Nuprl Lemma : accum_filter_rel_wf
∀[T,A:Type]. ∀[a,b:A]. ∀[X:T List]. ∀[P:{x:T| (x ∈ X)} ⟶ ℙ]. ∀[f:A ⟶ {x:T| (x ∈ X)} ⟶ A].
(b = accum(z,x.f[z;x],a,{x∈X|P[x]}) ∈ ℙ)
Proof
Definitions occuring in Statement :
accum_filter_rel: b = accum(z,x.f[z; x],a,{x∈X|P[x]}),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
accum_filter_rel: b = accum(z,x.f[z; x],a,{x∈X|P[x]}),
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Lemmas referenced :
l_member_wf,
list_wf,
equal_wf,
list_accum_wf,
all_wf,
list-subtype,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
hypothesisEquality,
setEquality,
lemma_by_obid,
isectElimination,
thin,
isect_memberEquality,
because_Cache,
cumulativity,
universeEquality,
productEquality,
lambdaEquality,
applyEquality,
productElimination,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T,A:Type]. \mforall{}[a,b:A]. \mforall{}[X:T List]. \mforall{}[P:\{x:T| (x \mmember{} X)\} {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:A {}\mrightarrow{} \{x:T| (x \mmember{} X)\} {}\mrightarrow{} A].
(b = accum(z,x.f[z;x],a,\{x\mmember{}X|P[x]\}) \mmember{} \mBbbP{})
Date html generated:
2016_05_15-PM-04_32_54
Last ObjectModification:
2015_12_27-PM-02_48_40
Theory : general
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