Nuprl Lemma : sq_stable__copathAgree
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀x,y:copath(a.B[a];w). SqStable(copathAgree(a.B[a];w;x;y))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
sq_stable: SqStable(P),
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),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
gt: i > j
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
sq_stable__coPathAgree,
coPath_subtype,
le_weakening2,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not-gt-2,
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,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
lambdaEquality,
applyEquality,
dependent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])].
\mforall{}x,y:copath(a.B[a];w). SqStable(copathAgree(a.B[a];w;x;y))
Date html generated:
2018_07_25-PM-01_40_56
Last ObjectModification:
2018_06_08-PM-04_18_35
Theory : co-recursion
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