Nuprl Lemma : reg-seq-mul_functionality_wrt_bdd-diff
∀x1:ℝ. ∀[x2,y1:ℕ+ ⟶ ℤ]. ∀y2:ℝ. (bdd-diff(y1;y2) ⇒ bdd-diff(x1;x2) ⇒ bdd-diff(reg-seq-mul(x1;y1);reg-seq-mul(x2;y2)))
Proof
Definitions occuring in Statement :
reg-seq-mul: reg-seq-mul(x;y),
real: ℝ,
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
reg-seq-mul: reg-seq-mul(x;y),
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
int_upper: {i...},
so_lambda: λ2x.t[x],
real: ℝ,
nat_plus: ℕ+,
so_apply: x[s],
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
canonical-bound_wf,
int_upper_wf,
all_wf,
nat_plus_wf,
le_wf,
absval_wf,
add_nat_wf,
false_wf,
multiply_nat_wf,
subtype_rel_set,
nat_wf,
int_upper_subtype_nat,
nat_properties,
decidable__le,
add-is-int-iff,
multiply-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
mul_cancel_in_le,
subtract_wf,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
absval_nat_plus,
equal-wf-T-base,
squash_wf,
true_wf,
absval_mul,
iff_weakening_equal,
less_than'_wf,
equal-wf-base,
bdd-diff_wf,
real_wf,
nequal_wf,
rem_bounds_absval,
less_than_wf,
set_wf,
left_mul_subtract_distrib,
div_rem_sum2,
sq_stable__less_than,
le_functionality,
le_weakening,
add_functionality_wrt_le,
int-triangle-inequality,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
add_functionality_wrt_eq,
multiply_functionality_wrt_le,
sq_stable__le,
absval_pos,
nat_plus_subtype_nat,
int_upper_properties,
absval_sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
multiplyEquality,
setElimination,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
setEquality,
because_Cache,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseClosed,
baseApply,
closedConclusion,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
divideEquality,
imageElimination,
imageMemberEquality,
universeEquality,
independent_pairEquality,
axiomEquality,
functionExtensionality,
functionEquality,
remainderEquality,
minusEquality,
instantiate,
cumulativity
Latex:
\mforall{}x1:\mBbbR{}
\mforall{}[x2,y1:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
\mforall{}y2:\mBbbR{}. (bdd-diff(y1;y2) {}\mRightarrow{} bdd-diff(x1;x2) {}\mRightarrow{} bdd-diff(reg-seq-mul(x1;y1);reg-seq-mul(x2;y2)))
Date html generated:
2017_10_02-PM-07_15_06
Last ObjectModification:
2017_07_28-AM-07_20_19
Theory : reals
Home
Index