Nuprl Lemma : gcd_functionality_wrt_eqmod
∀a,a',m:ℤ. ((a' ≡ a mod m) ⇒ (gcd(a';m) ~ gcd(a;m)))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
assoced: a ~ b,
gcd: gcd(a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
gcd_p: GCD(a;b;y),
assoced: a ~ b,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
gcd_elim,
assoced_wf,
gcd_wf,
eqmod_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
divisor_of_sum,
divisor_of_mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
hypothesis,
independent_pairFormation,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
isectElimination,
sqequalRule,
intEquality,
instantiate,
cumulativity,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
minusEquality,
equalityTransitivity
Latex:
\mforall{}a,a',m:\mBbbZ{}. ((a' \mequiv{} a mod m) {}\mRightarrow{} (gcd(a';m) \msim{} gcd(a;m)))
Date html generated:
2016_10_21-AM-11_09_12
Last ObjectModification:
2016_07_12-AM-06_01_44
Theory : num_thy_1
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