### 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: `ℕ` uall: `∀[x:A]. B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat-inf: `ℕ∞` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b 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: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f 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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