### Nuprl Lemma : transitive-closure-cases

`∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].  ∀x,y:A.  ((x TC(R) y) `` ((x R y) ∨ (∃z:A. ((x R z) ∧ (z TC(R) y)))))`

Proof

Definitions occuring in Statement :  transitive-closure: `TC(R)` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `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` transitive-closure: `TC(R)` infix_ap: `x f y` member: `t ∈ T` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` cons: `[a / b]` false: `False` and: `P ∧ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` length: `||as||` list_ind: list_ind nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rel_path: `rel_path(A;L;x;y)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` pi1: `fst(t)` pi2: `snd(t)` guard: `{T}` exists: `∃x:A. B[x]` cand: `A c∧ B` ge: `i ≥ j ` decidable: `Dec(P)` le: `A ≤ B` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A`
Lemmas referenced :  list-cases product_subtype_list transitive-closure_wf exists_wf list_ind_cons_lemma list_ind_nil_lemma length_of_cons_lemma length_of_nil_lemma equal_wf cons_wf non_neg_length decidable__lt length_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf intformle_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_le_lemma int_formula_prop_wf rel_path_wf less_than_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule rename setElimination thin cut productEquality hypothesisEquality applyEquality hypothesis lambdaEquality universeEquality introduction extract_by_obid isectElimination dependent_functionElimination unionElimination promote_hyp hypothesis_subsumption productElimination cumulativity functionExtensionality functionEquality imageElimination voidElimination inlFormation because_Cache isect_memberEquality voidEquality hyp_replacement equalityTransitivity equalitySymmetry applyLambdaEquality instantiate inrFormation dependent_pairFormation independent_pairFormation dependent_set_memberEquality dependent_pairEquality natural_numberEquality addEquality independent_isectElimination int_eqEquality intEquality computeAll

Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].    \mforall{}x,y:A.    ((x  TC(R)  y)  {}\mRightarrow{}  ((x  R  y)  \mvee{}  (\mexists{}z:A.  ((x  R  z)  \mwedge{}  (z  TC(R)  y)))))

Date html generated: 2017_04_17-AM-09_25_49
Last ObjectModification: 2017_02_27-PM-05_26_27

Theory : relations2

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