Nuprl Lemma : p-compose_wf
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:B ⟶ (C + Top)]. (f o g ∈ A ⟶ (C + Top))
Proof
Definitions occuring in Statement :
p-compose: f o g,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
p-compose: f o g,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
can-apply: can-apply(f;x),
isl: isl(x),
not: ¬A,
true: True,
prop: ℙ
Lemmas referenced :
can-apply_wf,
eqtt_to_assert,
do-apply_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
istype-universe,
istype-top,
true_wf,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIsType1,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
axiomEquality,
functionIsType,
unionIsType,
isect_memberEquality_alt,
universeIsType,
universeEquality,
natural_numberEquality,
inrEquality_alt
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:B {}\mrightarrow{} (C + Top)]. (f o g \mmember{} A {}\mrightarrow{} (C + Top))
Date html generated:
2019_10_15-AM-11_07_16
Last ObjectModification:
2018_10_11-PM-09_47_40
Theory : general
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