Nuprl Lemma : finite-set-type
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. SqStable(P[x])) ⇒ (finite-type({x:T| P[x]} ) ⇐⇒ ∃L:T List. ∀x:T. (P[x] ⇐⇒ (x ∈ L))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
l_member: (x ∈ l),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
subtype_rel_list,
l_member_wf,
list_wf,
subtype_rel_self,
finite-type-iff-list,
finite-type_wf,
sq_stable_wf,
l_member-settype,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
list-set-type2,
l_all_iff,
less_than_wf,
length_wf,
select_member,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
Error :dependent_pairFormation_alt,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
setEquality,
hypothesis,
because_Cache,
sqequalRule,
independent_isectElimination,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :setIsType,
Error :universeIsType,
Error :inhabitedIsType,
Error :functionIsType,
Error :productIsType,
instantiate,
universeEquality,
independent_functionElimination,
promote_hyp,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
Error :equalityIsType1
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:T. SqStable(P[x])) {}\mRightarrow{} (finite-type(\{x:T| P[x]\} ) \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. \mforall{}x:T. (P[x] \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))))
Date html generated:
2019_06_20-PM-01_32_36
Last ObjectModification:
2018_10_15-PM-01_40_42
Theory : list_1
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