Nuprl Lemma : reg-seq-mul_wf2
∀[x,y:ℝ]. (reg-seq-mul(x;y) ∈ {f:ℕ+ ⟶ ℤ| imax(|x 1|;|y 1|) + 4-regular-seq(f)} )
Proof
Definitions occuring in Statement :
reg-seq-mul: reg-seq-mul(x;y),
real: ℝ,
regular-int-seq: k-regular-seq(f),
imax: imax(a;b),
absval: |i|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
canon-bnd: canon-bnd(x),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
reg-seq-mul_wf,
regular-int-seq_wf,
imax_wf,
absval_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
real_wf,
ifthenelse_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
istype-le,
intformand_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-is-int-iff,
false_wf,
add_functionality_wrt_eq,
imax_unfold,
iff_weakening_equal,
reg-seq-mul-regular,
canon-bnd_wf,
imax_nat_plus,
subtype_rel_set,
int_upper_wf,
nat_plus_wf,
istype-int_upper,
subtype_rel_sets_simple,
less_than_wf,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
nat_plus_properties,
imax_ub,
decidable__le,
mul_preserves_le,
nat_plus_subtype_nat,
le_functionality,
le_weakening,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
universeIsType,
addEquality,
applyEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
hypothesisEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectIsTypeImplies,
intEquality,
lambdaFormation_alt,
equalityElimination,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
int_eqEquality,
independent_pairFormation,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
sqequalIntensionalEquality,
functionEquality,
multiplyEquality,
applyLambdaEquality,
inlFormation_alt,
imageMemberEquality,
imageElimination,
inrFormation_alt
Latex:
\mforall{}[x,y:\mBbbR{}]. (reg-seq-mul(x;y) \mmember{} \{f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| imax(|x 1|;|y 1|) + 4-regular-seq(f)\} )
Date html generated:
2019_10_16-PM-03_06_48
Last ObjectModification:
2019_01_31-PM-04_48_18
Theory : reals
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