Nuprl Lemma : composition-structure-subset
∀[Y,X:j⊢]. ∀[B:{X ⊢ _}]. (X ⊢ Compositon(B) ⊆r Y ⊢ Compositon(B)) supposing sub_cubical_set{j:l}(Y; X)
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(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-structure: Gamma ⊢ Compositon(A),
prop: ℙ,
composition-function: composition-function{j:l,i:l}(Gamma;A),
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T},
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
cubical-type: {X ⊢ _},
csm-ap-term: (t)s,
interval-type: 𝕀,
csm+: tau+,
csm-comp: G o F,
interval-1: 1(𝕀),
csm-ap-type: (AF)s,
csm-ap: (s)x,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-adjoin: (s;u),
compose: f o g
Lemmas referenced :
uniform-comp-function_wf,
subset-cubical-type,
composition-structure_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,
cubical-term_wf,
face-type_wf,
cube_set_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
instantiate,
inhabitedIsType,
functionExtensionality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
dependent_functionElimination,
productElimination
Latex:
\mforall{}[Y,X:j\mvdash{}].
\mforall{}[B:\{X \mvdash{} \_\}]. (X \mvdash{} Compositon(B) \msubseteq{}r Y \mvdash{} Compositon(B)) supposing sub\_cubical\_set\{j:l\}(Y; X)
Date html generated:
2020_05_20-PM-04_23_13
Last ObjectModification:
2020_04_13-PM-00_34_00
Theory : cubical!type!theory
Home
Index