`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀m,n:ℕ.  ∀[x,y,z:T].  ((x R^m y) `` (y R^n z) `` (x R^m + 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]` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` nat: `ℕ` all: `∀x:A. B[x]` guard: `{T}` sq_stable: `SqStable(P)` squash: `↓T` so_apply: `x[s]` infix_ap: `x f y` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` int_seg: `{i..j-}` lelt: `i ≤ j < k` rel_exp: `R^n` or: `P ∨ Q` sq_type: `SQType(T)` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` subtract: `n - m` ge: `i ≥ j ` top: `Top` nat_plus: `ℕ+` less_than: `a < b` true: `True` exists: `∃x:A. B[x]` cand: `A c∧ B` bool: `𝔹` unit: `Unit` it: `⋅` decidable: `Dec(P)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination sqequalRule lambdaEquality hypothesis cumulativity hypothesisEquality because_Cache functionEquality universeEquality functionExtensionality applyEquality dependent_set_memberEquality addEquality setElimination rename lambdaFormation natural_numberEquality independent_functionElimination imageMemberEquality baseClosed imageElimination equalityTransitivity equalitySymmetry dependent_functionElimination independent_isectElimination independent_pairFormation productElimination intEquality unionElimination impliesFunctionality addLevel hyp_replacement applyLambdaEquality levelHypothesis multiplyEquality minusEquality isect_memberEquality voidElimination voidEquality promote_hyp dependent_pairFormation productEquality equalityElimination

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

Date html generated: 2017_04_14-AM-07_38_14
Last ObjectModification: 2017_02_27-PM-03_10_26

Theory : relations

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