### Nuprl Lemma : rel-exp-add-iff

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀a,b:ℕ. ∀x,z:T.  (x R^a + b z `⇐⇒` ∃y:T. ((x R^a y) ∧ (y R^b z)))`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` add: `n + m` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` nat: `ℕ` ge: `i ≥ j ` 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_lambda: `λ2x.t[x]` so_apply: `x[s]` infix_ap: `x f y` cand: `A c∧ B` 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 ` btrue: `tt` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` subtype_rel: `A ⊆r B` bool: `𝔹` unit: `Unit` it: `⋅`
Lemmas referenced :  infix_ap_wf rel_exp_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf intformless_wf int_formula_prop_less_lemma all_wf nat_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma exists_wf set_wf less_than_wf primrec-wf2 zero-add false_wf eq_int_wf intformeq_wf int_formula_prop_eq_lemma assert_wf bnot_wf not_wf equal-wf-T-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 general_arith_equation1 equal-wf-base int_subtype_base uiff_transitivity equal_wf rel_exp_add
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesisEquality because_Cache universeEquality dependent_set_memberEquality addEquality natural_numberEquality setElimination rename hypothesis dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll functionExtensionality applyEquality functionEquality productEquality equalityTransitivity equalitySymmetry baseClosed productElimination independent_functionElimination impliesFunctionality baseApply closedConclusion equalityElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}a,b:\mBbbN{}.  \mforall{}x,z:T.    (x  R\^{}a  +  b  z  \mLeftarrow{}{}\mRightarrow{}  \mexists{}y:T.  ((x  R\^{}a  y)  \mwedge{}  (y  rel\_exp(T;  R;  b)  z)))

Date html generated: 2017_04_17-AM-09_28_17
Last ObjectModification: 2017_02_27-PM-05_29_07

Theory : relations2

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