Nuprl Lemma : divisor_bound
∀[a:ℕ]. ∀[b:ℕ+]. a ≤ b supposing a | b
Proof
Definitions occuring in Statement :
divides: b | a,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
nat: ℕ,
prop: ℙ,
exists: ∃x:A. B[x],
divides: b | a,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j
Lemmas referenced :
less_than'_wf,
divides_wf,
nat_plus_wf,
nat_wf,
mul_preserves_le,
intformand_wf,
itermMultiply_wf,
intformeq_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
nat_plus_properties,
nat_properties,
decidable__or,
le_wf,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformor_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
isect_memberEquality,
voidElimination,
computeAll,
voidEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
independent_functionElimination,
natural_numberEquality,
lemma_by_obid,
independent_pairFormation
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[b:\mBbbN{}\msupplus{}]. a \mleq{} b supposing a | b
Date html generated:
2019_06_20-PM-02_20_21
Last ObjectModification:
2018_09_26-PM-05_45_50
Theory : num_thy_1
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