`∀[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 :  member: `t ∈ T` rel_exp_add complete_nat_ind_with_y complete_nat_measure_ind genrec: genrec bool_cases uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` uimplies: `b supposing a` strict4: `strict4(F)` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` has-value: `(a)↓` prop: `ℙ` guard: `{T}` or: `P ∨ Q` squash: `↓T` eq_int: `(i =z j)` btrue: `tt` bfalse: `ff` any: `any x` subtract: `n - m`
Lemmas referenced :  rel_exp_add lifting-strict-decide top_wf equal_wf has-value_wf_base base_wf is-exception_wf lifting-strict-int_eq complete_nat_ind_with_y complete_nat_measure_ind bool_cases
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution isectElimination baseClosed isect_memberEquality voidElimination voidEquality independent_isectElimination independent_pairFormation lambdaFormation callbyvalueDecide hypothesisEquality equalityTransitivity equalitySymmetry unionEquality unionElimination sqleReflexivity dependent_functionElimination independent_functionElimination baseApply closedConclusion decideExceptionCases inrFormation because_Cache imageMemberEquality imageElimination exceptionSqequal inlFormation

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_19
Last ObjectModification: 2017_02_27-PM-03_10_11

Theory : relations

Home Index