Nuprl Lemma : div_div
∀[a:ℤ]. ∀[n,m:ℤ-o]. (a ÷ n ÷ m ~ a ÷ n * m)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
int_nzero: ℤ-o,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
prop: ℙ,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
gt: i > j,
squash: ↓T,
int_lower: {...i},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
decidable__le,
int_nzero_wf,
istype-int,
div_div_nat,
le_wf,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
add-associates,
int_entire_a,
int_nzero_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-base,
neg_mul_arg_bounds,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
gt_wf,
equal_wf,
squash_wf,
true_wf,
div_4_to_1,
itermMultiply_wf,
int_term_value_mul_lemma,
iff_weakening_equal,
decidable__equal_int,
itermMinus_wf,
int_term_value_minus_lemma,
divide_wf,
not-le-2,
minus-one-mul,
minus-one-mul-top,
div_2_to_1,
div_lbound_1,
zero-mul,
div_anti_sym,
div_3_to_1,
member_wf,
full-omega-unsat,
istype-void,
mul_nat_plus,
istype-false,
subtype_rel_self,
div_bounds_2,
div_anti_sym2,
pos_mul_arg_bounds,
div_bounds_3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
because_Cache,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomSqEquality,
Error :inhabitedIsType,
sqequalRule,
Error :isect_memberEquality_alt,
Error :universeIsType,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
addEquality,
minusEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
divideEquality,
multiplyEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
baseClosed,
inrFormation,
productEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseApply,
closedConclusion,
inlFormation,
Error :lambdaEquality_alt,
Error :lambdaFormation_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
Error :dependent_set_memberEquality_alt,
Error :inrFormation_alt,
Error :productIsType
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n,m:\mBbbZ{}\msupminus{}\msupzero{}]. (a \mdiv{} n \mdiv{} m \msim{} a \mdiv{} n * m)
Date html generated:
2019_06_20-PM-01_14_36
Last ObjectModification:
2018_10_04-PM-01_07_08
Theory : int_2
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