Nuprl Lemma : decidable__rel_plus

`∀[T:Type]`
`  ((∀x,y:T.  Dec(x = y ∈ T))`
`  `` (∀[R:T ⟶ T ⟶ ℙ]. (SWellFounded(x R y) `` rel_finite(T;R) `` (∀x,y:T.  Dec(x R y)) `` (∀x,y:T.  Dec(x R+ y)))))`

Proof

Definitions occuring in Statement :  strongwellfounded: `SWellFounded(R[x; y])` rel_finite: `rel_finite(T;R)` rel_plus: `R+` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` 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]` member: `t ∈ T` implies: `P `` Q` strongwellfounded: `SWellFounded(R[x; y])` exists: `∃x:A. B[x]` pi1: `fst(t)` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` infix_ap: `x f y` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` uimplies: `b supposing a` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` nat: `ℕ` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` rel_plus: `R+` iff: `P `⇐⇒` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` rev_implies: `P `` Q`
Lemmas referenced :  less_than_wf rel_finite_wf decidable__rel_exp_finite decidable__exists_int_seg rel_plus_wf decidable_functionality infix_ap_wf int_seg_wf int_seg_subtype_nat_plus exists_wf false_wf int_seg_subtype_nat int_formula_prop_less_lemma intformless_wf decidable__lt nat_plus_wf nat_plus_subtype_nat rel_exp_wf int_formula_prop_wf int_term_value_add_lemma int_formula_prop_not_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma itermAdd_wf intformnot_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf satisfiable-full-omega-tt le_wf nat_properties decidable__le nat_plus_properties equal_wf decidable_wf all_wf strongwellfounded_wf strongwellfounded_rel_exp
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation promote_hyp rename productElimination sqequalRule lambdaEquality applyEquality universeEquality functionEquality cumulativity independent_isectElimination setElimination dependent_functionElimination because_Cache unionElimination equalityTransitivity equalitySymmetry setEquality intEquality natural_numberEquality dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll dependent_set_memberEquality introduction addEquality instantiate independent_functionElimination

Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T.    Dec(x  =  y))
{}\mRightarrow{}  (\mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}]
(SWellFounded(x  R  y)
{}\mRightarrow{}  rel\_finite(T;R)
{}\mRightarrow{}  (\mforall{}x,y:T.    Dec(x  R  y))
{}\mRightarrow{}  (\mforall{}x,y:T.    Dec(x  R\msupplus{}  y)))))

Date html generated: 2016_05_14-PM-03_52_29
Last ObjectModification: 2016_01_14-PM-11_11_07

Theory : relations2

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