Nuprl Lemma : length-concat-lower-bound
∀[ll:Top List+ List]. (||ll|| ≤ ||concat(ll)||)
Proof
Definitions occuring in Statement :
listp: A List+,
length: ||as||,
concat: concat(ll),
list: T List,
uall: ∀[x:A]. B[x],
top: Top,
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
not: ¬A,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
listp: A List+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
l_member: (x ∈ l),
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
less_than': less_than'(a;b),
cons: [a / b],
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subtract: n - m
Lemmas referenced :
le_wf,
length-concat,
iff_weakening_equal,
less_than'_wf,
length_wf,
top_wf,
concat_wf,
listp_wf,
list_wf,
l_sum-lower-bound,
map_wf,
non_neg_length,
nat_wf,
length_wf_nat,
map_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_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_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
l_all_iff,
l_member_wf,
select-map,
subtype_rel_list,
map-length,
and_wf,
equal_wf,
less_than_wf,
lelt_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
select_wf,
nat_properties,
listp_properties,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
intEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
setEquality,
lambdaFormation,
dependent_set_memberEquality,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
instantiate,
cumulativity,
promote_hyp,
hypothesis_subsumption,
addEquality,
minusEquality
Latex:
\mforall{}[ll:Top List\msupplus{} List]. (||ll|| \mleq{} ||concat(ll)||)
Date html generated:
2017_04_17-AM-08_40_46
Last ObjectModification:
2017_02_27-PM-05_01_04
Theory : list_1
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