Nuprl Lemma : int-with-rational-square-root
∀n:ℤ. ∀q:ℚ. (((q * q) = n ∈ ℚ) ⇒ (∃m:ℤ. ((m * m) = n ∈ ℤ)))
Proof
Definitions occuring in Statement :
qmul: r * s,
rationals: ℚ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
pi2: snd(t),
mk-rational: mk-rational(a;b),
rationals: ℚ,
quotient: x,y:A//B[x; y],
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
cand: A c∧ B,
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
sq_type: SQType(T),
not: ¬A,
false: False,
or: P ∨ Q,
divides: b | a,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
prime: prime(a),
assoced: a ~ b,
int_upper: {i...},
label: ...$L... t,
sq_exists: ∃x:A [B[x]],
cons: [a / b],
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind
Lemmas referenced :
qmul_wf,
int-subtype-rationals,
rationals_wf,
istype-int,
equals-qrep,
qrep-coprime,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
qrep_wf,
b-union_wf,
int_nzero_wf,
bool_wf,
qeq_wf2,
mk-rational_wf,
nat_plus_inc_int_nzero,
btrue_wf,
set_subtype_base,
less_than_wf,
int_subtype_base,
qeq-elim,
qmul-elim,
eqtt_to_assert,
assert_of_eq_int,
coprime_wf,
one_divs_any,
divides_wf,
gcd_is_gcd,
absval_ifthenelse,
gcd_wf,
lt_int_wf,
subtype_base_sq,
assert_wf,
bnot_wf,
not_wf,
istype-less_than,
istype-assert,
istype-void,
divides_invar_2,
bool_cases,
bool_subtype_base,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
prime_wf,
prime_divs_prod,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_plus_wf,
decidable__equal_nat_plus,
decidable__lt,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
prime-factors,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
istype-le,
set_wf,
int_upper_wf,
list-cases,
product_subtype_list,
mul_list_nil_lemma,
mul-list_wf,
subtype_rel_list,
istype-int_upper,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
applyEquality,
sqequalRule,
universeIsType,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
pertypeElimination,
promote_hyp,
productIsType,
intEquality,
productEquality,
sqequalBase,
baseApply,
closedConclusion,
isintReduceTrue,
multiplyEquality,
setElimination,
rename,
dependent_pairFormation_alt,
independent_pairFormation,
cumulativity,
functionIsType,
unionElimination,
voidElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
dependent_set_memberEquality_alt,
hypothesis_subsumption,
setEquality,
setIsType
Latex:
\mforall{}n:\mBbbZ{}. \mforall{}q:\mBbbQ{}. (((q * q) = n) {}\mRightarrow{} (\mexists{}m:\mBbbZ{}. ((m * m) = n)))
Date html generated:
2020_05_20-AM-09_31_06
Last ObjectModification:
2020_01_01-AM-11_44_42
Theory : rationals
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