Nuprl Lemma : CCC-omni
∀K:Type. (CCCNSet(K) ⇒ (∀P:K ⟶ ℙ. ((∀k:K. Dec(P[k])) ⇒ ((∃k:K. P[k]) ∨ (∀k:K. (¬P[k]))))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
ge: i ≥ j ,
contra-cc: CCC(T),
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_type: SQType(T),
prop: ℙ,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
or: P ∨ Q,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
decidable: Dec(P),
weakly-decidable-nset: WD(K),
exists: ∃x:A. B[x],
and: P ∧ Q,
ccc-nset: CCCNSet(K),
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
nat_properties,
le_wf,
decidable__not,
decidable__implies,
not_wf,
decidable__all_int_seg,
zero-le-nat,
subtype_rel_self,
int_formula_prop_le_lemma,
intformle_wf,
decidable__and2,
equal-wf-base,
decidable__exists_int_seg,
istype-universe,
ccc-nset_wf,
decidable_wf,
int_seg_wf,
lelt_wf,
set_subtype_base,
int_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermConstant_wf,
itermAdd_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
int_seg_properties,
istype-less_than,
istype-le,
decidable__lt,
nat_wf,
subtype_rel_transitivity,
istype-nat,
decidable__le,
ccc-nset-minimum,
ccc-nset-weakly-decidable
Rules used in proof :
Error :unionIsType,
unionEquality,
functionEquality,
baseClosed,
closedConclusion,
baseApply,
productEquality,
equalityTransitivity,
universeEquality,
addEquality,
equalitySymmetry,
sqequalBase,
Error :inrFormation_alt,
Error :inhabitedIsType,
Error :equalityIstype,
Error :functionIsType,
Error :inlFormation_alt,
cumulativity,
instantiate,
Error :universeIsType,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
natural_numberEquality,
imageElimination,
Error :productIsType,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
intEquality,
isectElimination,
rename,
setElimination,
Error :lambdaEquality_alt,
applyEquality,
sqequalRule,
productElimination,
because_Cache,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}K:Type. (CCCNSet(K) {}\mRightarrow{} (\mforall{}P:K {}\mrightarrow{} \mBbbP{}. ((\mforall{}k:K. Dec(P[k])) {}\mRightarrow{} ((\mexists{}k:K. P[k]) \mvee{} (\mforall{}k:K. (\mneg{}P[k]))))))
Date html generated:
2019_06_20-PM-03_02_50
Last ObjectModification:
2019_06_14-AM-09_51_32
Theory : continuity
Home
Index