Nuprl Lemma : div_lbound_1
∀[a:ℕ]. ∀[n:ℕ+]. ∀[k:ℕ]. uiff(k ≤ (a ÷ n);(k * n) ≤ a)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
multiply: n * m
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
prop: ℙ,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
le: A ≤ B,
div_nrel: Div(a;n;q),
lelt: i ≤ j < k,
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
less_than'_wf,
le_wf,
nat_properties,
nat_plus_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,
nat_wf,
nat_plus_wf,
div_elim,
uiff_wf,
nat_plus_subtype_nat,
mul_preserves_le,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
lt_transitivity_2,
less_than_wf,
squash_wf,
true_wf,
mul_com,
subtype_rel_self,
iff_weakening_equal,
int_term_value_add_lemma,
itermAdd_wf,
mul_cancel_in_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
multiplyEquality,
because_Cache,
divideEquality,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
baseClosed,
isect_memberFormation,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
computeAll,
unionElimination,
addEquality,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[k:\mBbbN{}]. uiff(k \mleq{} (a \mdiv{} n);(k * n) \mleq{} a)
Date html generated:
2019_06_20-PM-01_14_29
Last ObjectModification:
2018_09_17-PM-05_43_52
Theory : int_2
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