### Nuprl Lemma : rel_plus-restriction-equiv

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].  ((∀x,y:T.  ((P[y] ∧ (R x y)) `` P[x])) `` (∀x,y:T.  (R|P+ x y `⇐⇒` R+|P x y)))`

Proof

Definitions occuring in Statement :  rel_plus: `R+` rel-restriction: `R|P` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rel_implies: `R1 => R2` infix_ap: `x f y` nat_plus: `ℕ+` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` rel_exp: `R^n` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` rel-restriction: `R|P` cand: `A c∧ B` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` squash: `↓T` true: `True` trans: `Trans(T;x,y.E[x; y])` rel_plus: `R+`
Lemmas referenced :  rel_plus_wf rel-restriction_wf all_wf rel_plus-of-restriction nat_plus_properties 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 primrec-wf-nat-plus nat_plus_subtype_nat nat_plus_wf false_wf rel-rel-plus itermAdd_wf int_term_value_add_lemma eq_int_wf intformeq_wf int_formula_prop_eq_lemma assert_wf bnot_wf not_wf equal-wf-base int_subtype_base bool_cases subtype_base_sq bool_wf bool_subtype_base eqtt_to_assert assert_of_eq_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot add-subtract-cancel exists_wf equal_wf add-associates add-swap add-commutes zero-add squash_wf true_wf nat_wf and_wf less_than_wf rel_plus_trans
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation applyEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality functionExtensionality hypothesis lambdaEquality sqequalRule functionEquality universeEquality productEquality because_Cache dependent_functionElimination independent_functionElimination rename setElimination dependent_set_memberEquality natural_numberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll productElimination equalitySymmetry hyp_replacement applyLambdaEquality addEquality equalityTransitivity baseApply closedConclusion baseClosed instantiate impliesFunctionality imageElimination imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}x,y:T.    ((P[y]  \mwedge{}  (R  x  y))  {}\mRightarrow{}  P[x]))  {}\mRightarrow{}  (\mforall{}x,y:T.    (R|P\msupplus{}  x  y  \mLeftarrow{}{}\mRightarrow{}  R\msupplus{}|P  x  y)))

Date html generated: 2017_04_17-AM-09_28_04
Last ObjectModification: 2017_02_27-PM-05_28_30

Theory : relations2

Home Index