Nuprl Lemma : cubical-apply_wf

[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠB}]. ∀[u:{X ⊢ _:A}].  (cubical-apply(w;u) ∈ {X ⊢ _:(B)[u]})


Proof




Definitions occuring in Statement :  cubical-apply: cubical-apply(w;u) cubical-pi: ΠB csm-id-adjoin: [u] cube-context-adjoin: X.A cubical-term: {X ⊢ _:A} csm-ap-type: (AF)s cubical-type: {X ⊢ _} cubical_set: CubicalSet uall: [x:A]. B[x] member: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T cubical_set: CubicalSet cube-context-adjoin: X.A psc-adjoin: X.A I_cube: A(I) I_set: A(I) cubical-type-at: A(a) presheaf-type-at: A(a) cube-set-restriction: f(s) psc-restriction: f(s) cubical-type-ap-morph: (u f) presheaf-type-ap-morph: (u f) cubical-pi: ΠB presheaf-pi: ΠB cubical-pi-family: cubical-pi-family(X;A;B;I;a) presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a) cube-cat: CubeCat all: x:A. B[x] cc-adjoin-cube: (v;u) psc-adjoin-set: (v;u) csm-ap-type: (AF)s pscm-ap-type: (AF)s csm-ap: (s)x pscm-ap: (s)x csm-id-adjoin: [u] pscm-id-adjoin: [u] csm-adjoin: (s;u) pscm-adjoin: (s;u) csm-id: 1(X) pscm-id: 1(X) cubical-apply: cubical-apply(w;u) presheaf-apply: presheaf-apply(w;u) cubical-app: app(w; u) presheaf-app: app(w; u)
Lemmas referenced :  presheaf-apply_wf cube-cat_wf cubical-type-sq-presheaf-type cat_ob_pair_lemma cat_arrow_triple_lemma cat_comp_tuple_lemma cubical-term-sq-presheaf-term cat_id_tuple_lemma
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin hypothesis sqequalRule Error :memTop,  dependent_functionElimination

Latex:
\mforall{}[X:j\mvdash{}].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[w:\{X  \mvdash{}  \_:\mPi{}A  B\}].  \mforall{}[u:\{X  \mvdash{}  \_:A\}].
    (cubical-apply(w;u)  \mmember{}  \{X  \mvdash{}  \_:(B)[u]\})



Date html generated: 2020_05_20-PM-02_25_51
Last ObjectModification: 2020_04_03-PM-08_36_01

Theory : cubical!type!theory


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