Nuprl Lemma : div_elim
∀a:ℕ. ∀n:ℕ+. ∃q:ℕ. (Div(a;n;q) ∧ ((a ÷ n) = q ∈ ℤ))
Proof
Definitions occuring in Statement :
div_nrel: Div(a;n;q),
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
divide: n ÷ m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
nat: ℕ,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
nat_plus_wf,
nat_wf,
divide_wf,
div_fun_sat_div_nrel,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
div_nrel_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
dependent_pairFormation,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
divideEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
applyEquality,
baseClosed,
productEquality
Latex:
\mforall{}a:\mBbbN{}. \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}q:\mBbbN{}. (Div(a;n;q) \mwedge{} ((a \mdiv{} n) = q))
Date html generated:
2019_06_20-PM-01_14_20
Last ObjectModification:
2018_09_17-PM-05_45_37
Theory : int_2
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