Nuprl Lemma : cubical-sigma-p-p

X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀C:{X.T ⊢ _}.  (((Σ B)p)p = Σ ((A)p)p (B)(p p;q) ∈ {X.T.C ⊢ _})


Proof




Definitions occuring in Statement :  cubical-sigma: Σ B csm-adjoin: (s;u) cc-snd: q cc-fst: p cube-context-adjoin: X.A csm-ap-type: (AF)s cubical-type: {X ⊢ _} csm-comp: F cubical_set: CubicalSet all: x:A. B[x] equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T cubical_set: CubicalSet uall: [x:A]. B[x] 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 cubical-sigma: Σ B presheaf-sigma: Σ B cc-adjoin-cube: (v;u) psc-adjoin-set: (v;u) csm-ap: (s)x pscm-ap: (s)x cc-fst: p psc-fst: p csm-adjoin: (s;u) pscm-adjoin: (s;u) csm-comp: F pscm-comp: F cc-snd: q psc-snd: q
Lemmas referenced :  presheaf-sigma-p-p cube-cat_wf cubical-type-sq-presheaf-type
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin hypothesis sqequalRule isectElimination Error :memTop

Latex:
\mforall{}X:j\mvdash{}.  \mforall{}T,A:\{X  \mvdash{}  \_\}.  \mforall{}B:\{X.A  \mvdash{}  \_\}.  \mforall{}C:\{X.T  \mvdash{}  \_\}.    (((\mSigma{}  A  B)p)p  =  \mSigma{}  ((A)p)p  (B)(p  o  p  o  p;q))



Date html generated: 2020_05_20-PM-02_26_44
Last ObjectModification: 2020_04_03-PM-08_37_03

Theory : cubical!type!theory


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