Nuprl Lemma : posint_div_dec
∀a,b:|<ℤ+,*>|. Dec(a | b)
Proof
Definitions occuring in Statement :
posint_mul_mon: <ℤ+,*>,
mdivides: b | a,
decidable: Dec(P),
all: ∀x:A. B[x],
grp_car: |g|
Definitions unfolded in proof :
posint_mul_mon: <ℤ+,*>,
mdivides: b | a,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_plus_wf,
decidable__equal_int,
remainder_wfa,
nat_plus_inc_int_nzero,
div_rem_sum,
div_bounds_1,
nat_plus_subtype_nat,
nat_plus_properties,
divide_wfa,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int_subtype_base,
nequal_wf,
intformnot_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
decidable__lt,
istype-less_than,
set_subtype_base,
less_than_wf,
subtype_base_sq,
intformle_wf,
int_formula_prop_le_lemma,
rem-exact
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
lambdaFormation_alt,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
applyEquality,
natural_numberEquality,
unionElimination,
inlFormation_alt,
dependent_pairFormation_alt,
because_Cache,
multiplyEquality,
dependent_set_memberEquality_alt,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
intEquality,
equalityTransitivity,
productElimination,
functionIsType,
productIsType,
baseApply,
closedConclusion,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
equalityIsType4,
inrFormation_alt
Latex:
\mforall{}a,b:|<\mBbbZ{}\msupplus{},*>|. Dec(a | b)
Date html generated:
2019_10_16-PM-01_06_11
Last ObjectModification:
2019_06_20-PM-06_44_22
Theory : factor_1
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