### 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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