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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