Nuprl Lemma : name-comp-flip
∀I:Cname List. ∀x:nameset(I). ∀K:Cname List. ∀f:name-morph(I;K). ∀f1:name-morph(K;[]).
(f o flip(f1;f x)) = flip((f o f1);x) ∈ name-morph(I;[]) supposing ↑isname(f x)
Proof
Definitions occuring in Statement :
name-morph-flip: flip(f;y),
name-comp: (f o g),
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
coordinate_name: Cname,
nil: [],
list: T List,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
apply: f a,
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,
name-comp: (f o g),
name-morph-flip: flip(f;y),
compose: f o g,
uext: uext(g),
nameset: nameset(L),
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),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
respects-equality: respects-equality(S;T),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
name-morph-ext,
nil_wf,
coordinate_name_wf,
name-comp_wf,
name-morph-flip_wf,
assert-isname,
istype-assert,
isname_wf,
name-morph_wf,
nameset_wf,
list_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,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
isname-name,
member_wf,
respects-equality-set-trivial2,
l_member_wf,
extd-nameset-nil,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
nsub2_subtype_extd-nameset,
name-morph-one-one,
not-assert-isname
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
sqequalRule,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_pairFormation,
approximateComputation,
int_eqEquality,
Error :memTop,
productIsType
Latex:
\mforall{}I:Cname List. \mforall{}x:nameset(I). \mforall{}K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}f1:name-morph(K;[]).
(f o flip(f1;f x)) = flip((f o f1);x) supposing \muparrow{}isname(f x)
Date html generated:
2020_05_21-AM-10_50_07
Last ObjectModification:
2019_12_08-PM-07_06_09
Theory : cubical!sets
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