Nuprl Lemma : transport-fun_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)].
(transport-fun(Gamma;A;cA) ∈ {Gamma ⊢ _:((A)[0(𝕀)] ⟶ (A)[1(𝕀)])})
Proof
Definitions occuring in Statement :
transport-fun: transport-fun(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
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],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
transport-fun: transport-fun(Gamma;A;cA),
cubical-type: {X ⊢ _},
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),
cc-snd: q,
constant-cubical-type: (X),
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
interval-1: 1(𝕀),
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
true: True
Lemmas referenced :
composition-op_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_cumulativity-i-j,
cubical_set_wf,
cubical-lambda_wf,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
csm-id-adjoin_wf-interval-1,
cc-fst_wf,
transport_wf,
csm+_wf_interval,
csm-composition_wf,
cc-snd_wf,
cubical-fun-as-cubical-pi,
cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
inhabitedIsType,
lambdaFormation_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
cumulativity,
universeEquality,
equalityIstype,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
(transport-fun(Gamma;A;cA) \mmember{} \{Gamma \mvdash{} \_:((A)[0(\mBbbI{})] {}\mrightarrow{} (A)[1(\mBbbI{})])\})
Date html generated:
2020_05_20-PM-04_18_31
Last ObjectModification:
2020_04_11-PM-06_37_44
Theory : cubical!type!theory
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