Nuprl Lemma : csm-revfill
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[a1:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[Delta:j⊢].
∀[s:Delta j⟶ Gamma].
((revfill(Gamma;cA;a1))s+ = revfill(Delta;(cA)s+;(a1)s) ∈ {Delta.𝕀 ⊢ _:(A)s+})
Proof
Definitions occuring in Statement :
revfill: revfill(Gamma;cA;a1),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
interval-1: 1(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],
revfill: revfill(Gamma;cA;a1),
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
respects-equality: respects-equality(S;T),
squash: ↓T,
face-0: 0(𝔽),
csm-ap-term: (t)s,
discrete-cubical-term: discr(t),
interval-type: 𝕀,
csm+: tau+
Lemmas referenced :
csm-face-0,
empty-context-subset-lemma3,
cube-context-adjoin_wf,
interval-type_wf,
trivial-constrained-term,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-1,
csm-rev_fill_term,
face-0_wf,
subtype_rel_sets_simple,
cubical-term_wf,
cubical-type-cumulativity2,
equal_wf,
thin-context-subset,
context-subset_wf,
respects-equality-context-subset-term,
cube_set_map_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
instantiate,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
lambdaEquality_alt,
universeIsType,
independent_isectElimination,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)]. \mforall{}[a1:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})]\}].
\mforall{}[Delta:j\mvdash{}]. \mforall{}[s:Delta j{}\mrightarrow{} Gamma].
((revfill(Gamma;cA;a1))s+ = revfill(Delta;(cA)s+;(a1)s))
Date html generated:
2020_05_20-PM-04_53_05
Last ObjectModification:
2020_04_13-PM-09_50_09
Theory : cubical!type!theory
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