Nuprl Lemma : ni-min-com

[x,y:ℕ∞].  (ni-min(x;y) ni-min(y;x) ∈ ℕ∞)


Proof




Definitions occuring in Statement :  ni-min: ni-min(f;g) nat-inf: ℕ∞ uall: [x:A]. B[x] equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat-inf: ℕ∞ squash: T so_lambda: λ2x.t[x] implies:  Q prop: nat: ge: i ≥  all: x:A. B[x] 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_apply: x[s] ni-min: ni-min(f;g) bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) band: p ∧b q ifthenelse: if then else fi  true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q bfalse: ff sq_type: SQType(T) bnot: ¬bb assert: b
Lemmas referenced :  ni-min_wf all_wf nat_wf assert_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 bool_wf eqtt_to_assert equal_wf squash_wf true_wf band_tt_simp iff_weakening_equal eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bfalse_wf band_ff_simp
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis applyLambdaEquality setElimination rename sqequalRule imageMemberEquality baseClosed imageElimination dependent_set_memberEquality lambdaEquality functionEquality applyEquality functionExtensionality addEquality natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache axiomEquality lambdaFormation equalityElimination equalityTransitivity equalitySymmetry productElimination universeEquality independent_functionElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}[x,y:\mBbbN{}\minfty{}].    (ni-min(x;y)  =  ni-min(y;x))



Date html generated: 2017_10_01-AM-08_30_05
Last ObjectModification: 2017_07_26-PM-04_24_20

Theory : basic


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