Nuprl Lemma : no_repeats_mu_index

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[i:ℕ||L||].  mu(λi@0.(eq L[i] L[i@0])) = i ∈ ℤ supposing no_repeats(T;L)`

Proof

Definitions occuring in Statement :  mu: `mu(f)` no_repeats: `no_repeats(T;l)` select: `L[n]` length: `||as||` list: `T List` deq: `EqDecider(T)` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` deq: `EqDecider(T)` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` cand: `A c∧ B` eqof: `eqof(d)` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` l_member!: `(x ∈! l)` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` nat: `ℕ` ge: `i ≥ j `
Lemmas referenced :  int_formula_prop_eq_lemma intformeq_wf decidable__equal_int le_wf nat_properties false_wf int_seg_subtype_nat select_member no_repeats_member deq_wf list_wf no_repeats_wf assert_wf safe-assert-deq int_seg_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le length_wf int_seg_properties select_wf length_wf_nat mu-bound-unique
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis lambdaEquality applyEquality setElimination rename cumulativity independent_isectElimination natural_numberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll because_Cache imageElimination lambdaFormation axiomEquality equalityTransitivity equalitySymmetry universeEquality independent_functionElimination setEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[L:T  List].  \mforall{}[i:\mBbbN{}||L||].
mu(\mlambda{}i@0.(eq  L[i]  L[i@0]))  =  i  supposing  no\_repeats(T;L)

Date html generated: 2016_05_14-PM-03_32_14
Last ObjectModification: 2016_01_14-PM-11_19_53

Theory : decidable!equality

Home Index