Nuprl Lemma : W-path-lemma2
∀A1:Type. ∀B1:A1 ⟶ Type. ∀a:A1. ∀x1:B1[a] ⟶ W(A1;a.B1[a]). ∀n:ℕ+. ∀s:ℕn ⟶ cw-step(A1;a.B1[a]). ∀a1:A1.
∀w1:b:B1[a1] ⟶ (pco-W ⋅). ∀x:B1[a1]. ∀a2:A1. ∀z1:b:B1[a2] ⟶ (pco-W ⋅).
((∀k:ℕn. (W-rel(A1;a.B1[a];<a, x1>) k s (s k)))
⇒ ((s (n - 1)) = <⋅, <a1, w1>, inl x> ∈ cw-step(A1;a.B1[a]))
⇒ ((w1 x) = <a2, z1> ∈ (pco-W ⋅))
⇒ (z1 ∈ b:B1[a2] ⟶ W(A1;a.B1[a])))
Proof
Definitions occuring in Statement :
W-rel: W-rel(A;a.B[a];w),
W: W(A;a.B[a]),
cw-step: cw-step(A;a.B[a]),
param-co-W: pco-W,
int_seg: {i..j-},
nat_plus: ℕ+,
it: ⋅,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
inl: inl x,
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
implies: P ⇒ Q,
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
ext-eq: A ≡ B,
and: P ∧ Q,
respects-equality: respects-equality(S;T),
int_seg: {i..j-},
nat_plus: ℕ+,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
guard: {T},
squash: ↓T,
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pi2: snd(t),
pi1: fst(t),
sq_type: SQType(T),
W: W(A;a.B[a]),
istype: istype(T)
Lemmas referenced :
param-co-W-ext,
unit_wf2,
it_wf,
param-co-W_wf,
top_wf,
istype-void,
subtype-respects-equality,
cw-step_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-int,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
nat_plus_wf,
add-mul-special,
zero-mul,
le-add-cancel-alt,
istype-le,
istype-less_than,
int_seg_wf,
W-rel_wf,
int_seg_subtype_nat,
subtype_rel_function,
int_seg_subtype,
sq_stable__le,
le_weakening2,
subtype_rel_self,
W_wf,
istype-universe,
param-W-ext,
cw-pp-lemma,
nat_plus_subtype_nat,
subtype_rel_weakening,
param-W_wf,
W-ext,
subtype_base_sq,
unit_subtype_base,
equal-implies-member-param-W,
istype-top,
ext-eq_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
Error :lambdaEquality_alt,
hypothesisEquality,
Error :universeIsType,
because_Cache,
applyEquality,
dependent_functionElimination,
Error :equalityIstype,
Error :isect_memberEquality_alt,
voidElimination,
Error :dependent_pairEquality_alt,
Error :functionIsType,
productEquality,
functionEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
independent_pairFormation,
unionElimination,
addEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
universeEquality,
applyLambdaEquality,
cumulativity,
functionExtensionality,
Error :inlEquality_alt,
Error :unionIsType,
hypothesis_subsumption
Latex:
\mforall{}A1:Type. \mforall{}B1:A1 {}\mrightarrow{} Type. \mforall{}a:A1. \mforall{}x1:B1[a] {}\mrightarrow{} W(A1;a.B1[a]). \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} cw-step(A1;a.B1[a]).
\mforall{}a1:A1. \mforall{}w1:b:B1[a1] {}\mrightarrow{} (pco-W \mcdot{}). \mforall{}x:B1[a1]. \mforall{}a2:A1. \mforall{}z1:b:B1[a2] {}\mrightarrow{} (pco-W \mcdot{}).
((\mforall{}k:\mBbbN{}n. (W-rel(A1;a.B1[a];<a, x1>) k s (s k)))
{}\mRightarrow{} ((s (n - 1)) = <\mcdot{}, <a1, w1>, inl x>)
{}\mRightarrow{} ((w1 x) = <a2, z1>)
{}\mRightarrow{} (z1 \mmember{} b:B1[a2] {}\mrightarrow{} W(A1;a.B1[a])))
Date html generated:
2019_06_20-PM-00_36_29
Last ObjectModification:
2018_11_23-PM-03_54_16
Theory : co-recursion
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