Nuprl Lemma : ts-stable-star
∀ts:transition-system{i:l}
∀[P:ts-type(ts) ⟶ ℙ]. (ts-stable(ts;x.P[x]) ⇒ (∀x,y:ts-type(ts). (P[x] ⇒ (x (ts-rel(ts)^*) y) ⇒ P[y])))
Proof
Definitions occuring in Statement :
ts-stable: ts-stable(ts;x.P[x]),
ts-rel: ts-rel(ts),
ts-type: ts-type(ts),
transition-system: transition-system{i:l},
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
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 :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rel_star: R^*,
infix_ap: x f y,
exists: ∃x:A. B[x],
rel_exp: R^n,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
ts-stable: ts-stable(ts;x.P[x])
Lemmas referenced :
all_wf,
ts-type_wf,
infix_ap_wf,
rel_exp_wf,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
ts-rel_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
rel_star_wf,
ts-stable_wf,
transition-system_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
and_wf,
equal_wf,
bnot_wf,
not_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
exists_wf,
int_subtype_base,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
rename,
setElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
because_Cache,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality,
instantiate,
cumulativity,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
productEquality,
baseApply,
closedConclusion,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}ts:transition-system\{i:l\}
\mforall{}[P:ts-type(ts) {}\mrightarrow{} \mBbbP{}]
(ts-stable(ts;x.P[x]) {}\mRightarrow{} (\mforall{}x,y:ts-type(ts). (P[x] {}\mRightarrow{} (x (ts-rel(ts)\^{}*) y) {}\mRightarrow{} P[y])))
Date html generated:
2018_05_21-PM-08_01_47
Last ObjectModification:
2017_07_26-PM-05_38_34
Theory : general
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