Nuprl Lemma : cubical-sigma_wf-level-type
∀[K:⊢''']. ∀[a,b:ℕ4]. ∀[A:{K ⊢a _}]. ∀[B:{K.A ⊢b _}]. (Σ A B ∈ K ⊢levelsup(a;b) )
Proof
Definitions occuring in Statement :
levelsup: levelsup(x;y),
ctt-level-type: {X ⊢lvl _},
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical_set: CubicalSet,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
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,
levelsup: levelsup(x;y),
imax: imax(a;b),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
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-sigma_wf,
int_seg_subtype_special,
int_seg_cases,
cubical-type-cumulativity,
cubical-type-cumulativity2,
cube-context-adjoin_wf,
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,
cube-context-adjoin_wf-level-type,
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,
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,
hypothesis_subsumption,
applyEquality,
voidElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
dependent_set_memberEquality_alt,
imageElimination
Latex:
\mforall{}[K:\mvdash{}''']. \mforall{}[a,b:\mBbbN{}4]. \mforall{}[A:\{K \mvdash{}a \_\}]. \mforall{}[B:\{K.A \mvdash{}b \_\}]. (\mSigma{} A B \mmember{} K \mvdash{}levelsup(a;b) )
Date html generated:
2020_05_20-PM-07_49_06
Last ObjectModification:
2020_05_07-AM-11_19_16
Theory : cubical!type!theory
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