Nuprl Lemma : append_split2
∀[T:Type]
∀L:T List
∀[P:ℕ||L|| ⟶ ℙ]
((∀x:ℕ||L||. Dec(P x))
⇒ (∀i,j:ℕ||L||. ((P i) ⇒ P j supposing i < j))
⇒ (∃L_1,L_2:T List. ((L = (L_1 @ L_2) ∈ (T List)) ∧ (∀i:ℕ||L||. (P i ⇐⇒ ||L_1|| ≤ i)))))
Proof
Definitions occuring in Statement :
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
guard: {T},
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
so_apply: x[s],
cand: A c∧ B,
int_iseg: {i...j},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
nat: ℕ,
less_than': less_than'(a;b),
ge: i ≥ j ,
label: ...$L... t
Lemmas referenced :
length_wf,
all_wf,
int_seg_wf,
not_wf,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__exists_int_seg,
decidable__and2,
decidable__all_int_seg,
decidable__not,
less_than_wf,
decidable_wf,
list_wf,
firstn_wf,
nth_tl_wf,
equal_wf,
squash_wf,
true_wf,
append_firstn_lastn,
subtype_rel_sets,
le_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
iff_weakening_equal,
less_than'_wf,
append_wf,
length-append,
length_firstn,
exists_wf,
iff_wf,
decidable__equal_int,
itermConstant_wf,
intformeq_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
nil_wf,
append_back_nil,
nat_wf,
false_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_wf,
primrec-wf2,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
natural_numberEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
because_Cache,
sqequalRule,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
isect_memberFormation,
lambdaFormation,
instantiate,
independent_functionElimination,
functionEquality,
universeEquality,
isectEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
axiomEquality,
hyp_replacement,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||. Dec(P x))
{}\mRightarrow{} (\mforall{}i,j:\mBbbN{}||L||. ((P i) {}\mRightarrow{} P j supposing i < j))
{}\mRightarrow{} (\mexists{}L$_{1}$,L$_{2}$:T List. ((L = (L$_{1\mbackslash{}\000Cff7d$ @ L$_{2}$)) \mwedge{} (\mforall{}i:\mBbbN{}||L||. (P i \mLeftarrow{}{}\mRightarrow{} ||L$_{1}$|| \mleq{} i\000C)))))
Date html generated:
2017_04_14-AM-09_25_30
Last ObjectModification:
2017_02_27-PM-04_00_30
Theory : list_1
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