Nuprl Lemma : accum_split_iseg
∀[T,A:Type].
∀x:A. ∀g:(T List × A) ⟶ A. ∀f:(T List × A) ⟶ 𝔹. ∀L1,L2:T List.
(L1 ≤ L2
⇒ let LL1,X,z1 = accum_split(g;x;f;L1) in
let LL2,Y,z2 = accum_split(g;x;f;L2) in
((LL1 = LL2 ∈ ((T List × A) List)) ∧ X ≤ Y ∧ (z1 = z2 ∈ A))
∨ (∃Z:T List. ∃ZZ:(T List × A) List. (((LL1 @ [<Z, z1> / ZZ]) = LL2 ∈ ((T List × A) List)) ∧ X ≤ Z)))
Proof
Definitions occuring in Statement :
accum_split: accum_split(g;x;f;L),
iseg: l1 ≤ l2,
append: as @ bs,
cons: [a / b],
list: T List,
bool: 𝔹,
spreadn: spread3,
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
spreadn: spread3,
squash: ↓T,
prop: ℙ,
iseg: l1 ≤ l2,
exists: ∃x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
accum_split: accum_split(g;x;f;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
pi1: fst(t),
pi2: snd(t),
not: ¬A,
false: False,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3]
Lemmas referenced :
accum_split_wf,
iseg_wf,
list_wf,
bool_wf,
istype-universe,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
list_accum_append,
subtype_rel_list,
top_wf,
list_accum_wf,
list-cases,
null_nil_lemma,
cons_wf,
nil_wf,
product_subtype_list,
null_cons_lemma,
eqtt_to_assert,
append_wf,
last_induction,
all_wf,
or_wf,
length_wf,
exists_wf,
length-append,
list_accum_nil_lemma,
iseg_weakening,
list_accum_cons_lemma,
null_wf3,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
length_of_nil_lemma,
uiff_transitivity,
assert_of_null,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
pi1_wf_top,
pi2_wf,
iseg_nil,
nil_iseg,
append_assoc,
list_ind_cons_lemma,
iseg_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
setElimination,
rename,
productElimination,
applyLambdaEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
functionIsType,
productIsType,
instantiate,
universeEquality,
applyEquality,
lambdaEquality_alt,
productEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
Error :memTop,
hyp_replacement,
unionElimination,
independent_pairEquality,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
functionEquality,
unionIsType,
inlFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
voidElimination,
inrFormation_alt,
dependent_pairFormation_alt
Latex:
\mforall{}[T,A:Type].
\mforall{}x:A. \mforall{}g:(T List \mtimes{} A) {}\mrightarrow{} A. \mforall{}f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}. \mforall{}L1,L2:T List.
(L1 \mleq{} L2
{}\mRightarrow{} let LL1,X,z1 = accum\_split(g;x;f;L1) in
let LL2,Y,z2 = accum\_split(g;x;f;L2) in
((LL1 = LL2) \mwedge{} X \mleq{} Y \mwedge{} (z1 = z2))
\mvee{} (\mexists{}Z:T List. \mexists{}ZZ:(T List \mtimes{} A) List. (((LL1 @ [<Z, z1> / ZZ]) = LL2) \mwedge{} X \mleq{} Z)))
Date html generated:
2020_05_20-AM-08_12_25
Last ObjectModification:
2020_01_08-PM-02_52_19
Theory : general
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