Nuprl Lemma : implies-bigrel
∀[T:Type]
∀F:(T ⟶ T ⟶ ℙ) ⟶ T ⟶ T ⟶ ℙ. (rel-monotone{i:l}(T;R.F[R]) ⇒ (∀R':T ⟶ T ⟶ ℙ. (R' => F[R'] ⇒ R' => νR.F[R])))
Proof
Definitions occuring in Statement :
bigrel: νR.F[R],
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
rel_implies: R1 => R2,
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
bigrel: νR.F[R],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
lt_int: i <z j,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
rel_implies: R1 => R2,
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
rel_rev_implies: R1 ⇐ R2,
isect-rel: ⋂i:T. R[i]
Lemmas referenced :
rel_implies_wf,
rel-monotone_wf,
istype-universe,
primrec-unroll,
btrue_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
assert_wf,
lt_int_wf,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
le_int_wf,
le_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
int_subtype_base,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
le-add-cancel2,
primrec_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-swap,
le-add-cancel,
istype-le,
true_wf,
int_seg_wf,
istype-int,
istype-less_than,
primrec-wf2,
istype-nat,
subtype_rel_self,
rel_implies_functionality,
rel_implies_weakening,
rel_equivalent_inversion,
rel_equivalent_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
functionIsType,
because_Cache,
universeEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
instantiate,
natural_numberEquality,
Error :memTop,
unionElimination,
equalityElimination,
baseClosed,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
voidElimination,
equalityIstype,
dependent_functionElimination,
rename,
setElimination,
baseApply,
closedConclusion,
independent_pairFormation,
imageElimination,
addEquality,
functionEquality,
cumulativity,
dependent_set_memberEquality_alt,
minusEquality,
setIsType
Latex:
\mforall{}[T:Type]
\mforall{}F:(T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}) {}\mrightarrow{} T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}
(rel-monotone\{i:l\}(T;R.F[R]) {}\mRightarrow{} (\mforall{}R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (R' => F[R'] {}\mRightarrow{} R' => \mnu{}R.F[R])))
Date html generated:
2020_05_19-PM-09_36_31
Last ObjectModification:
2020_01_04-PM-07_56_52
Theory : relations
Home
Index