Nuprl Lemma : assert-ble
∀[n:ℤ]. ∀[m:ℕ]. (↑ble(n;m) ⇐⇒ n ≤ m)
Proof
Definitions occuring in Statement :
ble: ble(n;m),
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
ble: ble(n;m),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
true: True,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
less_than: a < b,
less_than': less_than'(a;b),
squash: ↓T,
le: A ≤ B
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
true_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
intformless_wf,
int_formula_prop_less_lemma,
false_wf,
less_than'_wf,
assert_wf,
ble_wf,
nat_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
int_eqReduceTrueSq,
independent_pairFormation,
dependent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
int_eqReduceFalseSq,
lessCases,
isect_memberFormation,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[m:\mBbbN{}]. (\muparrow{}ble(n;m) \mLeftarrow{}{}\mRightarrow{} n \mleq{} m)
Date html generated:
2017_04_20-AM-07_29_18
Last ObjectModification:
2017_02_27-PM-06_00_02
Theory : continuity
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