Nuprl Lemma : context-subset-map-equal
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[X:j⊢]. ∀[f,g:X j⟶ H.𝕀]. ((f = g ∈ X j⟶ H.𝕀) ⇒ (f = g ∈ X, ((phi)p)f j⟶ H, phi.𝕀))
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
cube-context-adjoin_wf,
interval-type_wf,
context-subset-map,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
cube_set_map_wf,
cubical-term_wf,
cubical_set_wf,
squash_wf,
true_wf,
context-subset_wf,
csm-subtype-iso-instance1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
Error :memTop,
universeIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
productElimination,
applyLambdaEquality
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[X:j\mvdash{}]. \mforall{}[f,g:X j{}\mrightarrow{} H.\mBbbI{}]. ((f = g) {}\mRightarrow{} (f = g))
Date html generated:
2020_05_20-PM-03_06_16
Last ObjectModification:
2020_04_06-PM-00_49_14
Theory : cubical!type!theory
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