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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