### 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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