Nuprl Lemma : equipollent-nat-decidable-subset
∀P:ℕ ⟶ ℙ. ((∀n:ℕ. Dec(P[n])) ⇒ (∀m:ℕ. ∃n:ℕ. (P[n] ∧ (m ≤ n))) ⇒ ℕ ~ {n:ℕ| P[n]} )
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
isl: isl(x),
iff: P ⇐⇒ Q,
and: P ∧ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
false: False,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
true: True,
not: ¬A,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
equipollent: A ~ B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
less_than': less_than'(a;b),
bool-size: 𝔹size(k;f),
enumerate: enumerate(P;n),
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
bnot: ¬bb,
cand: A c∧ B,
has-value: (a)↓,
sq_stable: SqStable(P)
Lemmas referenced :
btrue_wf,
bfalse_wf,
istype-nat,
istype-true,
istype-void,
istype-assert,
istype-le,
subtype_rel_self,
decidable_wf,
enumerate_wf,
subtype_rel_sets_simple,
nat_wf,
assert_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
biject_wf,
decidable__lt,
enumerate-increases,
subtype_base_sq,
nat_properties,
decidable__equal_nat,
full-omega-unsat,
intformless_wf,
itermVar_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
intformand_wf,
intformnot_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__assert,
bool-size_wf,
subtype_rel_function,
bool_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
ge_wf,
istype-less_than,
subtract-1-ge-0,
primrec0_lemma,
zero-add,
mu_wf,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
minus-one-mul,
add-commutes,
add-associates,
add-mul-special,
zero-mul,
primrec-unroll,
lt_int_wf,
uiff_transitivity,
equal-wf-base,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
le_int_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal-wf-T-base,
not_wf,
assert_of_bnot,
int_seg_properties,
add-is-int-iff,
false_wf,
mu-unroll,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
add-zero,
assert_elim,
not_assert_elim,
btrue_neq_bfalse,
value-type-has-value,
int-value-type,
add-swap,
equal_wf,
true_wf,
istype-universe,
add_nat_wf,
bool_cases,
sq_stable__assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
rename,
sqequalHypSubstitution,
sqequalRule,
dependent_pairFormation_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
inhabitedIsType,
hypothesis,
thin,
unionElimination,
introduction,
extract_by_obid,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
voidElimination,
isectElimination,
because_Cache,
natural_numberEquality,
universeIsType,
functionIsType,
productIsType,
productElimination,
promote_hyp,
setElimination,
instantiate,
universeEquality,
independent_isectElimination,
setIsType,
intEquality,
sqequalBase,
setEquality,
cumulativity,
imageElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
dependent_set_memberEquality_alt,
addEquality,
intWeakElimination,
axiomEquality,
functionIsTypeImplies,
productEquality,
functionExtensionality_alt,
imageMemberEquality,
baseClosed,
equalityElimination,
baseApply,
closedConclusion,
applyLambdaEquality,
pointwiseFunctionality,
minusEquality,
callbyvalueReduce,
hyp_replacement,
functionEquality
Latex:
\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}n:\mBbbN{}. Dec(P[n])) {}\mRightarrow{} (\mforall{}m:\mBbbN{}. \mexists{}n:\mBbbN{}. (P[n] \mwedge{} (m \mleq{} n))) {}\mRightarrow{} \mBbbN{} \msim{} \{n:\mBbbN{}| P[n]\} )
Date html generated:
2020_05_20-AM-08_11_08
Last ObjectModification:
2020_01_04-PM-11_12_30
Theory : general
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