Nuprl Lemma : ts-reachable-induction3

ts:transition-system{i:l}
  ∀[P:ts-reachable(ts) ⟶ ℙ]
    (P[ts-init(ts)]
     (∀x:ts-type(ts). ((ts-init(ts) (ts-rel(ts)^*) x)  (∀y:ts-reachable(ts). (P[x]  (x ts-rel(ts) y)  P[y]))))
     {∀x:ts-type(ts). ((ts-init(ts) (ts-rel(ts)^*) x)  P[x])})


Proof




Definitions occuring in Statement :  ts-reachable: ts-reachable(ts) ts-rel: ts-rel(ts) ts-init: ts-init(ts) ts-type: ts-type(ts) transition-system: transition-system{i:l} rel_star: R^* uall: [x:A]. B[x] prop: guard: {T} infix_ap: y so_apply: x[s] all: x:A. B[x] implies:  Q function: x:A ⟶ B[x]
Definitions unfolded in proof :  guard: {T} all: x:A. B[x] uall: [x:A]. B[x] implies:  Q member: t ∈ T ts-reachable: ts-reachable(ts) infix_ap: y subtype_rel: A ⊆B rel_star: R^* exists: x:A. B[x] prop: so_lambda: λ2x.t[x] so_apply: x[s] uimplies: supposing a and: P ∧ Q cand: c∧ B or: P ∨ Q iff: ⇐⇒ Q rev_implies:  Q less_than: a < b squash: T less_than': less_than'(a;b) false: False nat: le: A ≤ B not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top decidable: Dec(P)
Lemmas referenced :  rel_star_wf ts-type_wf ts-rel_wf ts-init_wf all_wf infix_ap_wf ts-reachable_wf subtype_rel_set subtype_rel_wf subtype_rel_dep_function ts-init_wf_reachable transition-system_wf equal_wf rel_exp_iff rel_star_weakening rel_exp_wf false_wf le_wf satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf decidable__le intformnot_wf intformle_wf int_formula_prop_not_lemma int_formula_prop_le_lemma subtract_wf itermSubtract_wf int_term_value_subtract_lemma set_wf less_than_wf primrec-wf2 nat_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation isect_memberFormation cut dependent_set_memberEquality hypothesisEquality hypothesis applyEquality introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache productElimination lambdaEquality functionEquality instantiate cumulativity universeEquality functionExtensionality independent_isectElimination setElimination rename setEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination unionElimination imageElimination voidElimination addLevel hyp_replacement levelHypothesis natural_numberEquality independent_pairFormation dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll

Latex:
\mforall{}ts:transition-system\{i:l\}
    \mforall{}[P:ts-reachable(ts)  {}\mrightarrow{}  \mBbbP{}]
        (P[ts-init(ts)]
        {}\mRightarrow{}  (\mforall{}x:ts-type(ts)
                    ((ts-init(ts)  rel\_star(ts-type(ts);  ts-rel(ts))  x)
                    {}\mRightarrow{}  (\mforall{}y:ts-reachable(ts).  (P[x]  {}\mRightarrow{}  (x  ts-rel(ts)  y)  {}\mRightarrow{}  P[y]))))
        {}\mRightarrow{}  \{\mforall{}x:ts-type(ts).  ((ts-init(ts)  rel\_star(ts-type(ts);  ts-rel(ts))  x)  {}\mRightarrow{}  P[x])\})



Date html generated: 2018_05_21-PM-08_01_36
Last ObjectModification: 2017_07_26-PM-05_38_22

Theory : general


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