Nuprl Lemma : combine-combine-list-left
∀[T:Type]
∀f:T ⟶ T ⟶ T. ∀L:T List.
(∀a:T. uiff(f[a;combine-list(x,y.f[x;y];L)] = a ∈ T;(∀b∈L.f[a;b] = a ∈ T))) supposing
(0 < ||L|| and
(∀x,y,z:T. (f[x;f[y;z]] = x ∈ T ⇐⇒ (f[x;y] = x ∈ T) ∧ (f[x;z] = x ∈ T))))
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
less_than: a < b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: 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],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
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],
implies: P ⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
cand: A c∧ B,
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,
uiff_wf,
equal_wf,
list_accum_wf,
l_all_wf,
cons_wf,
l_member_wf,
list_wf,
list_accum_nil_lemma,
l_all_single,
int_seg_wf,
length_wf,
nil_wf,
list_accum_cons_lemma,
iff_weakening_uiff,
less_than_wf,
iff_wf,
combine-list_wf,
l_all_cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesisEquality,
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,
addLevel,
allFunctionality,
independent_isectElimination,
axiomEquality,
natural_numberEquality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}L:T List.
(\mforall{}a:T. uiff(f[a;combine-list(x,y.f[x;y];L)] = a;(\mforall{}b\mmember{}L.f[a;b] = a))) supposing
(0 < ||L|| and
(\mforall{}x,y,z:T. (f[x;f[y;z]] = x \mLeftarrow{}{}\mRightarrow{} (f[x;y] = x) \mwedge{} (f[x;z] = x))))
Date html generated:
2017_04_17-AM-07_39_13
Last ObjectModification:
2017_02_27-PM-04_13_14
Theory : list_1
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