Nuprl Lemma : div_div_nat
∀[a:ℕ]. ∀[n,m:ℕ+]. (a ÷ n ÷ m ~ a ÷ n * m)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
div_nrel: Div(a;n;q),
lelt: i ≤ j < k,
top: Top,
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
prop: ℙ
Lemmas referenced :
decidable__le,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
nat_properties,
mul_cancel_in_lt,
add-commutes,
one-mul,
mul-commutes,
mul-swap,
mul-associates,
mul-distributes,
nat_wf,
nat_plus_wf,
div_elim,
mul_nat_plus,
divide_wf,
div_unique2,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
productElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
sqequalAxiom,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
multiplyEquality,
addEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n,m:\mBbbN{}\msupplus{}]. (a \mdiv{} n \mdiv{} m \msim{} a \mdiv{} n * m)
Date html generated:
2016_05_14-AM-07_24_37
Last ObjectModification:
2016_01_14-PM-10_02_25
Theory : int_2
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