Nuprl Lemma : cond_rel_exp_monotone
∀n:ℕ. ∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,R2:T ⟶ T ⟶ ℙ]. (when P, R1 => R2 ⇒ R1 preserves P ⇒ when P, R1^n => R2^n)
Proof
Definitions occuring in Statement :
cond_rel_implies: when P, R1 => R2,
preserved_by: R preserves P,
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
cond_rel_implies: when P, R1 => R2,
rel_exp: R^n,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
infix_ap: x f y,
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
preserved_by: R preserves P
Lemmas referenced :
uall_wf,
cond_rel_implies_wf,
preserved_by_wf,
rel_exp_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
equal_wf,
all_wf,
subtype_rel_self,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
bnot_wf,
not_wf,
infix_ap_wf,
exists_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
because_Cache,
functionExtensionality,
applyEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
Error :isect_memberFormation_alt,
axiomEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
baseApply,
closedConclusion,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
productEquality,
equalityElimination
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(when P, R1 => R2 {}\mRightarrow{} R1 preserves P {}\mRightarrow{} when P, rel\_exp(T; R1; n) => rel\_exp(T; R2; n))
Date html generated:
2019_06_20-PM-00_30_32
Last ObjectModification:
2018_09_26-PM-00_49_24
Theory : relations
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