### Nuprl Lemma : rel_exp_iff

`∀n:ℕ`
`  ∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`    ∀x,y:T.  (x R^n y `⇐⇒` (∃z:T. (0 < n c∧ ((x R^n - 1 z) ∧ (z R y)))) ∨ ((n = 0 ∈ ℤ) ∧ (x = y ∈ T)))`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` cand: `A c∧ B` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` or: `P ∨ Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` subtract: `n - m` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` 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` and: `P ∧ Q` cand: `A c∧ B` subtype_rel: `A ⊆r B` so_apply: `x[s]` ge: `i ≥ j ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` guard: `{T}` infix_ap: `x f y` rel_exp: `R^n` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` btrue: `tt` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` le: `A ≤ B` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` subtract: `n - m`
Lemmas referenced :  uall_wf all_wf iff_wf infix_ap_wf rel_exp_wf decidable__le subtract_wf 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 or_wf exists_wf less_than_wf equal-wf-base int_subtype_base equal_wf set_wf primrec-wf2 nat_properties equal-wf-T-base nat_wf false_wf eq_int_wf bool_wf 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 intformeq_wf int_formula_prop_eq_lemma less_than_transitivity1 le_weakening less_than_irreflexivity and_wf decidable__equal_int subtype_base_sq bool_cases bool_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin rename setElimination instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination universeEquality sqequalRule lambdaEquality functionEquality cumulativity hypothesisEquality because_Cache dependent_set_memberEquality dependent_functionElimination natural_numberEquality hypothesis unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality applyEquality productEquality baseApply closedConclusion baseClosed isect_memberFormation inrFormation imageElimination productElimination equalityTransitivity equalitySymmetry equalityElimination independent_functionElimination impliesFunctionality inlFormation addLevel hyp_replacement applyLambdaEquality levelHypothesis

Latex:
\mforall{}n:\mBbbN{}
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}x,y:T.
(x  R\^{}n  y  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}z:T.  (0  <  n  c\mwedge{}  ((x  R\^{}n  -  1  z)  \mwedge{}  (z  R  y))))  \mvee{}  ((n  =  0)  \mwedge{}  (x  =  y)))

Date html generated: 2017_04_17-AM-09_26_24
Last ObjectModification: 2017_02_27-PM-05_28_16

Theory : relations2

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