Nuprl Lemma : pcw-path-coPath_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:Path].
∀[n:ℕ]
((pcw-path-coPath(n;p) ∈ copath(a.B[a];w))
∧ ((copath-length(pcw-path-coPath(n;p)) = n ∈ ℤ)
⇒ (copath-at(w;pcw-path-coPath(n;p)) = (fst(snd((p n)))) ∈ coW(A;a.B[a]))))
supposing StepAgree(p 0;⋅;w)
Proof
Definitions occuring in Statement :
pcw-path-coPath: pcw-path-coPath(n;p),
copath-length: copath-length(p),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
pcw-path: Path,
pcw-step-agree: StepAgree(s;p1;w),
nat: ℕ,
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
coW-item: coW-item(w;b),
ext-eq: A ≡ B,
squash: ↓T,
coW-dom: coW-dom(a.B[a];w),
pcw-steprel: StepRel(s1;s2),
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
cand: A c∧ B,
let: let,
coPath-at: coPath-at(n;w;p),
pi1: fst(t),
pi2: snd(t),
copath-nil: (),
spreadn: spread3,
pcw-step-agree: StepAgree(s;p1;w),
copath-at: copath-at(w;p),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
pcw-path-coPath: pcw-path-coPath(n;p),
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_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,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
and: P ∧ Q,
so_apply: x[s],
prop: ℙ,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-item_wf,
copath-at-extend,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
coW-dom_wf,
pi1_wf,
subtype_rel_weakening,
coW-ext,
copath-extend_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
le_weakening2,
pcw-steprel_wf,
not_wf,
bnot_wf,
le_weakening,
assert_wf,
bool_wf,
eq_int_wf,
int_subtype_base,
copath-length_wf,
subtract-add-cancel,
not-le-2,
equal-wf-base,
and_wf,
top_wf,
param-co-W_wf,
subtype_rel-equal,
equal_wf,
equal-wf-T-base,
pcw-step_wf,
copath-nil_wf,
copath_length_nil_lemma,
coW_wf,
pcw-path_wf,
le_wf,
it_wf,
unit_wf2,
pcw-step-agree_wf,
nat_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties
Rules used in proof :
imageMemberEquality,
imageElimination,
hypothesis_subsumption,
hyp_replacement,
impliesFunctionality,
equalityElimination,
closedConclusion,
baseApply,
applyLambdaEquality,
baseClosed,
productEquality,
functionExtensionality,
independent_pairEquality,
universeEquality,
functionEquality,
dependent_set_memberEquality,
cumulativity,
instantiate,
equalitySymmetry,
equalityTransitivity,
because_Cache,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
applyEquality,
addEquality,
productElimination,
independent_pairFormation,
unionElimination,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:Path].
\mforall{}[n:\mBbbN{}]
((pcw-path-coPath(n;p) \mmember{} copath(a.B[a];w))
\mwedge{} ((copath-length(pcw-path-coPath(n;p)) = n)
{}\mRightarrow{} (copath-at(w;pcw-path-coPath(n;p)) = (fst(snd((p n)))))))
supposing StepAgree(p 0;\mcdot{};w)
Date html generated:
2018_07_25-PM-01_41_40
Last ObjectModification:
2018_07_23-AM-11_52_52
Theory : co-recursion
Home
Index