Nuprl Lemma : csm-transport
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transport(Gamma;a))s = transport(H;(a)s) ∈ {H ⊢ _:((A)[1(𝕀)])s})
Proof
Definitions occuring in Statement :
transport: transport(Gamma;a),
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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,
transport: transport(Gamma;a),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
true: True,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
interval-1: 1(𝕀)
Lemmas referenced :
composition-term-uniformity,
face-0_wf,
empty-context-subset-lemma4,
interval-type_wf,
empty-context-subset-lemma3,
subset-cubical-term,
context-subset_wf,
context-subset-is-subset,
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf-interval-0,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term_wf,
cubical-type-cumulativity2,
csm-id-adjoin_wf-interval-1,
cube_set_map_wf,
cubical_set_cumulativity-i-j,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
csm-face-0,
csm-ap-term_wf,
csm-context-subset-subtype2,
csm-discrete-cubical-term,
transport_wf,
csm+_wf_interval,
subtype_rel_self,
csm-composition_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
applyEquality,
because_Cache,
independent_isectElimination,
instantiate,
sqequalRule,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
setElimination,
rename,
productElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[a:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}]. \mforall{}[H:j\mvdash{}].
\mforall{}[s:H j{}\mrightarrow{} Gamma].
((transport(Gamma;a))s = transport(H;(a)s))
Date html generated:
2020_05_20-PM-04_25_56
Last ObjectModification:
2020_04_10-PM-11_08_51
Theory : cubical!type!theory
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