Nuprl Lemma : csm-rev-type-line-comp
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G.𝕀 ⊢ _}]. ∀[cA:G.𝕀 ⊢ CompOp(A)].
(((cA)-)tau+ = ((cA)tau+)- ∈ K.𝕀 ⊢ CompOp(((A)-)tau+))
Proof
Definitions occuring in Statement :
rev-type-line-comp: (cA)-,
rev-type-line: (A)-,
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
interval-type: 𝕀,
csm+: tau+,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rev-type-line-comp: (cA)-,
rev-type-line: (A)-,
subtype_rel: A ⊆r B,
csm-composition: (comp)sigma,
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
cc-snd: q,
interval-rev: 1-(r),
cc-fst: p,
csm-adjoin: (s;u),
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-comp: G o F,
cubical-term-at: u(a),
pi2: snd(t),
pi1: fst(t),
compose: f o g,
uimplies: b supposing a,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C)
Lemmas referenced :
composition-op_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
interval-rev_wf,
cc-snd_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
cc-fst_wf,
csm-interval-type,
csm-composition_wf,
csm+_wf_interval,
subtype_rel_self,
csm-adjoin_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
Error :memTop
Latex:
\mforall{}[G,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} G]. \mforall{}[A:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:G.\mBbbI{} \mvdash{} CompOp(A)]. (((cA)-)tau+ = ((cA)tau+)-)
Date html generated:
2020_05_20-PM-04_18_02
Last ObjectModification:
2020_04_10-AM-04_52_19
Theory : cubical!type!theory
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