### Nuprl Lemma : qeq-functionality

`∀[r,s,x:ℤ ⋃ (ℤ × ℤ-o)].  qeq(r;x) = qeq(s;x) supposing qeq(r;s) = tt`

Proof

Definitions occuring in Statement :  qeq: `qeq(r;s)` int_nzero: `ℤ-o` b-union: `A ⋃ B` btrue: `tt` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` product: `x:A × B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` b-union: `A ⋃ B` tunion: `⋃x:A.B[x]` bool: `𝔹` unit: `Unit` ifthenelse: `if b then t else f fi ` pi2: `snd(t)` qeq: `qeq(r;s)` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` all: `∀x:A. B[x]` int_nzero: `ℤ-o` btrue: `tt` iff: `P `⇐⇒` Q` and: `P ∧ Q` uiff: `uiff(P;Q)` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` nequal: `a ≠ b ∈ T ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` true: `True` squash: `↓T`
Lemmas referenced :  valueall-type-has-valueall int-valueall-type evalall-reduce int_nzero_wf product-valueall-type set-valueall-type nequal_wf bool_wf qeq_wf btrue_wf iff_imp_equal_bool eq_int_wf eqtt_to_assert assert_of_eq_int iff_weakening_uiff assert_wf equal-wf-base int_subtype_base istype-assert set_subtype_base subtype_base_sq int_nzero_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermMultiply_wf itermVar_wf istype-int int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_var_lemma int_formula_prop_wf mul_cancel_in_eq equal_wf squash_wf true_wf istype-universe mul_com subtype_rel_self iff_weakening_equal intformand_wf int_formula_prop_and_lemma mul-associates mul-commutes mul-swap mul_assoc int_entire itermConstant_wf int_term_value_constant_lemma zero-mul zero_ann_a intformor_wf int_formula_prop_or_lemma mul_nzero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution imageElimination productElimination thin unionElimination equalityElimination sqequalRule extract_by_obid isectElimination intEquality independent_isectElimination hypothesis hypothesisEquality callbyvalueReduce because_Cache productEquality lambdaEquality_alt inhabitedIsType independent_functionElimination lambdaFormation_alt natural_numberEquality independent_pairEquality equalityIstype universeIsType isect_memberEquality_alt axiomEquality isectIsTypeImplies isintReduceTrue independent_pairFormation equalityTransitivity equalitySymmetry applyEquality promote_hyp multiplyEquality setElimination rename dependent_set_memberEquality_alt productIsType applyLambdaEquality baseApply closedConclusion baseClosed sqequalBase instantiate cumulativity dependent_functionElimination approximateComputation dependent_pairFormation_alt int_eqEquality Error :memTop,  voidElimination universeEquality imageMemberEquality inlFormation_alt

Latex:
\mforall{}[r,s,x:\mBbbZ{}  \mcup{}  (\mBbbZ{}  \mtimes{}  \mBbbZ{}\msupminus{}\msupzero{})].    qeq(r;x)  =  qeq(s;x)  supposing  qeq(r;s)  =  tt

Date html generated: 2020_05_20-AM-09_12_48
Last ObjectModification: 2020_01_28-PM-02_40_46

Theory : rationals

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