Nuprl Lemma : cubical-pi-p

X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}.  ((ΠB)p X.T ⊢ Π(A)p (B)(p p;q) ∈ {X.T ⊢ _})


Proof




Definitions occuring in Statement :  cubical-pi: ΠB 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-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 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-pi-p cube-cat_wf cubical-type-sq-presheaf-type 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 dependent_functionElimination thin hypothesis sqequalRule isectElimination Error :memTop

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



Date html generated: 2020_05_20-PM-02_00_24
Last ObjectModification: 2020_04_03-PM-08_33_28

Theory : cubical!type!theory


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