Nuprl Lemma : range_sublist
∀[T:Type]
∀L:T List. ∀n:ℕ. ∀f:ℕn ⟶ ℕ||L||.
∃L1:T List. ((||L1|| = n ∈ ℤ) c∧ sublist_occurence(T;L1;L;f)) supposing increasing(f;n)
Proof
Definitions occuring in Statement :
sublist_occurence: sublist_occurence(T;L1;L2;f),
length: ||as||,
list: T List,
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sublist_occurence: sublist_occurence(T;L1;L2;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
increasing: increasing(f;k),
less_than: a < b,
uiff: uiff(P;Q),
subtract: n - m,
sq_type: SQType(T),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
fadd: fadd(f;g)
Lemmas referenced :
list_induction,
all_wf,
nat_wf,
int_seg_wf,
length_wf,
isect_wf,
increasing_wf,
exists_wf,
list_wf,
equal_wf,
length_wf_nat,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
select_wf,
int_seg_properties,
itermConstant_wf,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
non_neg_length,
member-less_than,
length_of_nil_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-member-int_seg2,
subtract_wf,
nil_wf,
length_of_cons_lemma,
cons_wf,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
stuck-spread,
base_wf,
equal-wf-T-base,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
increasing_lower_bound,
itermAdd_wf,
int_term_value_add_lemma,
select-cons-hd,
select_cons_tl,
subtract-add-cancel,
fadd_increasing,
const_nondecreasing,
less_than_wf,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
hypothesis,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
cumulativity,
functionExtensionality,
applyEquality,
productEquality,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
dependent_set_memberEquality,
applyLambdaEquality,
independent_functionElimination,
universeEquality,
instantiate,
baseClosed,
imageElimination,
imageMemberEquality,
addEquality,
minusEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}||L||.
\mexists{}L1:T List. ((||L1|| = n) c\mwedge{} sublist\_occurence(T;L1;L;f)) supposing increasing(f;n)
Date html generated:
2017_10_01-AM-08_35_32
Last ObjectModification:
2017_07_26-PM-04_25_51
Theory : list!
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