Nuprl Lemma : constrained-cubical-term-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[t:Top]. ∀[x:{Gamma ⊢ _:A}]. (x ∈ {Gamma ⊢ _:A[0(𝔽) |⟶ t]})
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,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
face-0: 0(𝔽),
cubical-term-at: u(a),
not: ¬A,
implies: P ⇒ Q,
false: False,
respects-equality: respects-equality(S;T),
guard: {T}
Lemmas referenced :
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
istype-top,
cubical-type_wf,
cubical_set_wf,
I_cube_wf,
context-subset_wf,
face-0_wf,
fset_wf,
nat_wf,
cubical-term-equal,
thin-context-subset,
context-subset-term-subtype,
I_cube_pair_redex_lemma,
face-lattice-0-not-1,
context-subset-term-0,
respects-equality-context-subset-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
hypothesisEquality,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
equalitySymmetry,
functionExtensionality,
because_Cache,
equalityTransitivity,
independent_isectElimination,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
independent_functionElimination,
voidElimination,
equalityIstype
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[t:Top]. \mforall{}[x:\{Gamma \mvdash{} \_:A\}]. (x \mmember{} \{Gamma \mvdash{} \_:A[0(\mBbbF{}) |{}\mrightarrow{} t]\})
Date html generated:
2020_05_20-PM-02_58_28
Last ObjectModification:
2020_04_04-PM-05_13_12
Theory : cubical!type!theory
Home
Index