Nuprl Lemma : fill-type-up_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(fill-type-up(Gamma;A;cA) ∈ {Gamma.𝕀 ⊢ _:(((A)[0(𝕀)])p ⟶ A)})
Proof
Definitions occuring in Statement :
fill-type-up: fill-type-up(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm-id-adjoin: [u],
cc-fst: p,
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],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
fill-type-up: fill-type-up(Gamma;A;cA),
and: P ∧ Q,
cand: A c∧ B,
cubical-type: {X ⊢ _},
cc-snd: q,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
cc-fst: p,
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
constant-cubical-type: (X),
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cc-fst_wf,
composition-op_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
cubical-lambda_wf,
filling_term_wf,
face-0_wf,
csm+_wf_interval,
csm-composition_wf,
csm-face-0,
context-subset-term-0,
constrained-cubical-term-0,
csm+_wf,
csm-id-adjoin_wf,
csm-interval-type,
interval-0_wf,
cc-snd_wf,
csm-ap-term_wf,
swap-interval_wf,
cubical-term_wf,
squash_wf,
true_wf,
p+-swap-interval,
cubical-type-cumulativity,
cubical-fun-as-cubical-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
universeIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairFormation,
Error :memTop,
setElimination,
rename,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
hyp_replacement,
natural_numberEquality,
inhabitedIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
(fill-type-up(Gamma;A;cA) \mmember{} \{Gamma.\mBbbI{} \mvdash{} \_:(((A)[0(\mBbbI{})])p {}\mrightarrow{} A)\})
Date html generated:
2020_05_20-PM-04_54_32
Last ObjectModification:
2020_04_13-PM-02_49_09
Theory : cubical!type!theory
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