### Nuprl Lemma : l_all-set

`∀[T:Type]. ∀[P:T ⟶ ℙ].  ((∀x:T. SqStable(P[x])) `` (∀L:{x:T| P[x]}  List. (∀x∈L.P[x])))`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` list: `T List` sq_stable: `SqStable(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` 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` all: `∀x:A. B[x]` l_all: `(∀x∈L.P[x])` member: `t ∈ T` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` so_lambda: `λ2x.t[x]` sq_stable: `SqStable(P)`
Lemmas referenced :  sq_stable_wf all_wf list_wf int_seg_wf equal_wf set_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le length_wf int_seg_properties select_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin setEquality cumulativity hypothesisEquality applyEquality hypothesis because_Cache sqequalRule setElimination rename independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination introduction imageMemberEquality baseClosed equalityTransitivity equalitySymmetry universeEquality functionEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    ((\mforall{}x:T.  SqStable(P[x]))  {}\mRightarrow{}  (\mforall{}L:\{x:T|  P[x]\}    List.  (\mforall{}x\mmember{}L.P[x])))

Date html generated: 2016_05_14-AM-07_49_55
Last ObjectModification: 2016_01_15-AM-08_32_06

Theory : list_1

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