Nuprl Lemma : decidable__equal_function
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀i,j:ℤ. ∀f,g:{i..j-} ⟶ T. Dec(f = g ∈ ({i..j-} ⟶ T))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
false: False,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
uimplies: b supposing a,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P)
Lemmas referenced :
decidable_wf,
equal_wf,
istype-universe,
istype-le,
subtract_wf,
int_seg_wf,
istype-int,
istype-less_than,
primrec-wf2,
le_wf,
istype-nat,
istype-void,
le-add-cancel,
mul-distributes,
mul-associates,
omega-shadow,
add-zero,
minus-zero,
zero-add,
zero-mul,
mul-distributes-right,
two-mul,
add-mul-special,
add-swap,
one-mul,
add-associates,
minus-one-mul-top,
le_reflexive,
add_functionality_wrt_le,
less-iff-le,
int_subtype_base,
add-is-int-iff,
add-commutes,
minus-one-mul,
int_seg_properties,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-minus,
le-add-cancel-alt,
subtype_rel_function,
int_seg_subtype,
le-add-cancel2,
subtype_rel_self,
decidable__lt,
not-lt-2,
mul-swap,
mul-commutes,
decidable__equal_int,
subtype_base_sq,
not-equal-2,
iff_weakening_equal,
true_wf,
squash_wf,
equal_functionality_wrt_subtype_rel2,
sq_stable__le,
subtract_nat_wf,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
instantiate,
universeEquality,
rename,
setElimination,
Error :inhabitedIsType,
natural_numberEquality,
functionEquality,
Error :setIsType,
Error :lambdaEquality_alt,
intEquality,
Error :equalityIstype,
Error :functionExtensionality_alt,
Error :inlFormation_alt,
independent_functionElimination,
imageMemberEquality,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
multiplyEquality,
addEquality,
dependent_functionElimination,
independent_isectElimination,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
voidElimination,
Error :isect_memberEquality_alt,
productElimination,
unionElimination,
minusEquality,
Error :productIsType,
cumulativity,
applyLambdaEquality,
imageElimination,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}i,j:\mBbbZ{}. \mforall{}f,g:\{i..j\msupminus{}\} {}\mrightarrow{} T. Dec(f = g)))
Date html generated:
2019_06_20-PM-00_42_04
Last ObjectModification:
2019_01_02-PM-00_24_16
Theory : list_0
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