Nuprl Lemma : ni-min-nat

[n,m:ℕ].  (ni-min(n∞;m∞imin(n;m)∞ ∈ ℕ∞)


Proof




Definitions occuring in Statement :  ni-min: ni-min(f;g) nat2inf: n∞ nat-inf: ℕ∞ imin: imin(a;b) nat: uall: [x:A]. B[x] equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat-inf: ℕ∞ all: x:A. B[x] implies:  Q prop: subtype_rel: A ⊆B nat: ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False not: ¬A top: Top and: P ∧ Q so_lambda: λ2x.t[x] so_apply: x[s] nat2inf: n∞ ni-min: ni-min(f;g) iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) true: True bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff sq_type: SQType(T) guard: {T} bnot: ¬bb assert: b squash: T
Lemmas referenced :  assert_wf ni-min_wf nat2inf_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf all_wf nat_wf iff_imp_equal_bool band_wf lt_int_wf imin_wf less_than_wf iff_wf iff_transitivity iff_weakening_uiff assert_of_band assert_of_lt_int le_int_wf bool_wf eqtt_to_assert assert_of_le_int decidable__lt intformless_wf int_formula_prop_less_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot squash_wf true_wf imin_unfold iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality hypothesis because_Cache sqequalRule addEquality setElimination rename natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionEquality functionExtensionality axiomEquality productEquality productElimination addLevel impliesFunctionality independent_functionElimination equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity imageElimination universeEquality imageMemberEquality baseClosed

Latex:
\mforall{}[n,m:\mBbbN{}].    (ni-min(n\minfty{};m\minfty{})  =  imin(n;m)\minfty{})



Date html generated: 2017_10_01-AM-08_30_03
Last ObjectModification: 2017_07_26-PM-04_24_18

Theory : basic


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