Nuprl Lemma : rel-preserving-star-reachable
∀[T1,T2:Type]. ∀[i2:T2]. ∀[R1:T1 ⟶ T1 ⟶ Type]. ∀[R2:T2 ⟶ T2 ⟶ Type].
∀f:T2 ⟶ T1
((∀x,y:{s:T2| i2 (R2^*) s} . ((x R2 y) ⇒ ((f x) (R1^*) (f y))))
⇒ {∀x,y:{s:T2| i2 (R2^*) s} . ((x (R2^*) y) ⇒ ((f x) (R1^*) (f y)))})
Proof
Definitions occuring in Statement :
rel_star: R^*,
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
rel_star: R^*,
infix_ap: x f y,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
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),
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
bfalse: ff
Lemmas referenced :
rel_star_wf,
subtype_rel_self,
istype-universe,
rel_exp_wf,
istype-void,
istype-le,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-less_than,
primrec-wf2,
rel_star_weakening,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
istype-assert,
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,
rel_rel_star
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
functionExtensionality,
hypothesis,
instantiate,
functionEquality,
cumulativity,
universeEquality,
setElimination,
rename,
inhabitedIsType,
setIsType,
because_Cache,
functionIsType,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
setEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
intEquality,
equalityIstype,
sqequalBase
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[i2:T2]. \mforall{}[R1:T1 {}\mrightarrow{} T1 {}\mrightarrow{} Type]. \mforall{}[R2:T2 {}\mrightarrow{} T2 {}\mrightarrow{} Type].
\mforall{}f:T2 {}\mrightarrow{} T1
((\mforall{}x,y:\{s:T2| i2 rel\_star(T2; R2) s\} . ((x R2 y) {}\mRightarrow{} ((f x) rel\_star(T1; R1) (f y))))
{}\mRightarrow{} \{\mforall{}x,y:\{s:T2| i2 (R2\^{}*) s\} .
((x (R2\^{}*) y) {}\mRightarrow{} ((f x) (R1\^{}*) (f y)))\})
Date html generated:
2020_05_20-AM-08_11_45
Last ObjectModification:
2020_01_26-PM-00_16_16
Theory : general
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