Nuprl Lemma : trivial-constrained-term
∀[Gamma:j⊢]. ∀[B:{Gamma ⊢ _}]. ∀[xx:Top]. ({Gamma ⊢ _:B} ⊆r {Gamma ⊢ _:B[0(𝔽) |⟶ xx]})
Proof
Definitions occuring in Statement :
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
face-0: 0(𝔽),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
guard: {T},
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
context-subset: Gamma, phi,
uimplies: b supposing a,
face-0: 0(𝔽),
cubical-term-at: u(a),
not: ¬A,
false: False
Lemmas referenced :
cubical-term_wf,
context-subset_wf,
face-0_wf,
thin-context-subset,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
empty-context-subset-lemma3,
respects-equality-context-subset-term,
istype-top,
cubical-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
face-lattice-0-not-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
hypothesisEquality,
equalityIstype,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyEquality,
sqequalRule,
Error :memTop,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionExtensionality,
setElimination,
rename,
independent_isectElimination,
voidElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[B:\{Gamma \mvdash{} \_\}]. \mforall{}[xx:Top]. (\{Gamma \mvdash{} \_:B\} \msubseteq{}r \{Gamma \mvdash{} \_:B[0(\mBbbF{}) |{}\mrightarrow{} xx]\})
Date html generated:
2020_05_20-PM-04_52_35
Last ObjectModification:
2020_04_10-AM-11_29_39
Theory : cubical!type!theory
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