Nuprl Lemma : csm-path-ap-q_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ∀[B:{G ⊢ _}]. ∀[a,b:{G, phi ⊢ _:B}]. ∀[t:{G, phi ⊢ _:(Path_B a b)}].
(csm-path-ap-q(H;G;s;t) ∈ {H.𝕀, ((phi)p)s+ ⊢ _:((B)p)s+})
Proof
Definitions occuring in Statement :
csm-path-ap-q: csm-path-ap-q(H;G;s;t),
path-type: (Path_A a b),
context-subset: Gamma, phi,
face-type: 𝔽,
interval-type: 𝕀,
csm+: tau+,
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 ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
csm-path-ap-q: csm-path-ap-q(H;G;s;t),
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
csm+: tau+,
csm-comp: G o F,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
same-cubical-type: Gamma ⊢ A = B,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X)
Lemmas referenced :
csm-ap-term_wf,
context-subset_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
path-type_wf,
thin-context-subset,
context-subset-map,
csm+_wf_interval,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
istype-cubical-term,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
cubical-path-app_wf,
csm-context-subset-subtype2,
csm-ap-term-wf-subset,
face-term-implies-same,
cubical-term-eqcd,
csm-path-type,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
subset-cubical-type,
context-subset-is-subset,
cc-snd_wf,
subset-cubical-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
sqequalRule,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
applyEquality,
because_Cache,
independent_isectElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
lambdaEquality_alt,
hyp_replacement,
cumulativity,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G]. \mforall{}[B:\{G \mvdash{} \_\}]. \mforall{}[a,b:\{G, phi \mvdash{} \_:B\}].
\mforall{}[t:\{G, phi \mvdash{} \_:(Path\_B a b)\}].
(csm-path-ap-q(H;G;s;t) \mmember{} \{H.\mBbbI{}, ((phi)p)s+ \mvdash{} \_:((B)p)s+\})
Date html generated:
2020_05_20-PM-05_36_54
Last ObjectModification:
2020_04_18-PM-11_05_45
Theory : cubical!type!theory
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