Nuprl Lemma : ts-reachable-induction
∀[ts:transition-system{i:l}]. ∀[P:ts-reachable(ts) ⟶ ℙ].
((∀x:ts-reachable(ts). SqStable(P[x]))
⇒ (∀x,y:ts-reachable(ts). (P[x] ⇒ (x ts-rel(ts) y) ⇒ P[y])) ⇒ {∀x:ts-reachable(ts). P[x]}
supposing P[ts-init(ts)])
Proof
Definitions occuring in Statement :
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts),
ts-init: ts-init(ts),
transition-system: transition-system{i:l},
sq_stable: SqStable(P),
uimplies: b supposing a,
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},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
ts-reachable: ts-reachable(ts),
sq_stable: SqStable(P),
rel_star: R^*,
infix_ap: x f y,
exists: ∃x:A. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
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 :
ts-reachable_wf,
subtype_rel_wf,
ts-type_wf,
all_wf,
infix_ap_wf,
subtype_rel_set,
ts-rel_wf,
subtype_rel_dep_function,
ts-init_wf_reachable,
sq_stable_wf,
transition-system_wf,
equal_wf,
rel_exp_iff,
ts-init_wf,
rel_star_weakening,
rel_star_wf,
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,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
introduction,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
setEquality,
universeEquality,
cumulativity,
because_Cache,
functionEquality,
functionExtensionality,
instantiate,
independent_isectElimination,
unionElimination,
voidElimination,
hyp_replacement,
dependent_set_memberEquality,
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{}].
((\mforall{}x:ts-reachable(ts). SqStable(P[x]))
{}\mRightarrow{} (\mforall{}x,y:ts-reachable(ts). (P[x] {}\mRightarrow{} (x ts-rel(ts) y) {}\mRightarrow{} P[y])) {}\mRightarrow{} \{\mforall{}x:ts-reachable(ts). P[x]\}
supposing P[ts-init(ts)])
Date html generated:
2018_05_21-PM-08_01_23
Last ObjectModification:
2017_07_26-PM-05_38_11
Theory : general
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