### Nuprl Lemma : transitive-closure-transitive

`∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].  UniformlyTrans(A;x,y.x TC(R) y)`

Proof

Definitions occuring in Statement :  transitive-closure: `TC(R)` utrans: `UniformlyTrans(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` utrans: `UniformlyTrans(T;x,y.E[x; y])` implies: `P `` Q` transitive-closure: `TC(R)` infix_ap: `x f y` member: `t ∈ T` prop: `ℙ` subtype_rel: `A ⊆r B` and: `P ∧ Q` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` rel_path: `rel_path(A;L;x;y)` pi1: `fst(t)` pi2: `snd(t)` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` guard: `{T}`
Lemmas referenced :  transitive-closure_wf append_wf list_induction all_wf rel_path_wf list_wf list_ind_nil_lemma and_wf equal_wf list_ind_cons_lemma length-append decidable__lt length_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf less_than_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation rename sqequalHypSubstitution sqequalRule setElimination thin applyEquality cut introduction extract_by_obid isectElimination cumulativity hypothesisEquality functionExtensionality hypothesis functionEquality universeEquality dependent_set_memberEquality productEquality lambdaEquality productElimination because_Cache independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality addLevel hyp_replacement equalitySymmetry independent_pairFormation applyLambdaEquality levelHypothesis natural_numberEquality addEquality unionElimination imageElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality computeAll

Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].    UniformlyTrans(A;x,y.x  TC(R)  y)

Date html generated: 2017_04_17-AM-09_25_45
Last ObjectModification: 2017_02_27-PM-05_26_20

Theory : relations2

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