Nuprl Lemma : csm-comp-structure_wf
∀[Gamma,Delta:j⊢]. ∀[tau:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
((cA)tau ∈ Delta ⊢ Compositon((A)tau))
Proof
Definitions occuring in Statement :
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(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],
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
csm-comp-structure: (cA)tau,
composition-function: composition-function{j:l,i:l}(Gamma;A),
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
uimplies: b supposing a,
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
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),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
interval-type: 𝕀,
csm-comp: G o F,
csm+: tau+,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
prop: ℙ
Lemmas referenced :
composition-structure_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
cube-context-adjoin_wf,
interval-type_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
csm-ap-term_wf,
context-subset_wf,
csm-context-subset-subtype3,
cubical-term-eqcd,
face-type_wf,
csm-comp-type,
csm-id-adjoin_wf-interval-1,
csm-comp_wf,
subtype_rel_self,
cubical-term_wf,
uniform-comp-function_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
inhabitedIsType,
instantiate,
functionExtensionality,
sqequalRule,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
Error :memTop,
cumulativity,
lambdaFormation_alt,
dependent_functionElimination
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}]. \mforall{}[tau:Delta j{}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)].
((cA)tau \mmember{} Delta \mvdash{} Compositon((A)tau))
Date html generated:
2020_05_20-PM-04_35_22
Last ObjectModification:
2020_04_19-PM-01_54_35
Theory : cubical!type!theory
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