Nuprl Lemma : sbdecode-code
∀[m,n:ℕ+]. (sbdecode(sbcode(m;n)) ~ <m ÷ gcd(m;n), n ÷ gcd(m;n)>)
Proof
Definitions occuring in Statement :
sbdecode: sbdecode(L),
sbcode: sbcode(m;n),
gcd: gcd(a;b),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
pair: <a, b>,
divide: n ÷ m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
sbcode: sbcode(m;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
true: True,
squash: ↓T,
sbdecode: sbdecode(L),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nat_plus: ℕ+,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
reduce_cons_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
reduce_nil_lemma,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_plus_subtype_nat,
nat_plus_properties,
nat_plus_wf,
gcd-positive,
int_subtype_base,
gcd_sym_nat,
gcd_subtract,
gcd_wf,
squash_wf,
true_wf,
add-div-when-divides,
equal-wf-base,
nequal_wf,
gcd_is_divisor_1,
iff_weakening_equal,
minus-one-mul,
add-commutes,
minus-one-mul-top,
add-associates,
add-mul-special,
zero-mul,
zero-add,
divides_subtract,
gcd_is_divisor_2,
subtract-add-cancel,
set_subtype_base,
gcd_eq_args,
div-self
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
because_Cache,
productElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
equalityElimination,
lessCases,
imageMemberEquality,
baseClosed,
imageElimination,
promote_hyp,
instantiate,
cumulativity,
addEquality,
universeEquality,
baseApply,
closedConclusion,
divideEquality,
minusEquality
Latex:
\mforall{}[m,n:\mBbbN{}\msupplus{}]. (sbdecode(sbcode(m;n)) \msim{} <m \mdiv{} gcd(m;n), n \mdiv{} gcd(m;n)>)
Date html generated:
2018_05_21-PM-11_40_00
Last ObjectModification:
2017_07_26-PM-06_42_49
Theory : rationals
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