Nuprl Lemma : gcd-mod
∀x:ℕ+. ∀y:ℕ. (gcd(x;y mod x) = gcd(x;y) ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
gcd: gcd(a;b),
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
gcd: gcd(a;b),
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_wf,
nat_plus_wf,
gcd_wf,
squash_wf,
true_wf,
modulus_base,
false_wf,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
equal_wf,
iff_weakening_equal,
modulus-is-rem,
subtype_rel_sets,
less_than_wf,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
gcd_com,
nat_plus_subtype_nat,
int_upper_subtype_nat,
le_wf,
nequal-le-implies,
zero-add,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_upper_properties,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
independent_pairFormation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
setEquality,
applyLambdaEquality,
hypothesis_subsumption,
equalityElimination,
promote_hyp,
remainderEquality
Latex:
\mforall{}x:\mBbbN{}\msupplus{}. \mforall{}y:\mBbbN{}. (gcd(x;y mod x) = gcd(x;y))
Date html generated:
2018_05_21-PM-08_57_29
Last ObjectModification:
2017_07_26-PM-06_21_15
Theory : general
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