Nuprl Lemma : select-from-upto

`∀[n,m:ℤ]. ∀[k:ℕm - n].  ([n, m)[k] ~ n + k)`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` select: `L[n]` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` subtract: `n - m` add: `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` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` from-upto: `[n, m)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` select: `L[n]` nil: `[]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` cons: `[a / b]` has-value: `(a)↓` subtype_rel: `A ⊆r B`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 int_seg_wf subtract_wf le_wf subtract-1-ge-0 nat_wf int_seg_properties itermSubtract_wf int_term_value_subtract_lemma lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf stuck-spread istype-base intformnot_wf int_formula_prop_not_lemma decidable__equal_int int_subtype_base add-zero value-type-has-value int-value-type decidable__le itermAdd_wf int_term_value_add_lemma intformeq_wf int_formula_prop_eq_lemma decidable__lt select-cons-tl select_wf from-upto_wf length-from-upto satisfiable-full-omega-tt
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomSqEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  productElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry Error :equalityIsType1,  promote_hyp instantiate cumulativity because_Cache baseClosed intEquality callbyvalueReduce addEquality Error :dependent_set_memberEquality_alt,  Error :productIsType,  applyEquality Error :isect_memberFormation_alt,  computeAll voidEquality isect_memberEquality lambdaEquality dependent_pairFormation dependent_set_memberEquality

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[k:\mBbbN{}m  -  n].    ([n,  m)[k]  \msim{}  n  +  k)

Date html generated: 2019_06_20-PM-01_34_11
Last ObjectModification: 2018_10_04-PM-02_28_32

Theory : list_1

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