Nuprl Lemma : ni-min-zero

[x:ℕ∞]. (ni-min(x;0∞0∞ ∈ ℕ∞)


Proof




Definitions occuring in Statement :  ni-min: ni-min(f;g) nat2inf: n∞ nat-inf: ℕ∞ uall: [x:A]. B[x] natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: ¬A implies:  Q prop: nat-inf: ℕ∞ squash: T so_lambda: λ2x.t[x] 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] top: Top so_apply: x[s] nat2inf: n∞ 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  bfalse: ff sq_type: SQType(T) guard: {T} bnot: ¬bb assert: b iff: ⇐⇒ Q rev_implies:  Q
Lemmas referenced :  ni-min_wf nat2inf_wf false_wf le_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 nat-inf_wf bool_wf eqtt_to_assert lt_int_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot iff_imp_equal_bool bfalse_wf intformless_wf int_formula_prop_less_lemma less_than_wf assert_of_lt_int iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation lambdaFormation hypothesis applyLambdaEquality setElimination rename imageMemberEquality baseClosed imageElimination lambdaEquality functionEquality applyEquality functionExtensionality addEquality dependent_functionElimination because_Cache unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll equalityElimination equalityTransitivity equalitySymmetry productElimination promote_hyp instantiate cumulativity independent_functionElimination addLevel impliesFunctionality

Latex:
\mforall{}[x:\mBbbN{}\minfty{}].  (ni-min(x;0\minfty{})  =  0\minfty{})



Date html generated: 2017_10_01-AM-08_30_11
Last ObjectModification: 2017_07_26-PM-04_24_25

Theory : basic


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