Nuprl Lemma : csm-ap-comp-term
∀[Gamma,Delta,Z:CubicalSet]. ∀[s1:Z ⟶ Delta]. ∀[s2:Delta ⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}].
((t)s2 o s1 = ((t)s2)s1 ∈ {Z ⊢ _:(A)s2 o s1})
Proof
Definitions occuring in Statement :
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
csm-comp: G o F,
csm-ap-term: (t)s,
type-cat: TypeCat,
trans-comp: t1 o t2,
csm-ap: (s)x,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
implies: P ⇒ Q,
prop: ℙ,
cubical-term: {X ⊢ _:AF}
Lemmas referenced :
cubical-term-equal2,
csm-ap-type_wf,
csm-comp_wf,
csm-ap-term_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-term_wf,
cubical-type_wf,
cube-set-map_wf,
ap_mk_nat_trans_lemma,
cat_comp_tuple_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
rename,
functionExtensionality
Latex:
\mforall{}[Gamma,Delta,Z:CubicalSet]. \mforall{}[s1:Z {}\mrightarrow{} Delta]. \mforall{}[s2:Delta {}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}].
\mforall{}[t:\{Gamma \mvdash{} \_:A\}].
((t)s2 o s1 = ((t)s2)s1)
Date html generated:
2017_10_05-AM-10_13_12
Last ObjectModification:
2017_07_28-AM-11_18_45
Theory : cubical!sets
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