Nuprl Lemma : ctt-level-type-subtype
∀[X:⊢''']. ∀[lvl:ℕ4]. ({X ⊢lvl _} ⊆r {X ⊢''' _})
Proof
Definitions occuring in Statement :
ctt-level-type: {X ⊢lvl _},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
ctt-level-type: {X ⊢lvl _},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
nat: ℕ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
cubical-type-cumulativity2,
int_seg_subtype_special,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
ctt-level-type_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
int_seg_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
hypothesisEquality,
sqequalRule,
applyEquality,
hypothesis_subsumption,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
dependent_set_memberEquality_alt,
imageElimination
Latex:
\mforall{}[X:\mvdash{}''']. \mforall{}[lvl:\mBbbN{}4]. (\{X \mvdash{}lvl \_\} \msubseteq{}r \{X \mvdash{}''' \_\})
Date html generated:
2020_05_20-PM-07_46_11
Last ObjectModification:
2020_05_04-PM-05_03_47
Theory : cubical!type!theory
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