`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀a:ℕ+. ∀x,z:T.  (x R^a z `⇐⇒` ∃y:T. ((x R^a - 1 y) ∧ (y R z)))`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` nat_plus: `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]` subtract: `n - m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` infix_ap: `x f y` subtype_rel: `A ⊆r B` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` nat: `ℕ` nat_plus: `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` so_apply: `x[s]` cand: `A c∧ B`
Lemmas referenced :  equal_wf and_wf less_than_wf decidable__lt int_formula_prop_eq_lemma intformeq_wf rel_exp_iff nat_plus_wf le_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_plus_properties subtract_wf infix_ap_wf exists_wf nat_plus_subtype_nat rel_exp_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation applyEquality cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis sqequalRule cumulativity lambdaEquality productEquality instantiate because_Cache universeEquality dependent_set_memberEquality setElimination rename natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality productElimination independent_functionElimination inlFormation

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}a:\mBbbN{}\msupplus{}.  \mforall{}x,z:T.    (x  R\^{}a  z  \mLeftarrow{}{}\mRightarrow{}  \mexists{}y:T.  ((x  R\^{}a  -  1  y)  \mwedge{}  (y  R  z)))

Date html generated: 2016_05_14-PM-03_56_12
Last ObjectModification: 2016_01_14-PM-11_11_47

Theory : relations2

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