Nuprl Lemma : copath-nil-Agree
∀[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 :
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
guard: {T},
prop: ℙ,
false: False,
not: ¬A,
squash: ↓T,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
member: t ∈ T,
copath-nil: (),
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW_wf,
copath_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
less_than_irreflexivity,
less_than_transitivity1,
less_than_wf,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
coPathAgree0_lemma
Rules used in proof :
universeEquality,
functionEquality,
applyEquality,
lambdaEquality,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
independent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
sqequalAxiom,
lessCases,
because_Cache,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
natural_numberEquality,
hypothesisEquality,
rename,
setElimination,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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_18
Last ObjectModification:
2018_06_15-PM-05_28_51
Theory : co-recursion
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