Nuprl Lemma : list_split_iseg2
∀[T:Type]
∀f:(T List) ⟶ 𝔹. ∀L1,L2:T List.
(L1 ≤ L2
⇒ (∀LL1,LL2:T List List. ∀X,Y:T List.
(((LL1 = LL2 ∈ (T List List)) ∧ X ≤ Y)
∨ (∃Z:T List. ∃ZZ:T List List. (((LL1 @ [Z / ZZ]) = LL2 ∈ (T List List)) ∧ X ≤ Z))) supposing
((list_split(f;L2) = <LL2, Y> ∈ (T List List × (T List))) and
(list_split(f;L1) = <LL1, X> ∈ (T List List × (T List))))))
Proof
Definitions occuring in Statement :
list_split: list_split(f;L),
iseg: l1 ≤ l2,
append: as @ bs,
cons: [a / b],
list: T List,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
guard: {T},
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
list_split_iseg,
or_wf,
equal_wf,
list_wf,
length_wf,
iseg_wf,
exists_wf,
append_wf,
cons_wf,
length-append,
iff_weakening_equal,
list_split_wf,
is_list_splitting_wf,
bool_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
rename,
spreadEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
applyLambdaEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
productElimination,
hyp_replacement,
because_Cache,
functionExtensionality,
applyEquality,
setElimination,
setEquality,
independent_pairEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:(T List) {}\mrightarrow{} \mBbbB{}. \mforall{}L1,L2:T List.
(L1 \mleq{} L2
{}\mRightarrow{} (\mforall{}LL1,LL2:T List List. \mforall{}X,Y:T List.
(((LL1 = LL2) \mwedge{} X \mleq{} Y)
\mvee{} (\mexists{}Z:T List. \mexists{}ZZ:T List List. (((LL1 @ [Z / ZZ]) = LL2) \mwedge{} X \mleq{} Z))) supposing
((list\_split(f;L2) = <LL2, Y>) and
(list\_split(f;L1) = <LL1, X>))))
Date html generated:
2018_05_21-PM-08_05_26
Last ObjectModification:
2017_07_26-PM-05_41_24
Theory : general
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