Nuprl Lemma : deq-member-firstn
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[L:A List]. ∀[n:ℕ+].
∀[x:A]. (x ∈b firstn(n;L) ~ x ∈b firstn(n - 1;L) ∨b(eqof(eq) x L[n - 1])) supposing n - 1 < ||L||
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
deq-member: x ∈b L,
list: T List,
eqof: eqof(d),
deq: EqDecider(T),
bor: p ∨bq,
nat_plus: ℕ+,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
subtract: n - m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
deq-member_wf,
firstn_wf,
bor_wf,
select_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
iff_transitivity,
assert_wf,
or_wf,
l_member_wf,
equal_wf,
iff_weakening_uiff,
assert_of_bor,
subtract_wf,
assert-deq-member,
safe-assert-deq,
less_than_wf,
length_wf,
nat_plus_wf,
list_wf,
deq_wf,
firstn_decomp,
nat_plus_subtype_nat,
decidable__lt,
member_append,
cons_wf,
nil_wf,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
hypothesisEquality,
setElimination,
rename,
because_Cache,
applyEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
lambdaFormation,
independent_functionElimination,
orFunctionality,
productElimination,
equalityTransitivity,
equalitySymmetry,
sqequalAxiom,
universeEquality,
addLevel,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[L:A List]. \mforall{}[n:\mBbbN{}\msupplus{}].
\mforall{}[x:A]. (x \mmember{}\msubb{} firstn(n;L) \msim{} x \mmember{}\msubb{} firstn(n - 1;L) \mvee{}\msubb{}(eqof(eq) x L[n - 1])) supposing n - 1 < ||L||
Date html generated:
2017_09_29-PM-06_04_21
Last ObjectModification:
2017_07_26-PM-02_53_03
Theory : decidable!equality
Home
Index