Nuprl Lemma : equals-transprt
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 +⊢ Compositon(A)]. ∀[a:{Gamma ⊢ _:(A)[0(𝕀)]}]. ∀[xx:Top].
(comp cA [0(𝔽) ⊢→ xx] a = transprt(Gamma;cA;a) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
Proof
Definitions occuring in Statement :
transprt: transprt(G;cA;a0),
comp_term: comp cA [phi ⊢→ u] a0,
composition-structure: Gamma ⊢ Compositon(A),
face-0: 0(𝔽),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
transprt: transprt(G;cA;a0),
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T},
composition-structure: Gamma ⊢ Compositon(A),
all: ∀x:A. B[x],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
uimplies: b supposing a,
true: True
Lemmas referenced :
istype-top,
istype-cubical-term,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
comp_term_wf,
squash_wf,
true_wf,
constrained-cubical-term_wf,
csm-ap-term_wf,
context-subset_wf,
thin-context-subset-adjoin,
composition-function_wf,
face-type_wf,
face-0_wf,
composition-function-cumulativity,
empty-context-subset-lemma3,
subset-cubical-term,
context-subset-is-subset,
empty-context-subset-lemma4,
csm-id-adjoin_wf-interval-1,
empty-context-subset-lemma2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
universeIsType,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
dependent_set_memberEquality_alt,
equalityIstype,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} +\mvdash{} Compositon(A)]. \mforall{}[a:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}].
\mforall{}[xx:Top].
(comp cA [0(\mBbbF{}) \mvdash{}\mrightarrow{} xx] a = transprt(Gamma;cA;a))
Date html generated:
2020_05_20-PM-04_38_02
Last ObjectModification:
2020_04_14-PM-10_35_20
Theory : cubical!type!theory
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