Nuprl Lemma : pcw-path-copathAgree
∀[A:𝕌']. ∀[B:A ⟶ Type].
∀w:coW(A;a.B[a]). ∀path:Path.
(StepAgree(path 0;⋅;w)
⇒ (∀i:ℕ
((copath-length(pcw-path-coPath(i + 1;path)) = (i + 1) ∈ ℤ)
⇒ (copath-length(pcw-path-coPath(i;path)) = i ∈ ℤ)
⇒ copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i + 1;path)))))
Proof
Definitions occuring in Statement :
pcw-path-coPath: pcw-path-coPath(n;p),
copathAgree: copathAgree(a.B[a];w;x;y),
copath-length: copath-length(p),
coW: coW(A;a.B[a]),
pcw-path: Path,
pcw-step-agree: StepAgree(s;p1;w),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
exposed-it: exposed-it,
eq_int: (i =z j),
coPath-extend: coPath-extend(n;p;t),
coPath: coPath(a.B[a];w;n),
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
ge: i ≥ j ,
less_than: a < b,
copath-extend: copath-extend(q;t),
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
spreadn: spread3,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
let: let,
guard: {T},
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
pcw-path-coPath: pcw-path-coPath(n;p),
cand: A c∧ B,
coW: coW(A;a.B[a]),
pcw-path: Path,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
prop: ℙ,
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-dom_wf,
coW-item_wf,
coPath_wf,
le_weakening,
int_subtype_base,
minus-minus,
less-iff-le,
not-ge-2,
subtract_wf,
true_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
equal-wf-base,
member_wf,
squash_wf,
top_wf,
copath_wf,
copath-nil-Agree,
pcw-step_wf,
add-subtract-cancel,
le_antisymmetry_iff,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
copath-length_wf,
equal_wf,
sq_stable__copathAgree,
coW_wf,
pcw-path_wf,
it_wf,
unit_wf2,
pcw-step-agree_wf,
nat_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
pcw-path-coPath_wf
Rules used in proof :
closedConclusion,
baseApply,
functionExtensionality,
independent_pairEquality,
axiomEquality,
intWeakElimination,
multiplyEquality,
productEquality,
sqequalAxiom,
lessCases,
impliesFunctionality,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
universeEquality,
functionEquality,
cumulativity,
instantiate,
minusEquality,
because_Cache,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
applyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
voidElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
productElimination,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}w:coW(A;a.B[a]). \mforall{}path:Path.
(StepAgree(path 0;\mcdot{};w)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}
((copath-length(pcw-path-coPath(i + 1;path)) = (i + 1))
{}\mRightarrow{} (copath-length(pcw-path-coPath(i;path)) = i)
{}\mRightarrow{} copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i + 1;path)))))
Date html generated:
2018_07_25-PM-01_41_46
Last ObjectModification:
2018_07_23-PM-00_54_25
Theory : co-recursion
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