Nuprl Lemma : csm-ap-cubical-lambda

[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[H:j⊢]. ∀[s:H j⟶ X].  (((λb))s (b)s+) ∈ {H ⊢ _:(ΠB)s})


Proof




Definitions occuring in Statement :  cubical-lambda: b) cubical-pi: ΠB csm+: tau+ cube-context-adjoin: X.A csm-ap-term: (t)s cubical-term: {X ⊢ _:A} csm-ap-type: (AF)s cubical-type: {X ⊢ _} cube_set_map: A ⟶ B cubical_set: CubicalSet uall: [x:A]. B[x] equal: 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) cube_set_map: A ⟶ B csm-ap-type: (AF)s pscm-ap-type: (AF)s 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: (s)x pscm-ap: (s)x csm-ap-term: (t)s pscm-ap-term: (t)s cubical-lambda: b) presheaf-lambda: b) csm+: tau+ pscm+: tau+ csm-adjoin: (s;u) pscm-adjoin: (s;u) csm-comp: F pscm-comp: F cc-fst: p psc-fst: p cc-snd: q psc-snd: q
Lemmas referenced :  pscm-ap-presheaf-lambda cube-cat_wf cubical-type-sq-presheaf-type cubical-term-sq-presheaf-term cat_ob_pair_lemma cat_arrow_triple_lemma cat_comp_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{}[b:\{X.A  \mvdash{}  \_:B\}].  \mforall{}[H:j\mvdash{}].  \mforall{}[s:H  j{}\mrightarrow{}  X].
    (((\mlambda{}b))s  =  (\mlambda{}(b)s+))



Date html generated: 2020_05_20-PM-02_29_41
Last ObjectModification: 2020_04_03-PM-08_40_02

Theory : cubical!type!theory


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