Nuprl Lemma : rel_star-iff-path

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (x (R^*) y `⇐⇒` ∃L:T List. rel-path-between(T;R;x;y;L))`

Proof

Definitions occuring in Statement :  rel-path-between: `rel-path-between(T;R;x;y;L)` list: `T List` rel_star: `R^*` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` rel_star: `R^*` infix_ap: `x f y` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` so_apply: `x[s]` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` cand: `A c∧ B` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` rel-path-between: `rel-path-between(T;R;x;y;L)` less_than: `a < b` squash: `↓T`
Lemmas referenced :  rel-path-between_wf exists_wf nat_wf list_wf equal_wf length_wf rel_exp-iff-path rel_exp_wf iff_wf decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermVar_wf itermAdd_wf itermSubtract_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_wf subtract_wf decidable__le intformand_wf intformle_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_formula_prop_less_lemma le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule cut independent_pairFormation sqequalHypSubstitution productElimination thin dependent_pairFormation hypothesisEquality hypothesis introduction extract_by_obid isectElimination lambdaEquality productEquality intEquality addEquality setElimination rename natural_numberEquality addLevel independent_functionElimination because_Cache dependent_functionElimination cumulativity applyEquality functionEquality universeEquality unionElimination independent_isectElimination approximateComputation int_eqEquality isect_memberEquality voidElimination voidEquality dependent_set_memberEquality imageElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}x,y:T.    (x  (R\^{}*)  y  \mLeftarrow{}{}\mRightarrow{}  \mexists{}L:T  List.  rel-path-between(T;R;x;y;L))

Date html generated: 2019_06_20-PM-02_02_11
Last ObjectModification: 2018_08_24-PM-11_36_01

Theory : relations2

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