Nuprl Lemma : non-forking-rel_exp
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (non-forking(T;x,y.R[x;y]) ⇒ (∀n:ℕ. non-forking(T;x,y.x λx,y. R[x;y]^n y)))
Proof
Definitions occuring in Statement :
non-forking: non-forking(T;x,y.R[x; y]),
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
non-forking: non-forking(T;x,y.R[x; y]),
so_apply: x[s1;s2],
rel_exp: R^n,
eq_int: (i =z j),
infix_ap: x f y,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
infix_ap_wf,
rel_exp_wf,
equal_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
exists_wf,
nat_wf,
non-forking_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
instantiate,
cumulativity,
because_Cache,
universeEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
productElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
productEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(non-forking(T;x,y.R[x;y]) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. non-forking(T;x,y.x rel\_exp(T; \mlambda{}x,y. R[x;y]; n) y)))
Date html generated:
2018_05_21-PM-09_05_16
Last ObjectModification:
2017_07_26-PM-06_28_05
Theory : general
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