Nuprl Lemma : member-concat-map
∀[T,S:Type]. ∀f:T ⟶ (S List). ∀L:T List. ∀x:S. ((x ∈ concat(map(f;L))) ⇐⇒ (∃t∈L. (x ∈ f t)))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
concat: concat(ll),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
concat: concat(ll),
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q
Lemmas referenced :
member_append,
l_exists_cons,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
length_of_nil_lemma,
base_wf,
stuck-spread,
cons_wf,
append_wf,
concat-cons,
map_cons_lemma,
l_exists_wf_nil,
btrue_neq_bfalse,
nil_wf,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
reduce_nil_lemma,
map_nil_lemma,
l_exists_wf,
list_wf,
map_wf,
concat_wf,
l_member_wf,
iff_wf,
all_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionEquality,
universeEquality,
productElimination,
baseClosed,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[T,S:Type]. \mforall{}f:T {}\mrightarrow{} (S List). \mforall{}L:T List. \mforall{}x:S. ((x \mmember{} concat(map(f;L))) \mLeftarrow{}{}\mRightarrow{} (\mexists{}t\mmember{}L. (x \mmember{} f t)))
Date html generated:
2016_05_15-PM-02_17_08
Last ObjectModification:
2016_01_15-PM-00_18_08
Theory : monads
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