### Nuprl Lemma : from-upto-decomp-last

`∀[n,m:ℤ].  [n, m) = ([n, m - 1) @ [m - 1]) ∈ (ℤ List) supposing n < m`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtract: `n - m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` exists: `∃x:A. B[x]` nat: `ℕ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` guard: `{T}` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` ge: `i ≥ j ` less_than: `a < b` less_than': `less_than'(a;b)` from-upto: `[n, m)` has-value: `(a)↓` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` subtract: `n - m` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  decidable__le subtract_wf full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma equal-wf-base-T int_subtype_base subtype_base_sq less_than_wf squash_wf true_wf subtype_rel_self iff_weakening_equal nat_properties decidable__lt ge_wf lt_int_wf bool_wf value-type-has-value int-value-type eqtt_to_assert assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot cons_wf list_wf add-associates add-swap add-commutes zero-add nil_wf add-subtract-cancel list_ind_nil_lemma list_ind_cons_lemma list_subtype_base set_subtype_base from-upto_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_pairFormation dependent_set_memberEquality because_Cache extract_by_obid sqequalHypSubstitution dependent_functionElimination thin natural_numberEquality isectElimination hypothesisEquality hypothesis unionElimination independent_isectElimination approximateComputation independent_functionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation applyEquality addEquality setElimination rename productElimination instantiate cumulativity equalityTransitivity equalitySymmetry imageElimination imageMemberEquality baseClosed universeEquality intWeakElimination lambdaFormation axiomEquality callbyvalueReduce equalityElimination promote_hyp minusEquality setEquality productEquality

Latex:
\mforall{}[n,m:\mBbbZ{}].    [n,  m)  =  ([n,  m  -  1)  @  [m  -  1])  supposing  n  <  m

Date html generated: 2018_05_21-PM-00_40_26
Last ObjectModification: 2018_05_19-AM-06_46_05

Theory : list_1

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