Nuprl Lemma : csm-ap-comp-type
∀[Gamma,Delta,Z:CubicalSet]. ∀[s1:Z ⟶ Delta]. ∀[s2:Delta ⟶ Gamma]. ∀[A:{Gamma ⊢ _}].
((A)s2 o s1 = ((A)s2)s1 ∈ {Z ⊢ _})
Proof
Definitions occuring in Statement :
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,
cubical-type: {X ⊢ _},
uimplies: b supposing a,
csm-ap-type: (AF)s,
csm-comp: G o F,
csm-ap: (s)x,
type-cat: TypeCat,
trans-comp: t1 o t2,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
cube-set-map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
list: T List,
cat-ob: cat-ob(C),
pi1: fst(t),
name-cat: NameCat,
cat-arrow: cat-arrow(C),
pi2: snd(t),
I-cube: X(I),
functor-ob: ob(F),
squash: ↓T,
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-type-equal,
csm-ap-type_wf,
csm-comp_wf,
cubical-type_wf,
cube-set-map_wf,
ap_mk_nat_trans_lemma,
cat_comp_tuple_lemma,
I-cube_wf,
list_wf,
coordinate_name_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
csm-ap-restriction,
csm-ap_wf,
subtype_rel_self,
iff_weakening_equal,
squash_wf,
true_wf,
cubical-set_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
productElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
dependent_pairEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[Gamma,Delta,Z:CubicalSet]. \mforall{}[s1:Z {}\mrightarrow{} Delta]. \mforall{}[s2:Delta {}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}].
((A)s2 o s1 = ((A)s2)s1)
Date html generated:
2018_05_23-PM-06_28_49
Last ObjectModification:
2018_05_16-PM-02_30_29
Theory : cubical!sets
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