Nuprl Lemma : lcm-positive
∀[a,b:ℕ+].  0 < lcm(a;b)
Proof
Definitions occuring in Statement : 
lcm: lcm(a;b)
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
uall: ∀[x:A]. B[x]
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
nat_plus: ℕ+
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
lcm: lcm(a;b)
, 
has-value: (a)↓
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
nequal: a ≠ b ∈ T 
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
cand: A c∧ B
, 
gt: i > j
, 
int_nzero: ℤ-o
Lemmas referenced : 
gcd-positive, 
nat_plus_subtype_nat, 
nat_plus_properties, 
decidable__lt, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformless_wf, 
itermConstant_wf, 
itermVar_wf, 
istype-int, 
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, 
value-type-has-value, 
nat_plus_wf, 
set-value-type, 
less_than_wf, 
int-value-type, 
gcd_wf, 
eq_int_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
member-less_than, 
lcm_wf, 
gcd-property, 
set_subtype_base, 
int_subtype_base, 
istype-less_than, 
decidable__equal_int, 
itermMultiply_wf, 
int_term_value_mul_lemma, 
neg_mul_arg_bounds, 
div-cancel, 
nequal_wf, 
mul_bounds_1b
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
hypothesis, 
sqequalRule, 
independent_isectElimination, 
setElimination, 
rename, 
dependent_functionElimination, 
natural_numberEquality, 
unionElimination, 
approximateComputation, 
independent_functionElimination, 
Error :dependent_pairFormation_alt, 
Error :lambdaEquality_alt, 
int_eqEquality, 
Error :isect_memberEquality_alt, 
voidElimination, 
independent_pairFormation, 
Error :universeIsType, 
because_Cache, 
callbyvalueReduce, 
intEquality, 
Error :inhabitedIsType, 
Error :lambdaFormation_alt, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
Error :equalityIstype, 
promote_hyp, 
instantiate, 
cumulativity, 
Error :isectIsTypeImplies, 
baseApply, 
closedConclusion, 
baseClosed, 
sqequalBase, 
Error :inrFormation_alt, 
Error :productIsType, 
multiplyEquality, 
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}[a,b:\mBbbN{}\msupplus{}].    0  <  lcm(a;b)
Date html generated:
2019_06_20-PM-02_27_30
Last ObjectModification:
2019_03_06-AM-10_54_05
Theory : num_thy_1
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