### Nuprl Lemma : filter-filter

`∀[P1,P2,L:Top].  (filter(P2;filter(P1;L)) ~ filter(λt.((P1 t) ∧b (P2 t));L))`

Proof

Definitions occuring in Statement :  filter: `filter(P;l)` band: `p ∧b q` uall: `∀[x:A]. B[x]` top: `Top` apply: `f a` lambda: `λx.A[x]` sqequal: `s ~ t`
Definitions unfolded in proof :  band: `p ∧b q` filter: `filter(P;l)` reduce: `reduce(f;k;as)` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` list_ind: list_ind nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` ifthenelse: `if b then t else f fi ` cons: `[a / b]` nil: `[]` it: `⋅` fun_exp: `f^n` primrec: `primrec(n;b;c)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` nat_plus: `ℕ+` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` strict4: `strict4(F)` has-value: `(a)↓` guard: `{T}` squash: `↓T` subtype_rel: `A ⊆r B` bfalse: `ff`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf base_wf strictness-apply strictness-callbyvalue bottom-sqle decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma fun_exp_unroll_1 top_wf lifting-strict-callbyvalue has-value_wf_base is-exception_wf cbv_sqle int_subtype_base has-value-implies-dec-ispair-2 lifting-strict-ifthenelse has-value-ifthenelse-type equal_wf has-value-implies-dec-isaxiom-2 cbv_bottom_lemma
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut thin sqequalSqle lambdaFormation fixpointLeast extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomSqleEquality unionElimination dependent_set_memberEquality because_Cache sqequalAxiom baseClosed callbyvalueCallbyvalue callbyvalueReduce baseApply closedConclusion callbyvalueExceptionCases inrFormation imageMemberEquality imageElimination exceptionSqequal inlFormation applyEquality equalityTransitivity equalitySymmetry unionEquality sqleReflexivity decideExceptionCases sqleRule divergentSqle

Latex:
\mforall{}[P1,P2,L:Top].    (filter(P2;filter(P1;L))  \msim{}  filter(\mlambda{}t.((P1  t)  \mwedge{}\msubb{}  (P2  t));L))

Date html generated: 2017_04_14-AM-09_27_10
Last ObjectModification: 2017_02_27-PM-04_01_21

Theory : list_1

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