Nuprl Lemma : rel_exp_functionality_wrt_iff
∀[T:Type]. ∀[R,Q:T ⟶ T ⟶ ℙ]. ((∀x,y:T. (R x y ⇐⇒ Q x y)) ⇒ (∀n:ℕ. ∀x,y:T. (R^n x y ⇐⇒ Q^n x y)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
rel_exp: R^n,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B,
infix_ap: x f y
Lemmas referenced :
all_wf,
iff_wf,
rel_exp_wf,
subtract_wf,
decidable__le,
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,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
equal_wf,
bnot_wf,
not_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
exists_wf,
infix_ap_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
functionEquality,
universeEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productEquality,
instantiate,
independent_functionElimination,
equalityElimination,
productElimination,
impliesFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[R,Q:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. (R x y \mLeftarrow{}{}\mRightarrow{} Q x y)) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}x,y:T. (rel\_exp(T; R; n) x y \mLeftarrow{}{}\mRightarrow{} rel\_exp(T; Q; n) x y)))
Date html generated:
2017_04_17-AM-09_27_45
Last ObjectModification:
2017_02_27-PM-05_28_12
Theory : relations2
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