### Nuprl Lemma : final-iterate-property

`∀[A:Type]`
`  ∀f:A ⟶ (A + Top)`
`    (SWellFounded(p-graph(A;f) y x)`
`    `` (∀x:A`
`          ∃n:ℕ`
`           ((↑can-apply(f^n;x)) c∧ ((final-iterate(f;x) = do-apply(f^n;x) ∈ A) ∧ (¬↑can-apply(f;final-iterate(f;x)))))))`

Proof

Definitions occuring in Statement :  final-iterate: `final-iterate(f;x)` p-graph: `p-graph(A;f)` p-fun-exp: `f^n` do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` strongwellfounded: `SWellFounded(R[x; y])` nat: `ℕ` assert: `↑b` uall: `∀[x:A]. B[x]` top: `Top` cand: `A c∧ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` p-graph: `p-graph(A;f)` strongwellfounded: `SWellFounded(R[x; y])` exists: `∃x:A. B[x]` final-iterate: `final-iterate(f;x)` member: `t ∈ T` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` cand: `A c∧ B` so_apply: `x[s]` uimplies: `b supposing a` top: `Top` and: `P ∧ Q` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` guard: `{T}` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` assert: `↑b` ifthenelse: `if b then t else f fi ` can-apply: `can-apply(f;x)` isl: `isl(x)` p-fun-exp: `f^n` primrec: `primrec(n;b;c)` p-id: `p-id()` btrue: `tt` true: `True` do-apply: `do-apply(f;x)` outl: `outl(x)` less_than: `a < b` squash: `↓T` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff`
Lemmas referenced :  le_wf all_wf subtract_wf exists_wf nat_wf assert_wf can-apply_wf p-fun-exp_wf subtype_rel_dep_function top_wf subtype_rel_union equal_wf final-iterate_wf do-apply_wf not_wf set_wf less_than_wf primrec-wf2 decidable__le satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf strongwellfounded_wf p-graph_wf2 bool_wf nat_properties intformand_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_term_value_constant_lemma int_formula_prop_less_lemma equal-wf-T-base bnot_wf false_wf itermSubtract_wf int_term_value_subtract_lemma itermAdd_wf int_term_value_add_lemma p-fun-exp-add1-sq eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut hypothesis addLevel sqequalHypSubstitution sqequalRule thin productElimination introduction extract_by_obid isectElimination applyEquality functionExtensionality hypothesisEquality cumulativity because_Cache natural_numberEquality rename setElimination lambdaEquality functionEquality productEquality unionEquality independent_isectElimination isect_memberEquality voidElimination voidEquality equalityTransitivity equalitySymmetry intEquality dependent_functionElimination independent_functionElimination unionElimination dependent_pairFormation int_eqEquality computeAll levelHypothesis universeEquality independent_pairFormation applyLambdaEquality baseClosed dependent_set_memberEquality imageElimination addEquality equalityElimination

Latex:
\mforall{}[A:Type]
\mforall{}f:A  {}\mrightarrow{}  (A  +  Top)
(SWellFounded(p-graph(A;f)  y  x)
{}\mRightarrow{}  (\mforall{}x:A
\mexists{}n:\mBbbN{}
((\muparrow{}can-apply(f\^{}n;x))
c\mwedge{}  ((final-iterate(f;x)  =  do-apply(f\^{}n;x))  \mwedge{}  (\mneg{}\muparrow{}can-apply(f;final-iterate(f;x)))))))

Date html generated: 2018_05_21-PM-07_36_50
Last ObjectModification: 2017_07_26-PM-05_10_49

Theory : general

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