Nuprl Lemma : context-adjoin-subset4
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}].
∀T:{H ⊢j _}. ∀[psi:{H.T ⊢ _:𝔽}]. ((psi = (phi)p ∈ {H.T ⊢ _:𝔽}) ⇒ sub_cubical_set{j:l}(H, phi.T; H.T, psi))
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
context-adjoin-subset3,
csm-ap-term_wf,
cube-context-adjoin_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
cubical-term_wf,
cubical-type_wf,
cubical_set_wf,
context-subset_wf,
subset-cubical-type,
context-subset-is-subset,
sub_cubical_set_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
cut,
instantiate,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
equalityIstype,
because_Cache,
sqequalRule,
Error :memTop,
universeIsType,
applyEquality,
independent_isectElimination,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}].
\mforall{}T:\{H \mvdash{}j \_\}. \mforall{}[psi:\{H.T \mvdash{} \_:\mBbbF{}\}]. ((psi = (phi)p) {}\mRightarrow{} sub\_cubical\_set\{j:l\}(H, phi.T; H.T, psi))
Date html generated:
2020_05_20-PM-03_05_08
Last ObjectModification:
2020_04_13-PM-05_49_47
Theory : cubical!type!theory
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