Nuprl Lemma : stream-extensionality
∀[A:Type]. ∀[x,y:stream(A)]. x = y ∈ stream(A) supposing ∀n:ℕ. (s-nth(n;x) = s-nth(n;y) ∈ A)
Proof
Definitions occuring in Statement :
s-nth: s-nth(n;s),
stream: stream(A),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
guard: {T},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
top: Top,
s-nth: s-nth(n;s),
s-cons: x.s,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
has-value: (a)↓
Lemmas referenced :
stream-coinduction,
all_wf,
nat_wf,
equal_wf,
s-nth_wf,
stream_wf,
false_wf,
le_wf,
stream-decomp,
stream-subtype,
top_wf,
s_hd_cons_lemma,
s-hd_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
s_tl_cons_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_antisymmetry_iff,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-subtract-cancel,
value-type-has-value,
set-value-type,
int-value-type,
s-tl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
cumulativity,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
because_Cache,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
applyEquality,
voidElimination,
voidEquality,
independent_functionElimination,
callbyvalueReduce,
sqleReflexivity,
addEquality,
setElimination,
rename,
unionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[A:Type]. \mforall{}[x,y:stream(A)]. x = y supposing \mforall{}n:\mBbbN{}. (s-nth(n;x) = s-nth(n;y))
Date html generated:
2017_04_14-AM-07_47_26
Last ObjectModification:
2017_02_27-PM-03_17_36
Theory : co-recursion
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