### Nuprl Lemma : rel_plus_strongwellfounded

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (SWellFounded(x R y) `` SWellFounded(x R+ y))`

Proof

Definitions occuring in Statement :  strongwellfounded: `SWellFounded(R[x; y])` rel_plus: `R+` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  strongwellfounded: `SWellFounded(R[x; y])` uall: `∀[x:A]. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` rel_plus: `R+` infix_ap: `x f y` nat_plus: `ℕ+` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` 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` so_apply: `x[s]` rel_exp: `R^n` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` le: `A ≤ B` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` less_than: `a < b` squash: `↓T` true: `True`
Lemmas referenced :  nat_plus_properties all_wf infix_ap_wf rel_exp_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_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 le_wf less_than_wf primrec-wf-nat-plus nat_plus_subtype_nat nat_plus_wf rel_plus_wf exists_wf nat_wf false_wf itermAdd_wf int_term_value_add_lemma eq_int_wf bool_wf equal-wf-base int_subtype_base assert_wf bnot_wf not_wf uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot equal_wf intformeq_wf int_formula_prop_eq_lemma decidable__equal_int subtype_base_sq subtract_wf add-subtract-cancel decidable__lt add-associates add-swap add-commutes zero-add squash_wf true_wf and_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin dependent_pairFormation hypothesisEquality cut rename introduction extract_by_obid isectElimination hypothesis setElimination cumulativity lambdaEquality because_Cache functionEquality instantiate universeEquality dependent_set_memberEquality dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality applyEquality independent_functionElimination equalitySymmetry hyp_replacement applyLambdaEquality addEquality baseApply closedConclusion baseClosed equalityElimination impliesFunctionality equalityTransitivity productEquality imageElimination imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (SWellFounded(x  R  y)  {}\mRightarrow{}  SWellFounded(x  R\msupplus{}  y))

Date html generated: 2017_04_17-AM-09_26_42
Last ObjectModification: 2017_02_27-PM-05_27_57

Theory : relations2

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