Nuprl Lemma : member-concat-mklist
∀[T:Type]. ∀n:ℕ. ∀f:ℕn ⟶ (T List). ∀x:T. ((x ∈ concat(mklist(n;f))) ⇐⇒ ∃i:ℕn. (x ∈ f i))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
l_member: (x ∈ l),
concat: concat(ll),
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
nat: ℕ,
l_member: (x ∈ l),
cand: A c∧ B,
top: Top,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
exists_wf,
list_wf,
l_member_wf,
mklist_wf,
int_seg_wf,
member-concat,
concat_wf,
iff_wf,
nat_wf,
mklist_length,
equal_wf,
squash_wf,
true_wf,
mklist_select,
lelt_wf,
subtype_rel_self,
iff_weakening_equal,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
length_wf,
select_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
applyEquality,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
dependent_pairFormation,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
cumulativity,
functionExtensionality
Latex:
\mforall{}[T:Type]. \mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} (T List). \mforall{}x:T. ((x \mmember{} concat(mklist(n;f))) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (x \mmember{} f i))
Date html generated:
2018_05_21-PM-00_44_24
Last ObjectModification:
2018_05_19-AM-06_48_28
Theory : list_1
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