### Nuprl Lemma : rel_exp_functionality_wrt_iff

`∀[T:Type]. ∀[R,Q:T ⟶ T ⟶ ℙ].  ((∀x,y:T.  (R x y `⇐⇒` Q x y)) `` (∀n:ℕ. ∀x,y:T.  (R^n x y `⇐⇒` Q^n x y)))`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `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]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` 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` and: `P ∧ Q` so_apply: `x[s]` rel_exp: `R^n` subtype_rel: `A ⊆r B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` cand: `A c∧ B` infix_ap: `x f y`
Lemmas referenced :  all_wf iff_wf rel_exp_wf subtract_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_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_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf set_wf less_than_wf primrec-wf2 nat_wf eq_int_wf bool_wf equal-wf-base assert_wf equal_wf bnot_wf not_wf intformeq_wf int_formula_prop_eq_lemma exists_wf infix_ap_wf uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin rename setElimination introduction extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesisEquality sqequalRule lambdaEquality because_Cache applyEquality dependent_set_memberEquality natural_numberEquality hypothesis dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality functionEquality universeEquality baseClosed equalityTransitivity equalitySymmetry productEquality instantiate independent_functionElimination equalityElimination productElimination impliesFunctionality baseApply closedConclusion

Latex:
\mforall{}[T:Type].  \mforall{}[R,Q:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}x,y:T.    (R  x  y  \mLeftarrow{}{}\mRightarrow{}  Q  x  y))  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}x,y:T.    (rel\_exp(T;  R;  n)  x  y  \mLeftarrow{}{}\mRightarrow{}  rel\_exp(T;  Q;  n)  x  y)))

Date html generated: 2017_04_17-AM-09_27_45
Last ObjectModification: 2017_02_27-PM-05_28_12

Theory : relations2

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