Nuprl Lemma : copathAgree-nil
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀p:copath(a.B[a];w). copathAgree(a.B[a];w;();p)
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath-nil: (),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
copath: copath(a.B[a];w),
copathAgree: copathAgree(a.B[a];w;x;y),
copath-nil: (),
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
ge: i ≥ j ,
int_upper: {i...},
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
btrue_wf,
assert_of_eq_int,
eqff_to_assert,
eq_int_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
false_wf,
nat_properties,
nequal-le-implies,
zero-add,
le_wf,
sq_stable__le,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
copath_wf,
coW_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
hypothesis_subsumption,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
intEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}p:copath(a.B[a];w). copathAgree(a.B[a];w;();p)
Date html generated:
2018_07_25-PM-01_41_14
Last ObjectModification:
2018_06_04-PM-00_30_54
Theory : co-recursion
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