Nuprl Lemma : extend-name-morph-comp
∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀z,v,v1:Cname.
((¬(z ∈ I)) ⇒ (¬(v ∈ K)) ⇒ (¬(v1 ∈ J)) ⇒ ((f o g)[z:=v] = (f[z:=v1] o g[v1:=v]) ∈ name-morph([z / I];[v / K])))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
extend-name-morph: f[z1:=z2],
name-morph: name-morph(I;J),
coordinate_name: Cname,
l_member: (x ∈ l),
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
extend-name-morph: f[z1:=z2],
name-comp: (f o g),
compose: f o g,
uext: uext(g),
nameset: nameset(L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
prop: ℙ,
isname: isname(z),
true: True,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
top: Top,
select: L[n],
cons: [a / b],
cand: A c∧ B,
nat_plus: ℕ+,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_stable: SqStable(P),
ge: i ≥ j ,
respects-equality: respects-equality(S;T)
Lemmas referenced :
name-morphs-equal,
cons_wf,
coordinate_name_wf,
extend-name-morph_wf,
name-comp_wf,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
nameset_wf,
l_member_wf,
istype-void,
name-morph_wf,
list_wf,
iff_imp_equal_bool,
le_int_wf,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
assert_of_le_int,
iff_weakening_equal,
istype-true,
equal-wf-base,
istype-le,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
length_wf,
select_wf,
nat_properties,
sq_stable__le,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
nameset_subtype_extd-nameset,
cons_member,
isname_wf,
assert-isname,
respects-equality-set-trivial2,
extd-nameset_subtype,
l_subset_right_cons_trivial,
not-assert-isname,
nsub2_subtype_extd-nameset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
functionExtensionality,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
intEquality,
natural_numberEquality,
functionIsType,
universeIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
Error :memTop,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
productIsType,
sqequalBase
Latex:
\mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K). \mforall{}z,v,v1:Cname.
((\mneg{}(z \mmember{} I)) {}\mRightarrow{} (\mneg{}(v \mmember{} K)) {}\mRightarrow{} (\mneg{}(v1 \mmember{} J)) {}\mRightarrow{} ((f o g)[z:=v] = (f[z:=v1] o g[v1:=v])))
Date html generated:
2020_05_21-AM-10_49_56
Last ObjectModification:
2019_12_08-PM-07_06_12
Theory : cubical!sets
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