### Nuprl Lemma : retraction-fixedpoint

`∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) `` (∀x:T. ∃y:T. (((f y) = y ∈ T) ∧ y is f*(x))))`

Proof

Definitions occuring in Statement :  retraction: `retraction(T;f)` fun-connected: `y is f*(x)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  retraction: `retraction(T;f)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` and: `P ∧ Q` cand: `A c∧ B` nat: `ℕ` guard: `{T}` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` less_than: `a < b` squash: `↓T` fun-connected: `y is f*(x)` fun-path: `y=f*(x) via L` subtract: `n - m` last: `last(L)` select: `L[n]` cons: `[a / b]` true: `True` int_seg: `{i..j-}` sq_type: `SQType(T)` lelt: `i ≤ j < k`
Lemmas referenced :  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 fun-connected-test2 equal_wf fun-connected_wf less_than_wf all_wf subtract_wf exists_wf set_wf primrec-wf2 nat_wf add_nat_wf false_wf le_wf decidable__le add-is-int-iff intformnot_wf itermAdd_wf intformeq_wf int_formula_prop_not_lemma int_term_value_add_lemma int_formula_prop_eq_lemma decidable__lt or_wf itermSubtract_wf int_term_value_subtract_lemma fun-connected_transitivity cons_wf nil_wf fun-path_wf length_of_cons_lemma length_of_nil_lemma reduce_hd_cons_lemma decidable__equal_int subtype_base_sq int_subtype_base int_seg_properties select_wf int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin cut dependent_pairFormation because_Cache applyEquality functionExtensionality hypothesisEquality cumulativity introduction extract_by_obid isectElimination equalityTransitivity hypothesis equalitySymmetry applyLambdaEquality setElimination rename natural_numberEquality independent_isectElimination lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll productEquality functionEquality dependent_set_memberEquality addEquality unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed independent_functionElimination universeEquality imageElimination imageMemberEquality instantiate independent_pairEquality axiomEquality hyp_replacement

Latex:
\mforall{}[T:Type].  \mforall{}f:T  {}\mrightarrow{}  T.  (retraction(T;f)  {}\mRightarrow{}  (\mforall{}x:T.  \mexists{}y:T.  (((f  y)  =  y)  \mwedge{}  y  is  f*(x))))

Date html generated: 2018_05_21-PM-07_47_52
Last ObjectModification: 2017_07_26-PM-05_25_46

Theory : general

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