Nuprl Lemma : equivU_wf
∀[G:j⊢]. ∀[E:{G.𝕀 ⊢ _}]. ∀[cE:G.𝕀 ⊢ CompOp(E)]. (equivU(G;E;cE) ∈ {G ⊢ _:Equiv((E)[0(𝕀)];(E)[1(𝕀)])})
Proof
Definitions occuring in Statement :
equivU: equivU(G;E;cE),
composition-op: Gamma ⊢ CompOp(A),
cubical-equiv: Equiv(T;A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
cc-fst: p,
interval-1: 1(𝕀),
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
pi1: fst(t),
equivU: equivU(G;E;cE),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
cc-fst_wf_interval,
transport_wf,
cubical-equiv_wf,
csm-cubical-equiv,
subset-cubical-term2,
sub_cubical_set_self,
istype-cubical-term,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
equiv-comp_wf,
csm-composition_wf,
cubical-id-equiv_wf,
interval-1_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
subtype_rel_self,
iff_weakening_equal,
csm-ap-type-fst-id-adjoin,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
sqequalRule,
universeIsType,
setElimination,
rename,
productElimination,
dependent_functionElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
inhabitedIsType,
Error :memTop
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[E:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cE:G.\mBbbI{} \mvdash{} CompOp(E)].
(equivU(G;E;cE) \mmember{} \{G \mvdash{} \_:Equiv((E)[0(\mBbbI{})];(E)[1(\mBbbI{})])\})
Date html generated:
2020_05_20-PM-07_21_07
Last ObjectModification:
2020_04_25-PM-09_52_26
Theory : cubical!type!theory
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