### Nuprl Lemma : rel_inverse_exp

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀n:ℕ. ∀x,y:T.  (x R^n^-1 y `⇐⇒` x R^-1^n y)`

Proof

Definitions occuring in Statement :  rel_inverse: `R^-1` rel_exp: `R^n` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` 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` so_apply: `x[s]` infix_ap: `x f y` rel_inverse: `R^-1` rel_exp: `R^n` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` guard: `{T}` exists: `∃x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` bfalse: `ff` sq_type: `SQType(T)`
Lemmas referenced :  all_wf iff_wf infix_ap_wf rel_inverse_wf rel_exp_wf subtract_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 set_wf less_than_wf primrec-wf2 nat_wf equal_wf le_weakening2 eq_int_wf bool_wf equal-wf-base int_subtype_base assert_wf less_than_transitivity1 le_weakening less_than_irreflexivity bnot_wf not_wf exists_wf uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot subtype_base_sq rel_exp_add
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 instantiate universeEquality dependent_set_memberEquality natural_numberEquality hypothesis dependent_functionElimination unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination independent_isectElimination addEquality applyEquality isect_memberEquality voidEquality intEquality minusEquality functionExtensionality functionEquality equalitySymmetry independent_pairEquality axiomEquality baseApply closedConclusion baseClosed equalityTransitivity productEquality equalityElimination impliesFunctionality dependent_pairFormation

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}n:\mBbbN{}.  \mforall{}x,y:T.    (x  rel\_exp(T;  R;  n)\^{}-1  y  \mLeftarrow{}{}\mRightarrow{}  x  rel\_exp(T;  R\^{}-1;  n)  y)

Date html generated: 2017_04_14-AM-07_38_45
Last ObjectModification: 2017_02_27-PM-03_10_32

Theory : relations

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