Nuprl Lemma : qrep_wf
∀[r:ℚ]. (qrep(r) ∈ ℤ × ℕ+)
Proof
Definitions occuring in Statement :
qrep: qrep(r),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rationals: ℚ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
qrep: qrep(r),
qeq: qeq(r;s),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
bfalse: ff,
btrue: tt,
iff: P ⇐⇒ Q,
false: False,
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
nat: ℕ,
spreadn: spread3,
ge: i ≥ j ,
coprime: CoPrime(a,b),
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
gcd_p: GCD(a;b;y),
it: ⋅,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
gt: i > j,
pi1: fst(t),
true: True,
assert: ↑b,
bnot: ¬bb
Lemmas referenced :
nat_plus_wf,
bool_wf,
qeq_wf,
btrue_wf,
b-union_wf,
int_nzero_wf,
rationals_wf,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
assert_wf,
eq_int_wf,
equal-wf-base,
int_subtype_base,
istype-assert,
product-valueall-type,
set-valueall-type,
nequal_wf,
set_subtype_base,
iff_weakening_uiff,
assert_of_eq_int,
eqtt_to_assert,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
coprime_wf,
le_wf,
gcd_reduce_wf,
gcd_reduce_property,
decidable__equal_int,
nat_properties,
int_nzero_properties,
mul_cancel_in_eq,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
subtype_base_sq,
divides_wf,
one_divs_any,
bnot_wf,
less_than_wf,
lt_int_wf,
le_int_wf,
uiff_transitivity,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
int_formula_prop_le_lemma,
intformle_wf,
coprime-equiv-unique-pair,
int_formula_prop_or_lemma,
int_formual_prop_imp_lemma,
intformor_wf,
intformimplies_wf,
neg_mul_arg_bounds,
pi1_wf_top,
pi2_wf,
minus-minus,
divides_invar_2,
int_term_value_minus_lemma,
itermMinus_wf,
pos_mul_arg_bounds,
istype-universe,
true_wf,
squash_wf,
equal_wf,
mul_nzero,
mul-associates,
mul-swap,
mul-commutes,
subtype_rel_self,
iff_weakening_equal,
istype-le,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
ifthenelse_wf,
gt_wf,
product_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
productEquality,
intEquality,
thin,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
imageElimination,
unionElimination,
equalityElimination,
independent_functionElimination,
equalityIstype,
universeIsType,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
productIsType,
because_Cache,
sqequalBase,
axiomEquality,
independent_isectElimination,
callbyvalueReduce,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
lambdaEquality_alt,
natural_numberEquality,
independent_pairEquality,
multiplyEquality,
setElimination,
rename,
isintReduceTrue,
voidElimination,
isect_memberEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
applyLambdaEquality,
independent_pairFormation,
int_eqEquality,
cumulativity,
instantiate,
minusEquality,
imageMemberEquality,
Error :memTop,
universeEquality,
hyp_replacement,
functionIsType,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[r:\mBbbQ{}]. (qrep(r) \mmember{} \mBbbZ{} \mtimes{} \mBbbN{}\msupplus{})
Date html generated:
2020_05_20-AM-09_13_13
Last ObjectModification:
2019_12_31-PM-04_58_21
Theory : rationals
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