Nuprl Lemma : cubical-path-condition'_wf

[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}]. ∀[a1:A((i1)(rho))].
  (cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1) ∈ ℙ')


Proof




Definitions occuring in Statement :  cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1) cubical-term: {X ⊢ _:A} csm-ap-type: (AF)s cubical-type-at: A(a) cubical-type: {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-1: (i1) nc-s: s add-name: I+i fset-member: a ∈ s fset: fset(T) int-deq: IntDeq nat: uall: [x:A]. B[x] prop: not: ¬A member: t ∈ T set: {x:A| B[x]} 
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1) prop: all: x:A. B[x] nat: ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: supposing a not: ¬A implies:  Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False and: P ∧ Q subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q) 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 bdd-distributive-lattice: BoundedDistributiveLattice true: True squash: T guard: {T} iff: ⇐⇒ Q rev_implies:  Q context-map: <rho> subset-iota: iota csm-comp: F csm-ap: (s)x compose: g functor-arrow: arrow(F) cube-set-restriction: f(s)
Lemmas referenced :  cubical-subset_wf fset_wf nat_wf I_cube_wf cubical-subset-I_cube-member istype-cubical-type-at cube-set-restriction_wf add-name_wf nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int 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 nc-1_wf istype-cubical-term face-presheaf_wf2 nc-s_wf f-subset-add-name csm-ap-type_wf csm-comp_wf formal-cube_wf1 subset-iota_wf context-map_wf istype-nat fset-member_wf int-deq_wf strong-subtype-deq-subtype strong-subtype-set3 le_wf strong-subtype-self istype-void cubical-type_wf cubical_set_wf name-morph-satisfies-comp nh-comp_wf member-cubical-subset-I_cube cubical-term-at_wf subtype_rel_self 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 equal_wf lattice-meet_wf lattice-join_wf name-morph-satisfies_wf nh-id-right iff_weakening_equal squash_wf true_wf names-hom_wf nh-comp-assoc s-comp-nc-1 csm-ap-type-at cubical-type-at_wf cube-set-restriction-comp cubical-type-ap-morph_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt cut thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination because_Cache hypothesisEquality sqequalRule functionEquality cumulativity hypothesis dependent_set_memberEquality_alt setElimination rename dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  independent_pairFormation universeIsType voidElimination setIsType functionIsType applyEquality intEquality productElimination equalityTransitivity equalitySymmetry productEquality isectEquality lambdaFormation_alt imageElimination imageMemberEquality baseClosed inhabitedIsType universeEquality hyp_replacement

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{}[a1:A((i1)(rho))].
    (cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1)  \mmember{}  \mBbbP{}')



Date html generated: 2020_05_20-PM-03_45_28
Last ObjectModification: 2020_04_21-AM-01_14_41

Theory : cubical!type!theory


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