Nuprl Lemma : pdivisor_bound
∀a:ℕ. ∀b:ℕ+. ((a | b) ∧ (¬(b | a)) ⇐⇒ a < b ∧ (a | b))
Proof
Definitions occuring in Statement :
divides: b | a,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
squash: ↓T,
true: True,
assoced: a ~ b,
subtype_rel: A ⊆r B
Lemmas referenced :
divides_wf,
not_wf,
less_than_wf,
nat_plus_wf,
nat_wf,
divisor_bound,
decidable__equal_int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
assoced_nelim,
nat_plus_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
Error :productIsType,
Error :universeIsType,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
because_Cache,
independent_functionElimination,
voidElimination,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
equalityTransitivity,
Error :inhabitedIsType,
imageMemberEquality,
baseClosed
Latex:
\mforall{}a:\mBbbN{}. \mforall{}b:\mBbbN{}\msupplus{}. ((a | b) \mwedge{} (\mneg{}(b | a)) \mLeftarrow{}{}\mRightarrow{} a < b \mwedge{} (a | b))
Date html generated:
2019_06_20-PM-02_21_18
Last ObjectModification:
2018_10_03-AM-00_12_05
Theory : num_thy_1
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