Nuprl Lemma : concat_map_upto
∀[T:Type]. ∀f:ℕ ⟶ (T List). ∀t,t':ℕ. concat(map(f;upto(t))) @ (f t) ≤ concat(map(f;upto(t'))) supposing t < t'
Proof
Definitions occuring in Statement :
upto: upto(n),
iseg: l1 ≤ l2,
concat: concat(ll),
map: map(f;as),
append: as @ bs,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
prop: ℙ,
top: Top,
concat: concat(ll),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
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]
Lemmas referenced :
le_wf,
upto_iseg,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_subtype,
iseg-map,
upto_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
map_wf,
concat_iseg,
top_wf,
subtype_rel_list,
append_nil_sq,
reduce_nil_lemma,
concat-cons,
map_nil_lemma,
map_cons_lemma,
concat_append,
map_append_sq,
upto_add_1,
list_wf,
nat_wf,
less_than_wf,
member-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_isectElimination,
hypothesis,
functionEquality,
universeEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
applyEquality,
lambdaEquality,
because_Cache,
natural_numberEquality,
addEquality,
cumulativity,
independent_pairFormation,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:\mBbbN{} {}\mrightarrow{} (T List). \mforall{}t,t':\mBbbN{}.
concat(map(f;upto(t))) @ (f t) \mleq{} concat(map(f;upto(t'))) supposing t < t'
Date html generated:
2016_05_14-PM-03_10_13
Last ObjectModification:
2016_01_15-AM-07_17_36
Theory : list_1
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