Nuprl Lemma : face-map-property
∀[L:Cname List]
∀x:Cname. ∀p:ℕ2. ∀y:nameset([x / L]).
(((↑isname((x:=p) y)) ⇒ ((¬(y = x ∈ Cname)) ∧ (((x:=p) y) = y ∈ nameset(L))))
∧ ((¬↑isname((x:=p) y)) ⇒ ((y = x ∈ Cname) ∧ (((x:=p) y) = p ∈ ℕ2))))
Proof
Definitions occuring in Statement :
face-map: (x:=i),
isname: isname(z),
nameset: nameset(L),
coordinate_name: Cname,
cons: [a / b],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
face-map: (x:=i),
member: t ∈ T,
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
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,
not: ¬A,
rev_implies: P ⇐ Q,
assert: ↑b,
cand: A c∧ B,
false: False,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
squash: ↓T,
true: True
Lemmas referenced :
nameset_wf,
cons_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal_wf,
subtype_base_sq,
int_subtype_base,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
bool_subtype_base,
false_wf,
decidable__equal_int,
int_seg_properties,
bfalse_wf,
int_seg_subtype,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
isname-name,
intformeq_wf,
intformnot_wf,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
true_wf,
l_member_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
intEquality,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
voidElimination,
independent_pairEquality,
axiomEquality,
hypothesis_subsumption,
addEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
dependent_set_memberEquality
Latex:
\mforall{}[L:Cname List]
\mforall{}x:Cname. \mforall{}p:\mBbbN{}2. \mforall{}y:nameset([x / L]).
(((\muparrow{}isname((x:=p) y)) {}\mRightarrow{} ((\mneg{}(y = x)) \mwedge{} (((x:=p) y) = y)))
\mwedge{} ((\mneg{}\muparrow{}isname((x:=p) y)) {}\mRightarrow{} ((y = x) \mwedge{} (((x:=p) y) = p))))
Date html generated:
2017_10_05-AM-10_06_30
Last ObjectModification:
2017_07_28-AM-11_16_22
Theory : cubical!sets
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