Nuprl Lemma : list_split_prefix
∀[T:Type]. ∀[L:T List]. ∀[g:(T List) ⟶ 𝔹].
↑(g (snd(list_split(g;concat(fst(list_split(g;L))))))) supposing ¬↑null(fst(list_split(g;L)))
Proof
Definitions occuring in Statement :
list_split: list_split(f;L),
null: null(as),
concat: concat(ll),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s],
guard: {T},
list_split: list_split(f;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi1: fst(t),
concat: concat(ll),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi2: snd(t),
not: ¬A,
true: True,
false: False,
sq_stable: SqStable(P),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
last_induction,
uall_wf,
list_wf,
bool_wf,
isect_wf,
not_wf,
assert_wf,
null_wf3,
equal_wf,
subtype_rel_list,
top_wf,
list_split_wf,
concat_wf,
set_wf,
is_list_splitting_wf,
pi1_wf_top,
pi2_wf,
assert_witness,
subtype_rel_product,
list_accum_nil_lemma,
null_nil_lemma,
reduce_nil_lemma,
true_wf,
sq_stable__uall,
sq_stable__assert,
squash_wf,
list_accum_append,
list_accum_cons_lemma,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
assert_of_null,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
append_wf,
cons_wf,
nil_wf,
equal_functionality_wrt_subtype_rel2,
list_split_inverse,
concat_append,
reduce_cons_lemma,
append-nil
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesis,
because_Cache,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionExtensionality,
productEquality,
spreadEquality,
productElimination,
independent_pairEquality,
setElimination,
rename,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
equalityElimination,
independent_pairFormation,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[g:(T List) {}\mrightarrow{} \mBbbB{}].
\muparrow{}(g (snd(list\_split(g;concat(fst(list\_split(g;L))))))) supposing \mneg{}\muparrow{}null(fst(list\_split(g;L)))
Date html generated:
2018_05_21-PM-08_05_51
Last ObjectModification:
2017_07_26-PM-05_41_48
Theory : general
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