Nuprl Lemma : accum_split_prefix
∀[A,T:Type]. ∀[x:A]. ∀[g:(T List × A) ⟶ A]. ∀[f:(T List × A) ⟶ 𝔹]. ∀[L:T List].
↑(f (snd(accum_split(g;x;f;concat(map(λp.(fst(p));fst(accum_split(g;x;f;L))))))))
supposing ¬↑null(fst(accum_split(g;x;f;L)))
Proof
Definitions occuring in Statement :
accum_split: accum_split(g;x;f;L),
null: null(as),
concat: concat(ll),
map: map(f;as),
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,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s],
all: ∀x:A. B[x],
pi1: fst(t),
guard: {T},
accum_split: accum_split(g;x;f;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
concat: concat(ll),
pi2: snd(t),
not: ¬A,
true: True,
false: False,
spreadn: spread3,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T
Lemmas referenced :
last_induction,
not_wf,
assert_wf,
null_wf3,
list_wf,
accum_split_wf,
concat_wf,
map_wf,
pi1_wf_top,
set_wf,
is_accum_splitting_wf,
equal_wf,
pi2_wf,
assert_witness,
subtype_rel_product,
top_wf,
subtype_rel_list,
bool_wf,
list_accum_nil_lemma,
null_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
true_wf,
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,
list_accum_append,
accum_split_inverse,
map_append_sq,
map_cons_lemma,
concat_append,
concat-single,
squash_wf,
append_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
applyEquality,
hypothesis,
functionExtensionality,
productEquality,
cumulativity,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
spreadEquality,
lambdaFormation,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
universeEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
baseClosed,
independent_pairFormation,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality,
setEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[x:A]. \mforall{}[g:(T List \mtimes{} A) {}\mrightarrow{} A]. \mforall{}[f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
\muparrow{}(f (snd(accum\_split(g;x;f;concat(map(\mlambda{}p.(fst(p));fst(accum\_split(g;x;f;L))))))))
supposing \mneg{}\muparrow{}null(fst(accum\_split(g;x;f;L)))
Date html generated:
2018_05_21-PM-08_07_23
Last ObjectModification:
2017_07_26-PM-05_43_18
Theory : general
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