Nuprl Lemma : equipollent-nat-subset
∀[T:Type]. ∀P:T ⟶ ℙ. ((∀x:T. Dec(P[x])) ⇒ (∀L:T List. ∃x:T. (P[x] ∧ (¬(x ∈ L)))) ⇒ ℕ ~ T ⇒ ℕ ~ {x:T| P[x]} )
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
l_member: (x ∈ l),
list: T List,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
inject: Inj(A;B;f)
Lemmas referenced :
equipollent_transitivity,
int_term_value_constant_lemma,
itermConstant_wf,
sq_stable_from_decidable,
biject_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
member_upto,
equal_wf,
subtype_rel_list,
member_map,
decidable__le,
le_wf,
iff_weakening_equal,
upto_wf,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
map_wf,
decidable_wf,
l_member_wf,
not_wf,
and_wf,
exists_wf,
list_wf,
all_wf,
equipollent_wf,
nat_wf,
equipollent-nat-decidable-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
isectElimination,
functionEquality,
cumulativity,
universeEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productEquality,
unionElimination,
voidElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
setEquality,
functionExtensionality,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbP{}
((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (\mforall{}L:T List. \mexists{}x:T. (P[x] \mwedge{} (\mneg{}(x \mmember{} L)))) {}\mRightarrow{} \mBbbN{} \msim{} T {}\mRightarrow{} \mBbbN{} \msim{} \{x:T| P[x]\} )
Date html generated:
2016_05_15-PM-05_28_59
Last ObjectModification:
2016_01_16-PM-00_32_00
Theory : general
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