Nuprl Lemma : lcm-unique-nat
∀n,m,l:ℕ. ((((n | l) ∧ (m | l)) ∧ (∀v:ℤ. ((n | v) ⇒ (m | v) ⇒ (l | v)))) ⇒ (l = lcm(n;m) ∈ ℤ))
Proof
Definitions occuring in Statement :
lcm: lcm(a;b),
divides: b | a,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
divides: b | a,
exists: ∃x:A. B[x],
lcm: lcm(a;b),
has-value: (a)↓,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
divides_wf,
all_wf,
nat_wf,
value-type-has-value,
int-value-type,
set-value-type,
le_wf,
gcd_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lcm-unique,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
int_entire,
intformor_wf,
int_formula_prop_or_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityElimination,
promote_hyp,
dependent_set_memberEquality,
addEquality,
applyEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}n,m,l:\mBbbN{}. ((((n | l) \mwedge{} (m | l)) \mwedge{} (\mforall{}v:\mBbbZ{}. ((n | v) {}\mRightarrow{} (m | v) {}\mRightarrow{} (l | v)))) {}\mRightarrow{} (l = lcm(n;m)))
Date html generated:
2017_04_17-AM-09_46_51
Last ObjectModification:
2017_02_27-PM-05_41_09
Theory : num_thy_1
Home
Index