Nuprl Lemma : fill-type-down-0
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma ⊢ _:(A)[1(𝕀)]}].
((app(fill-type-down(Gamma;A;cA); (u)p))[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); u) ∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Proof
Definitions occuring in Statement :
fill-type-down: fill-type-down(Gamma;A;cA),
rev-transport-fun: rev-transport-fun(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-app: app(w; u),
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
rev-type-line: (A)-,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
fill-type-down: fill-type-down(Gamma;A;cA),
all: ∀x:A. B[x],
cubical-type: {X ⊢ _},
interval-rev: 1-(r),
csm-adjoin: (s;u),
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
interval-0: 0(𝕀),
csm-id: 1(X),
csm-ap: (s)x,
cubical-term-at: u(a),
pi1: fst(t),
pi2: snd(t),
rev-transport-fun: rev-transport-fun(Gamma;A;cA)
Lemmas referenced :
fill-type-up-1,
rev-type-line_wf,
csm-adjoin_wf,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf,
csm-interval-type,
interval-rev_wf,
cc-snd_wf,
csm-composition_wf,
subtype_rel_self,
composition-op_wf,
cubical-type-cumulativity2,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
rev-type-line-0,
csm-id-adjoin_wf-interval-1,
cubical-term_wf,
squash_wf,
true_wf,
cubical-type_wf,
rev-type-line-1,
cubical_set_wf,
csm-cubical-app,
csm_id_adjoin_fst_term_lemma,
dma-neg-dM0
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
universeIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
setElimination,
rename,
productElimination,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[u:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})]\}].
((app(fill-type-down(Gamma;A;cA); (u)p))[0(\mBbbI{})] = app(rev-transport-fun(Gamma;A;cA); u))
Date html generated:
2020_05_20-PM-04_55_52
Last ObjectModification:
2020_04_12-AM-08_42_19
Theory : cubical!type!theory
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