### Nuprl Lemma : decidable__fun-connected

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

Proof

Definitions occuring in Statement :  retraction: `retraction(T;f)` fun-connected: `y is f*(x)` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` retraction: `retraction(T;f)` exists: `∃x:A. B[x]` nat: `ℕ` guard: `{T}` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` le: `A ≤ B` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)`
Lemmas referenced :  all_wf decidable_wf equal_wf retraction_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 less_than_wf nat_wf subtract_wf fun-connected_wf set_wf primrec-wf2 fun-connected_weakening_eq not_wf fun-connected-fixedpoint decidable__lt intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma fun-connected-step fun-connected_transitivity fun-connected-step-back add_nat_wf false_wf le_wf decidable__le add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality sqequalRule lambdaEquality hypothesis functionExtensionality applyEquality functionEquality universeEquality productElimination equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache unionElimination inlFormation inrFormation independent_functionElimination imageElimination dependent_set_memberEquality addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed

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

Date html generated: 2018_05_21-PM-07_48_21
Last ObjectModification: 2017_07_26-PM-05_26_08

Theory : general

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