Nuprl Lemma : coPathAgree_transitivity
∀[A:𝕌']. ∀[B:A ⟶ Type].
∀n:ℕ
∀[w:coW(A;a.B[a])]
∀p,q,r:coPath(a.B[a];w;n).
(coPathAgree(a.B[a];n;w;p;q) ⇒ coPathAgree(a.B[a];n;w;q;r) ⇒ coPathAgree(a.B[a];n;w;p;r))
Proof
Definitions occuring in Statement :
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
coPath: coPath(a.B[a];w;n),
coW: coW(A;a.B[a]),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
coPath: coPath(a.B[a];w;n),
eq_int: (i =z j),
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
subtract: n - m,
nequal: a ≠ b ∈ T ,
not: ¬A,
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
squash: ↓T
Lemmas referenced :
btrue_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
true_wf,
top_wf,
coW_wf,
eqff_to_assert,
eq_int_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
coW-dom_wf,
coPathAgree_wf,
coW-item_wf,
coPath_wf,
uall_wf,
all_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
subtype_rel-equal,
not-equal-2,
and_wf,
squash_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
natural_numberEquality,
because_Cache,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
axiomEquality,
instantiate,
cumulativity,
applyEquality,
dependent_pairFormation,
promote_hyp,
independent_functionElimination,
voidElimination,
productEquality,
functionExtensionality,
rename,
setElimination,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
functionEquality,
imageElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}n:\mBbbN{}
\mforall{}[w:coW(A;a.B[a])]
\mforall{}p,q,r:coPath(a.B[a];w;n).
(coPathAgree(a.B[a];n;w;p;q) {}\mRightarrow{} coPathAgree(a.B[a];n;w;q;r) {}\mRightarrow{} coPathAgree(a.B[a];n;w;p;r))
Date html generated:
2018_07_25-PM-01_38_24
Last ObjectModification:
2018_06_04-PM-09_44_49
Theory : co-recursion
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