Nuprl Lemma : coPathAgree_le
∀[A:𝕌']. ∀[B:A ⟶ Type].
∀n:ℕ
∀[w:coW(A;a.B[a])]
∀p,q:coPath(a.B[a];w;n). (coPathAgree(a.B[a];n;w;p;q) ⇒ (∀m:ℕ. ((m ≤ n) ⇒ coPathAgree(a.B[a];m;w;p;q))))
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],
le: A ≤ B,
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],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
squash: ↓T,
coPath: coPath(a.B[a];w;n),
cand: A c∧ B
Lemmas referenced :
uall_wf,
coW_wf,
all_wf,
coPath_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,
coPathAgree_wf,
nat_wf,
coPath_subtype,
set_wf,
less_than_wf,
primrec-wf2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-equal-2,
sq_stable__le,
minus-zero,
true_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
top_wf,
coW-item_wf,
subtype_rel-equal,
and_wf,
coW-dom_wf,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
rename,
setElimination,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
hypothesis,
universeEquality,
dependent_set_memberEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
functionEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination,
applyLambdaEquality,
productEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}n:\mBbbN{}
\mforall{}[w:coW(A;a.B[a])]
\mforall{}p,q:coPath(a.B[a];w;n).
(coPathAgree(a.B[a];n;w;p;q) {}\mRightarrow{} (\mforall{}m:\mBbbN{}. ((m \mleq{} n) {}\mRightarrow{} coPathAgree(a.B[a];m;w;p;q))))
Date html generated:
2018_07_25-PM-01_38_19
Last ObjectModification:
2018_06_04-PM-10_02_41
Theory : co-recursion
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