Nuprl Lemma : lcm_wf_nat
∀[n,m:ℕ]. (lcm(n;m) ∈ ℕ)
Proof
Definitions occuring in Statement :
lcm: lcm(a;b),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
lcm: lcm(a;b),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True,
subtract: n - m
Lemmas referenced :
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
value-type-has-value,
int-value-type,
set-value-type,
le_wf,
gcd_wf,
false_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
int_upper_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lcm_wf,
le_weakening2,
lcm-positive,
decidable__lt,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
callbyvalueReduce,
lambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
equalityElimination,
productElimination,
hypothesis_subsumption,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
addEquality,
applyEquality,
minusEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. (lcm(n;m) \mmember{} \mBbbN{})
Date html generated:
2017_04_17-AM-09_46_57
Last ObjectModification:
2017_02_27-PM-05_40_54
Theory : num_thy_1
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