Nuprl Lemma : vexample_wf
∀n,a:ℕ. ∀b:{b:ℕ| (((a * a) + (b * b)) + 1) = ((a * b) * 3) ∈ ℤ} .
(vexample(n;a;b) ∈ a:ℕ × {b:ℕ| (((a * a) + (b * b)) + 1) = ((a * b) * 3) ∈ ℤ} )
Proof
Definitions occuring in Statement :
vexample: vexample(n;a;b),
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
vexample: vexample(n;a;b),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
set_subtype_base,
le_wf,
int_subtype_base,
istype-nat,
subtract-1-ge-0,
intformeq_wf,
int_formula_prop_eq_lemma,
value-type-has-value,
int-value-type,
subtract_wf,
decidable__equal_int,
intformnot_wf,
itermAdd_wf,
itermMultiply_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
nat_wf,
equal_wf,
decidable__le,
istype-le,
decidable__lt,
exp_preserves_lt,
less_than_wf,
squash_wf,
true_wf,
exp2,
subtype_rel_self,
iff_weakening_equal,
mul_bounds_1a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :setIsType,
because_Cache,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
Error :dependent_pairEquality_alt,
Error :dependent_set_memberEquality_alt,
int_eqReduceFalseSq,
callbyvalueReduce,
multiplyEquality,
unionElimination,
addEquality,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality,
productElimination
Latex:
\mforall{}n,a:\mBbbN{}. \mforall{}b:\{b:\mBbbN{}| (((a * a) + (b * b)) + 1) = ((a * b) * 3)\} .
(vexample(n;a;b) \mmember{} a:\mBbbN{} \mtimes{} \{b:\mBbbN{}| (((a * a) + (b * b)) + 1) = ((a * b) * 3)\} )
Date html generated:
2019_06_20-PM-02_43_31
Last ObjectModification:
2019_03_11-PM-05_55_04
Theory : num_thy_1
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