Nuprl Lemma : list_split_inverse
∀[T:Type]. ∀[f:(T List) ⟶ 𝔹]. ∀[L:T List]. ∀[LL:T List List]. ∀[X:T List].
L = (concat(LL) @ X) ∈ (T List) supposing list_split(f;L) = <LL, X> ∈ (T List List × (T List))
Proof
Definitions occuring in Statement :
list_split: list_split(f;L),
concat: concat(ll),
append: as @ bs,
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
top: Top,
pi1: fst(t),
pi2: snd(t),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
squash: ↓T,
true: True
Lemmas referenced :
bool_wf,
true_wf,
squash_wf,
append_wf,
concat_wf,
equal_wf,
and_wf,
length_wf,
pi2_wf,
top_wf,
subtype_rel_product,
pi1_wf_top,
is_list_splitting_wf,
list_wf,
set_wf,
list_split_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
productEquality,
sqequalRule,
lambdaEquality,
spreadEquality,
lambdaFormation,
setElimination,
rename,
productElimination,
independent_pairFormation,
applyEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
addLevel,
equalitySymmetry,
dependent_set_memberEquality,
setEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:(T List) {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[LL:T List List]. \mforall{}[X:T List].
L = (concat(LL) @ X) supposing list\_split(f;L) = <LL, X>
Date html generated:
2016_05_15-PM-05_52_36
Last ObjectModification:
2016_01_16-PM-00_33_24
Theory : general
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