Nuprl Lemma : face-map-comp
∀A,B:Cname List. ∀g:name-morph(A;B). ∀x:nameset(A). ∀i:ℕ2.
(g o (g x:=i)) = ((x:=i) o g) ∈ name-morph(A;B-[g x]) supposing ↑isname(g x)
Proof
Definitions occuring in Statement :
name-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
nameset: nameset(L),
face-map: (x:=i),
name-comp: (f o g),
uext: uext(g),
compose: f o g,
coordinate_name: Cname,
int_upper: {i...},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
isname: isname(z),
int_seg: {i..j-},
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
true: True,
decidable: Dec(P),
squash: ↓T,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
assert-isname,
istype-assert,
isname_wf,
int_seg_wf,
nameset_wf,
name-morph_wf,
name-morphs-equal,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
nil_wf,
name-comp_wf,
face-map_wf2,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
iff_imp_equal_bool,
le_int_wf,
bfalse_wf,
iff_functionality_wrt_iff,
assert_wf,
le_wf,
false_wf,
iff_weakening_uiff,
assert_of_le_int,
iff_weakening_equal,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
nsub2_subtype_extd-nameset,
btrue_wf,
true_wf,
decidable__assert,
not-assert-isname,
member-list-diff,
member_singleton,
l_member_wf,
nameset_subtype_extd-nameset,
nameset_subtype_base,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
equal_wf,
squash_wf,
istype-universe,
eq_int_eq_true,
extd-nameset_subtype_int,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
universeIsType,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
inhabitedIsType,
sqequalRule,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIsType3,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
equalityIsType1,
independent_pairFormation,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
intEquality,
closedConclusion,
universeEquality
Latex:
\mforall{}A,B:Cname List. \mforall{}g:name-morph(A;B). \mforall{}x:nameset(A). \mforall{}i:\mBbbN{}2.
(g o (g x:=i)) = ((x:=i) o g) supposing \muparrow{}isname(g x)
Date html generated:
2019_11_05-PM-00_24_42
Last ObjectModification:
2018_11_08-PM-00_27_32
Theory : cubical!sets
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