Nuprl Lemma : composition-function-subset
∀[Y,X:j⊢].
∀[B:{X ⊢ _}]. (composition-function{j:l,i:l}(X;B) ⊆r composition-function{j:l,i:l}(Y;B))
supposing sub_cubical_set{j:l}(Y; X)
Proof
Definitions occuring in Statement :
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-type: {X ⊢ _},
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
composition-function: composition-function{j:l,i:l}(Gamma;A),
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T}
Lemmas referenced :
composition-function_wf,
cubical-type_wf,
sub_cubical_set_wf,
cubical_set_wf,
cube_set_map_subtype3,
cube-context-adjoin_wf,
interval-type_wf,
sub_cubical_set_self,
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,
subset-cubical-type,
cubical-term_wf,
face-type_wf,
cube_set_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
inhabitedIsType,
functionExtensionality,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[Y,X:j\mvdash{}].
\mforall{}[B:\{X \mvdash{} \_\}]. (composition-function\{j:l,i:l\}(X;B) \msubseteq{}r composition-function\{j:l,i:l\}(Y;B))
supposing sub\_cubical\_set\{j:l\}(Y; X)
Date html generated:
2020_05_20-PM-04_22_58
Last ObjectModification:
2020_04_13-PM-00_33_41
Theory : cubical!type!theory
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