Nuprl Lemma : int_mod_union_int_mod
∀[n,m:ℕ+]. ℤ_n ⋃ ℤ_m ≡ ℤ_gcd(n;m)
Proof
Definitions occuring in Statement :
int_mod: ℤ_n,
gcd: gcd(a;b),
nat_plus: ℕ+,
b-union: A ⋃ B,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
bfalse: ff,
btrue: tt,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtract: n - m,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
eqmod: a ≡ b mod m,
divides: b | a,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
gcd_p: GCD(a;b;y),
cand: A c∧ B,
nat: ℕ,
uimplies: b supposing a,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
int_mod: ℤ_n,
quotient: x,y:A//B[x; y],
prop: ℙ,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
all: ∀x:A. B[x]
Lemmas referenced :
bfalse_wf,
btrue_wf,
quotient-member-eq,
eqmod_equiv_rel,
ifthenelse_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
minus-one-mul,
add-commutes,
mul-distributes-right,
mul-associates,
mul-swap,
mul-commutes,
one-mul,
add_functionality_wrt_eq,
mul_assoc,
subtype_rel_self,
iff_weakening_equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
assoced_nelim,
istype-le,
int_term_value_add_lemma,
itermAdd_wf,
mul_preserves_eq,
subtype_base_sq,
nat_properties,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
le_wf,
divides_wf,
gcd_unique,
gcd_sat_pred,
gcd-reduce,
gcd-positive,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
eqmod_wf,
istype-int,
subtype_rel_int_mod,
gcd_is_divisor_1,
gcd_is_divisor_2,
b-union_wf,
int_mod_wf,
gcd_wf,
nat_plus_wf
Rules used in proof :
dependent_pairEquality_alt,
minusEquality,
hyp_replacement,
universeEquality,
imageMemberEquality,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
addEquality,
multiplyEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
equalityIstype,
productIsType,
sqequalBase,
imageElimination,
unionElimination,
equalityElimination,
applyEquality,
independent_functionElimination,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
axiomEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[n,m:\mBbbN{}\msupplus{}]. \mBbbZ{}\_n \mcup{} \mBbbZ{}\_m \mequiv{} \mBbbZ{}\_gcd(n;m)
Date html generated:
2019_10_15-AM-10_25_32
Last ObjectModification:
2019_09_20-PM-03_53_56
Theory : num_thy_1
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