### Nuprl Lemma : fix-step

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

Proof

Definitions occuring in Statement :  fix: `f**(x)` retraction: `retraction(T;f)` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` implies: `P `` Q` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` fix: `f**(x)` ycomb: `Y` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` eqof: `eqof(d)` deq: `EqDecider(T)` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A`
Lemmas referenced :  retraction_wf equal_wf squash_wf true_wf fix_wf deq_wf iff_weakening_equal eqof_wf bool_wf eqtt_to_assert safe-assert-deq eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot assert_wf bnot_wf not_wf bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality functionExtensionality applyEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry lambdaFormation lambdaEquality imageElimination universeEquality independent_isectElimination functionEquality imageMemberEquality baseClosed natural_numberEquality productElimination independent_functionElimination unionElimination equalityElimination setElimination rename dependent_pairFormation promote_hyp dependent_functionElimination instantiate voidElimination independent_pairFormation impliesFunctionality

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

Date html generated: 2018_05_21-PM-07_47_07
Last ObjectModification: 2017_07_26-PM-05_24_39

Theory : general

Home Index