Nuprl Lemma : lift-reduce-face-map
∀[I:Cname List]. ∀[x:nameset(I)]. ∀[c,i:ℕ2].
((iota(fresh-cname(I)) o ((x:=i) o (fresh-cname(I):=c))) = (x:=i) ∈ name-morph(I;I-[x]))
Proof
Definitions occuring in Statement :
name-comp: (f o g),
iota: iota(x),
face-map: (x:=i),
name-morph: name-morph(I;J),
fresh-cname: fresh-cname(I),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
nameset: nameset(L),
prop: ℙ,
false: False,
coordinate_name: Cname,
int_upper: {i...},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
top: Top,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
select: L[n],
cons: [a / b],
cand: A c∧ B,
sq_stable: SqStable(P),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
label: ...$L... t
Lemmas referenced :
fresh-cname_wf,
int_seg_wf,
nameset_wf,
list_wf,
coordinate_name_wf,
l_member_wf,
cons_wf,
nil_wf,
member_singleton,
istype-le,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
list-diff-disjoint,
cname_deq_wf,
list-diff_wf,
l_disjoint_singleton,
member-list-diff,
equal_wf,
squash_wf,
true_wf,
istype-universe,
deq_wf,
list-diff-cons,
subtype_rel_self,
iff_weakening_equal,
deq_member_cons_lemma,
istype-void,
deq_member_nil_lemma,
bor_wf,
bfalse_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
deq-member_wf,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
length_wf,
ite_rw_true,
btrue_wf,
assert_of_tt,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
select_wf,
nat_properties,
sq_stable__le,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
name-comp-assoc,
iota_wf,
face-map_wf2,
list_subtype_base,
name-comp-id-right,
name-morph_wf,
name-comp_wf,
iota-identity,
iota-face-map,
cons_member,
name-morph_subtype,
nameset_subtype,
l_subset_wf,
l_subset_refl,
equal_functionality_wrt_subtype_rel2,
face-map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
natural_numberEquality,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
instantiate,
cumulativity,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
hyp_replacement,
independent_pairFormation,
productIsType,
applyLambdaEquality,
voidElimination,
because_Cache,
applyEquality,
universeEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
approximateComputation,
int_eqEquality,
Error :memTop,
sqequalBase,
inlFormation_alt
Latex:
\mforall{}[I:Cname List]. \mforall{}[x:nameset(I)]. \mforall{}[c,i:\mBbbN{}2].
((iota(fresh-cname(I)) o ((x:=i) o (fresh-cname(I):=c))) = (x:=i))
Date html generated:
2020_05_21-AM-10_49_12
Last ObjectModification:
2019_12_08-PM-07_06_33
Theory : cubical!sets
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