### Nuprl Lemma : length-filter-le

`∀[T:Type]. ∀[P1,P2:T ⟶ 𝔹]. ∀[L:T List].  ||filter(P1;L)|| ≤ ||filter(P2;L)|| supposing (∀x∈L.(↑(P1 x)) `` (↑(P2 x)))`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` length: `||as||` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` all: `∀x:A. B[x]` prop: `ℙ` implies: `P `` Q` so_apply: `x[s]` subtype_rel: `A ⊆r B` top: `Top` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` iff: `P `⇐⇒` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  list_induction isect_wf l_all_wf l_member_wf assert_wf le_wf length_wf filter_wf5 subtype_rel_dep_function bool_wf subtype_rel_self set_wf list_wf filter_nil_lemma length_of_nil_lemma false_wf less_than'_wf l_all_wf_nil filter_cons_lemma l_all_cons eqtt_to_assert length_of_cons_lemma decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermAdd_wf itermVar_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_add_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot cons_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity lambdaFormation hypothesis setElimination rename functionEquality applyEquality functionExtensionality because_Cache setEquality independent_isectElimination independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation natural_numberEquality productElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry unionElimination equalityElimination addEquality dependent_pairFormation int_eqEquality intEquality computeAll promote_hyp instantiate universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P1,P2:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].
||filter(P1;L)||  \mleq{}  ||filter(P2;L)||  supposing  (\mforall{}x\mmember{}L.(\muparrow{}(P1  x))  {}\mRightarrow{}  (\muparrow{}(P2  x)))

Date html generated: 2017_04_17-AM-07_48_24
Last ObjectModification: 2017_02_27-PM-04_22_52

Theory : list_1

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