### 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: `ℙ` guard: `{T}` infix_ap: `x f y` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  guard: `{T}` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` ts-reachable: `ts-reachable(ts)` infix_ap: `x f y` subtype_rel: `A ⊆r B` rel_star: `R^*` exists: `∃x:A. B[x]` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` and: `P ∧ Q` cand: `A c∧ B` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` nat: `ℕ` 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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