### Nuprl Lemma : rel_exp_add_iff

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀m,n:ℕ. ∀x,z:T.  (x R^m + n z `⇐⇒` ∃y:T. ((x R^m y) ∧ (y R^n 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]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` guard: `{T}` sq_stable: `SqStable(P)` squash: `↓T` so_apply: `x[s]` rel_exp: `R^n` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` infix_ap: `x f y` exists: `∃x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` cand: `A c∧ B` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` ge: `i ≥ j ` nat_plus: `ℕ+` less_than: `a < b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  all_wf nat_wf iff_wf infix_ap_wf rel_exp_wf subtract_wf add_nat_wf decidable__le false_wf not-le-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel le_wf sq_stable__le equal_wf exists_wf set_wf less_than_wf primrec-wf2 and_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int le_weakening2 eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int minus-zero add-mul-special zero-mul int_subtype_base le_reflexive one-mul two-mul mul-distributes-right mul-associates omega-shadow nat_properties general_arith_equation1 not-equal-2 less_than_transitivity1 le_weakening less_than_irreflexivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin hypothesisEquality because_Cache rename setElimination introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesis sqequalRule lambdaEquality cumulativity instantiate universeEquality dependent_set_memberEquality addEquality natural_numberEquality dependent_functionElimination unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination independent_isectElimination applyEquality isect_memberEquality voidEquality intEquality minusEquality imageMemberEquality baseClosed imageElimination equalityTransitivity equalitySymmetry functionExtensionality productEquality functionEquality dependent_pairFormation addLevel hyp_replacement applyLambdaEquality levelHypothesis equalityElimination promote_hyp multiplyEquality impliesFunctionality existsFunctionality andLevelFunctionality existsLevelFunctionality

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

Date html generated: 2017_04_14-AM-07_38_17
Last ObjectModification: 2017_02_27-PM-03_10_39

Theory : relations

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