Nuprl Lemma : cubical-pair_wf

[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}].  (cubical-pair(u;v) ∈ {X ⊢ _:Σ B})


Proof




Definitions occuring in Statement :  cubical-pair: cubical-pair(u;v) cubical-sigma: Σ B 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) 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-sigma: Σ B presheaf-sigma: Σ B cc-adjoin-cube: (v;u) psc-adjoin-set: (v;u) cubical-pair: cubical-pair(u;v) presheaf-pair: presheaf-pair(u;v)
Lemmas referenced :  presheaf-pair_wf cube-cat_wf cubical-type-sq-presheaf-type cubical-term-sq-presheaf-term
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin hypothesis sqequalRule Error :memTop

Latex:
\mforall{}[X:j\mvdash{}].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[u:\{X  \mvdash{}  \_:A\}].  \mforall{}[v:\{X  \mvdash{}  \_:(B)[u]\}].
    (cubical-pair(u;v)  \mmember{}  \{X  \mvdash{}  \_:\mSigma{}  A  B\})



Date html generated: 2020_05_20-PM-02_27_37
Last ObjectModification: 2020_04_03-PM-08_37_57

Theory : cubical!type!theory


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