Nuprl Lemma : name-morph-extend-comp
∀[I,J,K:Cname List]. ∀[f:name-morph(I;J)]. ∀[g:name-morph(J;K)]. (((f o g))+ = ((f)+ o (g)+) ∈ name-morph(I+;K+))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
name-morph-extend: (f)+,
name-morph: name-morph(I;J),
add-fresh-cname: I+,
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
uimplies: b supposing a,
name-morph-extend: (f)+,
name-comp: (f o g),
compose: f o g,
has-value: (a)↓,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq-cname: eq-cname(x;y),
uext: uext(g),
nameset: nameset(L),
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
coordinate_name: Cname,
int_upper: {i...},
isname: isname(z),
true: True,
assert: ↑b,
sq_type: SQType(T),
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
add-fresh-cname: I+,
or: P ∨ Q
Lemmas referenced :
name-morphs-equal,
add-fresh-cname_wf,
name-morph-extend_wf,
name-comp_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
value-type-has-value,
not_wf,
l_member_wf,
set-value-type,
coordinate_name-value-type,
fresh-cname_wf,
nameset_wf,
eq-cname_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal_wf,
bnot_wf,
assert_elim,
btrue_neq_bfalse,
uiff_transitivity,
eqtt_to_assert,
assert-eq-cname,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
set_wf,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
le_int_wf,
btrue_wf,
le_wf,
true_wf,
assert_of_le_int,
iff_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
cons_member,
nameset_subtype_extd-nameset,
isname_wf,
assert-isname,
set_subtype_base,
int_subtype_base,
fresh-cname-not-member,
extd-nameset_subtype,
l_subset_right_cons_trivial,
not-assert-isname,
nsub2_subtype_extd-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
callbyvalueReduce,
setEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
addLevel,
levelHypothesis,
independent_functionElimination,
voidElimination,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
instantiate,
cumulativity,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
dependent_set_memberEquality,
inlFormation
Latex:
\mforall{}[I,J,K:Cname List]. \mforall{}[f:name-morph(I;J)]. \mforall{}[g:name-morph(J;K)]. (((f o g))+ = ((f)+ o (g)+))
Date html generated:
2017_10_05-AM-10_07_08
Last ObjectModification:
2017_07_28-AM-11_16_32
Theory : cubical!sets
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