Nuprl Lemma : s-sub_transitivity
∀[T:Type]. ∀[s,t,r:stream(T)]. (s-sub(T;s;t) ⇒ s-sub(T;t;r) ⇒ s-sub(T;s;r))
Proof
Definitions occuring in Statement :
s-sub: s-sub(T;s;t),
stream: stream(A),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
s-sub: s-sub(T;s;t),
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
compose: f o g,
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
sq_type: SQType(T)
Lemmas referenced :
compose_wf,
nat_wf,
istype-nat,
istype-less_than,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
istype-le,
s-nth_wf,
s-sub_wf,
stream_wf,
istype-universe,
nat_plus_properties,
general_add_assoc,
less_than_transitivity2,
le_weakening2,
add_nat_wf,
primrec-wf-nat-plus,
less_than_wf,
nat_plus_subtype_nat,
nat_plus_wf,
istype-sqequal,
set_subtype_base,
le_wf,
int_subtype_base,
less-iff-le,
subtract_wf,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
nat_properties,
subtype_base_sq,
istype-int,
not-lt-2,
le-add-cancel-alt,
decidable__lt,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
Error :dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
Error :productIsType,
Error :functionIsType,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :isect_memberEquality_alt,
minusEquality,
Error :equalityIstype,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
promote_hyp,
multiplyEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[s,t,r:stream(T)]. (s-sub(T;s;t) {}\mRightarrow{} s-sub(T;t;r) {}\mRightarrow{} s-sub(T;s;r))
Date html generated:
2019_06_20-PM-00_37_59
Last ObjectModification:
2019_03_06-AM-11_06_38
Theory : co-recursion
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