Nuprl Lemma : ccc-nset-minimum
∀K:Type. (CCCNSet(K) ⇒ (∃n:K. ∀m:K. (n ≤ m)))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
squash: ↓T,
less_than: a < b,
cand: A c∧ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
contra-cc: CCC(T),
and: P ∧ Q,
ccc-nset: CCCNSet(K),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
add_nat_wf,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
primrec-wf2,
less_than_wf,
subtype_rel_self,
istype-less_than,
istype-le,
decidable__lt,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
lelt_wf,
subtype_base_sq,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
istype-universe,
ccc-nset_wf,
istype-nat,
nat_wf,
subtype_rel_transitivity,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
equal-wf-base
Rules used in proof :
imageElimination,
sqequalBase,
Error :equalityIstype,
addEquality,
Error :setIsType,
functionEquality,
hypothesis_subsumption,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
cumulativity,
unionElimination,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
universeEquality,
instantiate,
Error :universeIsType,
Error :functionIsType,
independent_functionElimination,
Error :inhabitedIsType,
setElimination,
because_Cache,
independent_isectElimination,
hypothesis,
natural_numberEquality,
intEquality,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
productEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
rename,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}K:Type. (CCCNSet(K) {}\mRightarrow{} (\mexists{}n:K. \mforall{}m:K. (n \mleq{} m)))
Date html generated:
2019_06_20-PM-03_01_40
Last ObjectModification:
2019_06_13-AM-11_28_18
Theory : continuity
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