraints-property
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut dependent_set_memberEquality_alt sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry setIsType extract_by_obid functionIsType universeIsType isectElimination thin applyEquality intEquality independent_isectElimination because_Cache lambdaEquality_alt natural_numberEquality hypothesisEquality isect_memberEquality_alt isectIsTypeImplies inhabitedIsType instantiate setElimination rename dependent_functionElimination unionElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality Error :memTop,  independent_pairFormation voidElimination lambdaFormation_alt productElimination hyp_replacement imageElimination universeEquality productEquality cumulativity isectEquality imageMemberEquality baseClosed equalityIstype

Latex:
\mforall{}[Gamma:j\mvdash{}].  \mforall{}[A:\{Gamma  \mvdash{}  \_\}].  \mforall{}[I:fset(\mBbbN{})].  \mforall{}[i:\{i:\mBbbN{}|  \mneg{}i  \mmember{}  I\}  ].  \mforall{}[rho:Gamma(I+i)].  \mforall{}[phi:\mBbbF{}(I)].
\mforall{}[u:\{I+i,s(phi)  \mvdash{}  \_:(A)<rho>  o  iota\}].  \mforall{}[a:cubical-path-0(Gamma;A;I;i;rho;phi;u)].  \mforall{}[J:fset(\mBbbN{})].
\mforall{}[g:J  {}\mrightarrow{}  I].  \mforall{}[j:\{j:\mBbbN{}|  \mneg{}j  \mmember{}  J\}  ].
    ((a  (i0)(rho)  g)
      \mmember{}  cubical-path-0(Gamma;A;J;j;g,i=j(rho);g(phi);(u)subset-trans(I+i;J+j;g,i=j;s(phi))))



Date html generated: 2020_05_20-PM-03_47_18
Last ObjectModification: 2020_04_09-AM-11_15_41

Theory : cubical!type!theory


Home Index