Nuprl Lemma : fill-type-down_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(fill-type-down(Gamma;A;cA) ∈ {Gamma.𝕀 ⊢ _:(((A)[1(𝕀)])p ⟶ A)})
Proof
Definitions occuring in Statement :
fill-type-down: fill-type-down(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-1: 1(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,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
rev-type-line: (A)-,
fill-type-down: fill-type-down(Gamma;A;cA),
squash: ↓T,
prop: ℙ,
true: True,
all: ∀x:A. B[x],
interval-rev: 1-(r),
cubical-type: {X ⊢ _},
interval-1: 1(𝕀),
csm-id-adjoin: [u],
interval-0: 0(𝕀),
cubical-term-at: u(a),
csm-adjoin: (s;u),
csm-ap: (s)x,
csm-id: 1(X),
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q
Lemmas referenced :
csm-adjoin_wf,
interval-type_wf,
cc-fst_wf,
csm-interval-type,
interval-rev_wf,
cube-context-adjoin_wf,
cc-snd_wf,
csm-composition_wf,
cubical_set_cumulativity-i-j,
subtype_rel_self,
composition-op_wf,
rev-type-line_wf,
cubical-type-cumulativity2,
fill-type-up_wf,
cubical-fun_wf,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type_wf,
cubical_set_wf,
cubical-term_wf,
squash_wf,
true_wf,
csm-ap-term_wf,
csm-cubical-fun,
dma-neg-dM0,
csm-id-adjoin_wf-interval-1,
rev-rev-type-line
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
instantiate,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
applyEquality,
universeIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
dependent_functionElimination,
setElimination,
rename,
productElimination,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
(fill-type-down(Gamma;A;cA) \mmember{} \{Gamma.\mBbbI{} \mvdash{} \_:(((A)[1(\mBbbI{})])p {}\mrightarrow{} A)\})
Date html generated:
2020_05_20-PM-04_55_22
Last ObjectModification:
2020_04_13-PM-02_48_50
Theory : cubical!type!theory
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