Nuprl Lemma : rel_exp_add_iff

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀m,n:ℕ. ∀x,z:T.  (x R^m ⇐⇒ ∃y:T. ((x R^m y) ∧ (y R^n z)))


Proof




Definitions occuring in Statement :  rel_exp: R^n nat: uall: [x:A]. B[x] prop: infix_ap: y all: x:A. B[x] exists: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q function: x:A ⟶ B[x] add: m universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] member: t ∈ T implies:  Q prop: so_lambda: λ2x.t[x] nat: decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q and: P ∧ Q not: ¬A rev_implies:  Q false: False uiff: uiff(P;Q) uimplies: supposing a subtract: m subtype_rel: A ⊆B top: Top le: A ≤ B less_than': less_than'(a;b) true: True guard: {T} sq_stable: SqStable(P) squash: T so_apply: x[s] rel_exp: R^n eq_int: (i =z j) ifthenelse: if then else fi  btrue: tt infix_ap: y exists: x:A. B[x] bool: 𝔹 unit: Unit it: cand: c∧ B bfalse: ff sq_type: SQType(T) bnot: ¬bb assert: b ge: i ≥  nat_plus: + less_than: a < b nequal: a ≠ b ∈ 
Lemmas referenced :  all_wf nat_wf iff_wf infix_ap_wf rel_exp_wf subtract_wf add_nat_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 sq_stable__le equal_wf exists_wf set_wf less_than_wf primrec-wf2 and_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int le_weakening2 eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int minus-zero add-mul-special zero-mul int_subtype_base le_reflexive one-mul two-mul mul-distributes-right mul-associates omega-shadow nat_properties general_arith_equation1 not-equal-2 less_than_transitivity1 le_weakening less_than_irreflexivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin hypothesisEquality because_Cache rename setElimination introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesis sqequalRule lambdaEquality cumulativity instantiate universeEquality dependent_set_memberEquality addEquality natural_numberEquality dependent_functionElimination unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination independent_isectElimination applyEquality isect_memberEquality voidEquality intEquality minusEquality imageMemberEquality baseClosed imageElimination equalityTransitivity equalitySymmetry functionExtensionality productEquality functionEquality dependent_pairFormation addLevel hyp_replacement applyLambdaEquality levelHypothesis equalityElimination promote_hyp multiplyEquality impliesFunctionality existsFunctionality andLevelFunctionality existsLevelFunctionality

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



Date html generated: 2017_04_14-AM-07_38_17
Last ObjectModification: 2017_02_27-PM-03_10_39

Theory : relations


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