Nuprl Lemma : can-apply-compose
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:B ⟶ (C + Top)]. ∀[x:A].
{(↑can-apply(g;x)) ∧ (↑can-apply(f;do-apply(g;x)))} supposing ↑can-apply(f o g;x)
Proof
Definitions occuring in Statement :
p-compose: f o g,
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
and: P ∧ Q,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
assert: ↑b,
cand: A c∧ B,
true: True,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
false: False,
top: Top
Lemmas referenced :
can-apply-compose-sq,
can-apply_wf,
bool_wf,
eqtt_to_assert,
assert_wf,
do-apply_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
assert_witness,
subtype_rel_union,
top_wf,
p-compose_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
cumulativity,
functionExtensionality,
applyEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
independent_pairEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
functionEquality,
unionEquality,
universeEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:B {}\mrightarrow{} (C + Top)]. \mforall{}[x:A].
\{(\muparrow{}can-apply(g;x)) \mwedge{} (\muparrow{}can-apply(f;do-apply(g;x)))\} supposing \muparrow{}can-apply(f o g;x)
Date html generated:
2017_10_01-AM-09_13_34
Last ObjectModification:
2017_07_26-PM-04_48_56
Theory : general
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