Nuprl Lemma : fix_wf

[T:Type]. ∀[eq:EqDecider(T)]. ∀[f:T ⟶ T].  ∀[x:T]. (f**(x) ∈ T) supposing retraction(T;f)


Proof




Definitions occuring in Statement :  fix: f**(x) retraction: retraction(T;f) deq: EqDecider(T) uimplies: supposing a uall: [x:A]. B[x] member: t ∈ T function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a retraction: retraction(T;f) exists: x:A. B[x] all: x:A. B[x] prop: nat: implies:  Q false: False ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) not: ¬A top: Top and: P ∧ Q subtype_rel: A ⊆B guard: {T} decidable: Dec(P) or: P ∨ Q le: A ≤ B less_than': less_than'(a;b) uiff: uiff(P;Q) fix: f**(x) ycomb: Y less_than: a < b squash: T bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  bfalse: ff iff: ⇐⇒ Q rev_implies:  Q
Lemmas referenced :  retraction_wf deq_wf 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 decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma add_nat_wf false_wf le_wf nat_wf add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma equal_wf decidable__lt eqof_wf bool_wf equal-wf-T-base assert_wf bnot_wf not_wf uiff_transitivity eqtt_to_assert safe-assert-deq iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin sqequalHypSubstitution productElimination dependent_functionElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry sqequalRule axiomEquality isect_memberEquality isectElimination because_Cache extract_by_obid cumulativity functionExtensionality applyEquality functionEquality universeEquality lambdaFormation setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination applyLambdaEquality unionElimination dependent_set_memberEquality addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed imageElimination equalityElimination impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[f:T  {}\mrightarrow{}  T].    \mforall{}[x:T].  (f**(x)  \mmember{}  T)  supposing  retraction(T;f)



Date html generated: 2018_05_21-PM-07_46_45
Last ObjectModification: 2017_07_26-PM-05_24_20

Theory : general


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