### Nuprl Lemma : length-from-upto

`∀[n,m:ℤ].  (||[n, m)|| ~ if n <z m then m - n else 0 fi )`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` length: `||as||` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` uall: `∀[x:A]. B[x]` subtract: `n - m` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` from-upto: `[n, m)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` decidable: `Dec(P)` or: `P ∨ Q` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` le: `A ≤ B` less_than': `less_than'(a;b)` has-value: `(a)↓`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf le_wf subtract_wf subtype_base_sq nat_wf set_subtype_base int_subtype_base lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int length_of_cons_lemma decidable__equal_int itermSubtract_wf int_term_value_subtract_lemma eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot length_of_nil_lemma intformnot_wf intformeq_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma false_wf decidable__le value-type-has-value int-value-type itermAdd_wf int_term_value_add_lemma non_neg_length from-upto_wf
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom instantiate cumulativity because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination addEquality dependent_set_memberEquality promote_hyp callbyvalueReduce isect_memberFormation setEquality productEquality

Latex:
\mforall{}[n,m:\mBbbZ{}].    (||[n,  m)||  \msim{}  if  n  <z  m  then  m  -  n  else  0  fi  )

Date html generated: 2017_04_17-AM-07_53_25
Last ObjectModification: 2017_02_27-PM-04_27_34

Theory : list_1

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