Nuprl Lemma : power-sum_functionality_wrt_eqmod
∀m:ℤ. ∀n:ℕ. ∀x,y:ℤ. ∀a,b:ℕn ⟶ ℤ.
((x ≡ y mod m) ⇒ (∀i:ℕn. (a[i] ≡ b[i] mod m)) ⇒ (Σi<n.a[i]*x^i ≡ Σi<n.b[i]*y^i mod m))
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
eqmod: a ≡ b mod m,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
power-sum: Σi<n.a[i]*x^i,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
int_seg_wf,
eqmod_wf,
istype-nat,
istype-int,
exp_wf2,
int_seg_subtype_nat,
istype-false,
istype-void,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
add_functionality_wrt_eqmod,
primrec_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
istype-le,
decidable__lt,
istype-less_than,
sum-as-primrec,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
primrec-wf2,
all_wf,
eqmod_weakening,
eqmod_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
exp_functionality_wrt_eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
inhabitedIsType,
lambdaEquality_alt,
multiplyEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
closedConclusion,
intEquality,
dependent_set_memberEquality_alt,
unionElimination,
addEquality,
productIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIsType4,
baseApply,
baseClosed,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
setIsType,
functionEquality
Latex:
\mforall{}m:\mBbbZ{}. \mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbZ{}. \mforall{}a,b:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}.
((x \mequiv{} y mod m) {}\mRightarrow{} (\mforall{}i:\mBbbN{}n. (a[i] \mequiv{} b[i] mod m)) {}\mRightarrow{} (\mSigma{}i<n.a[i]*x\^{}i \mequiv{} \mSigma{}i<n.b[i]*y\^{}i mod m))
Date html generated:
2019_10_15-AM-11_25_40
Last ObjectModification:
2018_10_19-AM-11_45_28
Theory : general
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