Nuprl Lemma : bigrel-induction-monotone
∀[T:Type]
∀P:(T ⟶ T ⟶ ℙ) ⟶ ℙ. ∀F:(T ⟶ T ⟶ ℙ) ⟶ T ⟶ T ⟶ ℙ.
(rel-monotone{i:l}(T;R.F[R])
⇒ (∀Rs:ℕ ⟶ T ⟶ T ⟶ ℙ. ((∀n:ℕ. Rs[n + 1] => Rs[n]) ⇒ (∀n:ℕ. P[Rs[n]]) ⇒ P[isect-rel(ℕ;n.Rs[n])]))
⇒ (∀R:T ⟶ T ⟶ ℙ. (P[R] ⇒ P[F[R]]))
⇒ P[λx,y. True]
⇒ P[∨R.F[R]])
Proof
Definitions occuring in Statement :
bigrel: ∨R.F[R],
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
isect-rel: isect-rel(T;i.R[i]),
rel_implies: R1 => R2,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
true: True,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bigrel: ∨R.F[R],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
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,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T,
eq_int: (i =z j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
rel_implies: R1 => R2,
infix_ap: x f y
Lemmas referenced :
primrec_wf,
true_wf,
int_seg_wf,
nat_wf,
rel_implies_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
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,
sq_stable__le,
primrec-unroll,
btrue_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
eq_int_wf,
equal_wf,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
all_wf,
isect-rel_wf,
rel-monotone_wf,
bfalse_wf,
int_subtype_base,
primrec0_lemma,
le_antisymmetry_iff,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
not-equal-2,
general_arith_equation1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}P:(T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}) {}\mrightarrow{} \mBbbP{}. \mforall{}F:(T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}) {}\mrightarrow{} T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(rel-monotone\{i:l\}(T;R.F[R])
{}\mRightarrow{} (\mforall{}Rs:\mBbbN{} {}\mrightarrow{} T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}
((\mforall{}n:\mBbbN{}. Rs[n + 1] => Rs[n]) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. P[Rs[n]]) {}\mRightarrow{} P[isect-rel(\mBbbN{};n.Rs[n])]))
{}\mRightarrow{} (\mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (P[R] {}\mRightarrow{} P[F[R]]))
{}\mRightarrow{} P[\mlambda{}x,y. True]
{}\mRightarrow{} P[\mvee{}R.F[R]])
Date html generated:
2017_04_14-AM-07_38_56
Last ObjectModification:
2017_02_27-PM-03_10_50
Theory : relations
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