Nuprl Lemma : rel-preserving-star
∀[T1,T2:Type]. ∀[R1:T1 ⟶ T1 ⟶ Type]. ∀[R2:T2 ⟶ T2 ⟶ Type].
∀f:T2 ⟶ T1. (λx.f[x]:T2->T1 takes R2 into R1*) ⇒ λx.f[x]:T2->T1 takes R2^* into R1*))
Proof
Definitions occuring in Statement :
rel-preserving: λx.f[x]:T2->T1 takes R2 into R1*),
rel_star: R^*,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
rel-preserving: λx.f[x]:T2->T1 takes R2 into R1*),
rel_star: R^*,
infix_ap: x f y,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
rel_star_wf,
rel-preserving_wf,
infix_ap_wf,
rel_exp_wf,
false_wf,
le_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
all_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
rel_star_weakening,
and_wf,
equal_wf,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
rel_star_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
instantiate,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
rename,
setElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
baseClosed,
impliesFunctionality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[R1:T1 {}\mrightarrow{} T1 {}\mrightarrow{} Type]. \mforall{}[R2:T2 {}\mrightarrow{} T2 {}\mrightarrow{} Type].
\mforall{}f:T2 {}\mrightarrow{} T1
(\mlambda{}x.f[x]:T2->T1 takes R2 into R1*) {}\mRightarrow{} \mlambda{}x.f[x]:T2->T1 takes R2\^{}* into R1*))
Date html generated:
2018_05_21-PM-08_00_45
Last ObjectModification:
2017_07_26-PM-05_37_36
Theory : general
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