Nuprl Lemma : combine-list-member
∀[T:Type]
∀f:T ⟶ T ⟶ T. ∀L:T List.
((∀x,y:T. ((f[x;y] = x ∈ T) ∨ (f[x;y] = y ∈ T))) ⇒ 0 < ||L|| ⇒ (combine-list(x,y.f[x;y];L) ∈ L))
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
combine-list: combine-list(x,y.f[x; y];L),
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
list_induction,
all_wf,
l_member_wf,
list_accum_wf,
cons_wf,
list_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
cons_member,
equal_wf,
less_than_wf,
length_wf,
or_wf,
member_wf,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
lambdaEquality,
cumulativity,
applyEquality,
functionExtensionality,
independent_functionElimination,
rename,
because_Cache,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
inrFormation,
natural_numberEquality,
functionEquality,
universeEquality,
addLevel,
allFunctionality,
inlFormation,
orFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}L:T List.
((\mforall{}x,y:T. ((f[x;y] = x) \mvee{} (f[x;y] = y))) {}\mRightarrow{} 0 < ||L|| {}\mRightarrow{} (combine-list(x,y.f[x;y];L) \mmember{} L))
Date html generated:
2016_10_21-AM-09_49_46
Last ObjectModification:
2016_07_12-AM-05_09_20
Theory : list_0
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