Nuprl Lemma : combine-combine-list-right
∀[T:Type]
∀f:T ⟶ T ⟶ T. ∀L:T List.
((∀x,y,z:T. (f[x;f[y;z]] = f[y;z] ∈ T ⇐⇒ (f[x;y] = y ∈ T) ∨ (f[x;z] = z ∈ T)))
⇒ 0 < ||L||
⇒ (∀a:T. (f[a;combine-list(x,y.f[x;y];L)] = combine-list(x,y.f[x;y];L) ∈ T ⇐⇒ (∃b∈L. f[a;b] = b ∈ T))))
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
l_exists: (∃x∈L. P[x]),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
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_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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,
iff_wf,
equal_wf,
list_accum_wf,
l_exists_wf,
cons_wf,
l_member_wf,
list_wf,
list_accum_nil_lemma,
l_exists_single,
nil_wf,
list_accum_cons_lemma,
or_wf,
l_exists_cons,
less_than_wf,
length_wf
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,
because_Cache,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
independent_functionElimination,
independent_pairFormation,
independent_pairEquality,
axiomEquality,
addLevel,
allFunctionality,
impliesFunctionality,
orFunctionality,
levelHypothesis,
inlFormation,
inrFormation,
natural_numberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}L:T List.
((\mforall{}x,y,z:T. (f[x;f[y;z]] = f[y;z] \mLeftarrow{}{}\mRightarrow{} (f[x;y] = y) \mvee{} (f[x;z] = z)))
{}\mRightarrow{} 0 < ||L||
{}\mRightarrow{} (\mforall{}a:T
(f[a;combine-list(x,y.f[x;y];L)] = combine-list(x,y.f[x;y];L) \mLeftarrow{}{}\mRightarrow{} (\mexists{}b\mmember{}L. f[a;b] = b))))
Date html generated:
2017_04_17-AM-07_39_22
Last ObjectModification:
2017_02_27-PM-04_13_20
Theory : list_1
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