Nuprl Lemma : split-gap_wf
∀[T:Type]
∀f:T ⟶ ℤ. ∀L:T List.
(split-gap(f;L) ∈ ∃XY:T List × (T List) [let X,Y = XY
in (L = (X @ Y) ∈ (T List))
∧ (∀i:ℕ||X|| - 1. ((f X[i + 1]) = ((f X[i]) + 1) ∈ ℤ))
∧ ((¬↑null(L))
⇒ ((¬↑null(X))
∧ ¬((f hd(Y)) = ((f last(X)) + 1) ∈ ℤ) supposing ||Y|| ≥ 1 ))])
Proof
Definitions occuring in Statement :
split-gap: split-gap(f;L),
last: last(L),
select: L[n],
hd: hd(l),
length: ||as||,
null: null(as),
append: as @ bs,
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
int_seg: {i..j-},
ge: i ≥ j ,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q),
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
split-gap: split-gap(f;L),
list_ind: list_ind,
split-at-first-gap-ext
Lemmas referenced :
split-at-first-gap-ext,
list_wf,
sq_exists_wf,
equal_wf,
append_wf,
length_wf,
length-append,
int_seg_wf,
subtract_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
ge_wf,
hd_wf,
last_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
isectEquality,
universeEquality,
functionEquality,
cumulativity,
intEquality,
sqequalHypSubstitution,
productEquality,
because_Cache,
productElimination,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
functionExtensionality,
addEquality,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
axiomEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} \mBbbZ{}. \mforall{}L:T List.
(split-gap(f;L) \mmember{} \mexists{}XY:T List \mtimes{} (T List) [let X,Y = XY
in (L = (X @ Y))
\mwedge{} (\mforall{}i:\mBbbN{}||X|| - 1. ((f X[i + 1]) = ((f X[i]) + 1)))
\mwedge{} ((\mneg{}\muparrow{}null(L))
{}\mRightarrow{} ((\mneg{}\muparrow{}null(X))
\mwedge{} \mneg{}((f hd(Y)) = ((f last(X)) + 1))
supposing ||Y|| \mgeq{} 1 ))])
Date html generated:
2018_05_21-PM-07_40_39
Last ObjectModification:
2017_07_26-PM-05_14_45
Theory : general
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