Nuprl Lemma : sbdecode_wf_gcd
∀[L:ℕ2 List]. (sbdecode(L) ∈ {p:ℕ+ × ℕ+| let m,n = p in gcd(m;n) = 1 ∈ ℤ} )
Proof
Definitions occuring in Statement :
sbdecode: sbdecode(L),
list: T List,
gcd: gcd(a;b),
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
pi2: snd(t),
pi1: fst(t),
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
nat: ℕ,
less_than': less_than'(a;b),
ge: i ≥ j
Lemmas referenced :
sbdecode_wf,
nat_plus_wf,
sbcode_wf,
sbcode-decode,
equal_wf,
equal-wf-T-base,
gcd_wf,
list_wf,
int_seg_wf,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
lelt_wf,
int_subtype_base,
sbdecode-code,
gcd-positive,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
div_rem_sum,
equal-wf-base,
nequal_wf,
less_than_wf,
rem_bounds_1,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
decidable__equal_int,
false_wf,
le_wf,
le_functionality,
add_functionality_wrt_le,
multiply_functionality_wrt_le,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
productEquality,
lambdaFormation,
productElimination,
applyLambdaEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
spreadEquality,
independent_pairEquality,
intEquality,
setElimination,
rename,
baseClosed,
axiomEquality,
natural_numberEquality,
instantiate,
cumulativity,
independent_isectElimination,
lambdaEquality,
applyEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseApply,
closedConclusion,
imageElimination,
addEquality,
multiplyEquality
Latex:
\mforall{}[L:\mBbbN{}2 List]. (sbdecode(L) \mmember{} \{p:\mBbbN{}\msupplus{} \mtimes{} \mBbbN{}\msupplus{}| let m,n = p in gcd(m;n) = 1\} )
Date html generated:
2018_05_21-PM-11_40_29
Last ObjectModification:
2017_07_26-PM-06_42_50
Theory : rationals
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