### Nuprl Lemma : rel_plus-iff-path

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (x R+ y `⇐⇒` ∃L:T List. (1 < ||L|| ∧ rel-path-between(T;R;x;y;L)))`

Proof

Definitions occuring in Statement :  rel-path-between: `rel-path-between(T;R;x;y;L)` rel_plus: `R+` length: `||as||` list: `T List` less_than: `a < b` 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` and: `P ∧ Q` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` rel_plus: `R+` infix_ap: `x f y` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` cand: `A c∧ B` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` less_than: `a < b` squash: `↓T`
Lemmas referenced :  nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformeq_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_formula_prop_wf less_than_wf length_wf rel-path-between_wf exists_wf nat_plus_wf list_wf equal_wf subtract_wf false_wf not-lt-2 less-iff-le condition-implies-le minus-add minus-minus minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-swap le-add-cancel decidable__equal_int itermSubtract_wf int_term_value_subtract_lemma rel_exp-iff-path nat_plus_subtype_nat rel_exp_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule cut independent_pairFormation sqequalHypSubstitution productElimination thin dependent_pairFormation hypothesisEquality introduction extract_by_obid isectElimination hypothesis setElimination rename dependent_functionElimination natural_numberEquality equalityTransitivity equalitySymmetry unionElimination independent_isectElimination approximateComputation independent_functionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality productEquality addEquality dependent_set_memberEquality minusEquality applyEquality because_Cache imageElimination addLevel cumulativity functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}x,y:T.    (x  R\msupplus{}  y  \mLeftarrow{}{}\mRightarrow{}  \mexists{}L:T  List.  (1  <  ||L||  \mwedge{}  rel-path-between(T;R;x;y;L)))

Date html generated: 2019_06_20-PM-02_02_13
Last ObjectModification: 2018_08_24-PM-11_35_59

Theory : relations2

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