Nuprl Lemma : csm-equivTerm
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ((equivTerm(G;A;B))s = equivTerm(H;(A)s;(B)s) ∈ {H ⊢ _:c𝕌})
Proof
Definitions occuring in Statement :
equivTerm: equivTerm(G;A;B),
cubical-universe: c𝕌,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
equivTerm: equivTerm(G;A;B),
all: ∀x:A. B[x]
Lemmas referenced :
csm-ap-term_wf,
cubical-universe_wf,
csm-cubical-universe,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
csm-cubical-equiv,
universe-decode_wf,
cubical-equiv_wf,
subtype_rel_self,
iff_weakening_equal,
csm-universe-decode,
csm-universe-encode,
equiv-comp_wf,
universe-comp-op_wf,
universe-encode_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cube_set_map_wf,
istype-cubical-universe-term,
cubical_set_wf,
csm-equiv-comp,
csm-composition_wf,
csm-universe-comp-op,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
because_Cache,
hypothesis,
sqequalRule,
Error :memTop,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
dependent_functionElimination,
hyp_replacement,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G]. ((equivTerm(G;A;B))s = equivTerm(H;(A)s;(B)s))
Date html generated:
2020_05_20-PM-07_33_50
Last ObjectModification:
2020_04_30-AM-10_02_23
Theory : cubical!type!theory
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