Nuprl Lemma : compU_wf
∀[G:j⊢]. (compU() ∈ G ⊢ Compositon'(c𝕌))
Proof
Definitions occuring in Statement :
compU: compU(),
cubical-universe: c𝕌,
composition-structure: Gamma ⊢ Compositon(A),
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),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
prop: ℙ,
compU: compU(),
squash: ↓T,
true: True,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
same-cubical-type: Gamma ⊢ A = B,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
rev-type-line: (A)-,
interval-rev: 1-(r),
cubical-term-at: u(a),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
universe-comp-op: compOp(t),
rev-type-line-comp: (cA)-,
csm-composition: (comp)sigma,
csm-comp-structure: (cA)tau,
composition-function: composition-function{j:l,i:l}(Gamma;A)
Lemmas referenced :
compU_wf1,
csm-cubical-universe,
cubical-term_wf,
cube-context-adjoin_wf,
context-subset_wf,
interval-type_wf,
cubical-universe_wf,
istype-cubical-term,
face-type_wf,
cube_set_map_wf,
uniform-comp-function_wf,
cubical_set_wf,
rev-type-line_wf,
universe-decode_wf,
equivU_wf,
rev-type-line-comp_wf,
universe-comp-op_wf,
csm-ap-term_wf,
csm-id-adjoin_wf,
interval-1_wf,
csm-universe-decode,
cubical-equiv_wf,
squash_wf,
true_wf,
cubical-type_wf,
cubical-term-eqcd,
equiv-fun_wf,
thin-context-subset,
glue-type_wf,
glue-type-constraint,
comp-op-to-comp-fun_wf2,
cubical_set_cumulativity-i-j,
csm-composition_wf,
glue-comp_wf2,
rev-type-line-0,
rev-type-line-1,
cubical-universe-cumulativity,
csm-universe-encode,
comp-fun-to-comp-op_wf,
csm-ap-type_wf,
csm-face-type,
context-subset-map,
csm-equivU,
dma-neg-dM0,
dma-neg-dM1,
istype-cubical-universe-term,
csm-ap-term-universe,
composition-op_wf,
cubical-type-cumulativity2,
equal_wf,
istype-universe,
csm-rev-type-line,
csm+_wf_interval,
subtype_rel_self,
iff_weakening_equal,
csm-equiv-fun,
csm-glue-type,
cubical-fun_wf,
universe-encode_wf,
csm-glue-comp,
subtype_rel-equal,
csm-comp-structure_wf,
cube_set_map_cumulativity-i-j,
csm-universe-comp-op,
csm-comp-op-to-comp-fun-sq,
composition-structure_wf,
csm-comp-fun-to-comp-op,
subset-cubical-term,
context-subset-is-subset,
csm-id-adjoin_wf-interval-1,
interval-0_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaFormation_alt,
sqequalRule,
Error :memTop,
because_Cache,
instantiate,
universeIsType,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
rename,
setElimination,
applyLambdaEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
cumulativity,
universeEquality,
hyp_replacement,
dependent_functionElimination,
equalityIstype,
independent_functionElimination,
productElimination,
independent_pairFormation,
productIsType,
functionExtensionality,
setEquality,
setIsType
Latex:
\mforall{}[G:j\mvdash{}]. (compU() \mmember{} G \mvdash{} Compositon'(c\mBbbU{}))
Date html generated:
2020_05_20-PM-07_23_17
Last ObjectModification:
2020_04_28-PM-00_05_23
Theory : cubical!type!theory
Home
Index