### Nuprl Lemma : fun-path-no_repeats

`∀[T:Type]. ∀[f:T ⟶ T].  ∀[L:T List]. ∀[x,y:T].  no_repeats(T;L) supposing x=f*(y) via L supposing retraction(T;f)`

Proof

Definitions occuring in Statement :  retraction: `retraction(T;f)` fun-path: `y=f*(x) via L` no_repeats: `no_repeats(T;l)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` retraction: `retraction(T;f)` exists: `∃x:A. B[x]` implies: `P `` Q` prop: `ℙ` all: `∀x:A. B[x]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` subtype_rel: `A ⊆r B` nat: `ℕ` so_apply: `x[s]` so_apply: `x[s1;s2;s3]` uiff: `uiff(P;Q)` sq_type: `SQType(T)` select: `L[n]` cons: `[a / b]` ge: `i ≥ j ` le: `A ≤ B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` l_before: `x before y ∈ l` sublist: `L1 ⊆ L2` increasing: `increasing(f;k)`
Lemmas referenced :  no_repeats_witness fun-path_wf list_wf retraction_wf fun-path-induction all_wf int_seg_wf length_wf less_than_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_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_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma nat_wf length_of_cons_lemma length_of_nil_lemma cons_wf nil_wf itermAdd_wf int_term_value_add_lemma add-is-int-iff false_wf member-less_than not_wf equal_wf decidable__equal_int subtype_base_sq int_subtype_base squash_wf true_wf select_cons_tl intformeq_wf int_formula_prop_eq_lemma non_neg_length iff_weakening_equal subtract_wf itermSubtract_wf int_term_value_subtract_lemma lelt_wf le_weakening2 no_repeats_iff length_wf_nat nat_properties l_before_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin extract_by_obid isectElimination hypothesisEquality independent_functionElimination hypothesis cumulativity functionExtensionality applyEquality sqequalRule isect_memberEquality because_Cache equalityTransitivity equalitySymmetry functionEquality universeEquality dependent_functionElimination lambdaEquality natural_numberEquality setElimination rename independent_isectElimination unionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination lambdaFormation addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed instantiate imageMemberEquality hyp_replacement applyLambdaEquality dependent_set_memberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  T].
\mforall{}[L:T  List].  \mforall{}[x,y:T].    no\_repeats(T;L)  supposing  x=f*(y)  via  L  supposing  retraction(T;f)

Date html generated: 2018_05_21-PM-07_47_31
Last ObjectModification: 2017_07_26-PM-05_25_26

Theory : general

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