Nuprl Lemma : nc-e'-r
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ].
(r_i ⋅ f,i=j = f,i=j ⋅ r_j ∈ J+j ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j
,
nc-r: r_i
,
add-name: I+i
,
nh-comp: g ⋅ f
,
names-hom: I ⟶ J
,
fset-member: a ∈ s
,
fset: fset(T)
,
int-deq: IntDeq
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
not: ¬A
,
set: {x:A| B[x]}
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
names-hom: I ⟶ J
,
prop: ℙ
,
so_lambda: λ2x.t[x]
,
subtype_rel: A ⊆r B
,
uimplies: b supposing a
,
nat: ℕ
,
so_apply: x[s]
,
top: Top
,
compose: f o g
,
nc-r: r_i
,
nc-e': g,i=j
,
names: names(I)
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
ifthenelse: if b then t else f fi
,
sq_type: SQType(T)
,
guard: {T}
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
bnot: ¬bb
,
assert: ↑b
,
false: False
,
DeMorgan-algebra: DeMorganAlgebra
,
true: True
,
squash: ↓T
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
nequal: a ≠ b ∈ T
,
ge: i ≥ j
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
not: ¬A
,
sq_stable: SqStable(P)
,
decidable: Dec(P)
Lemmas referenced :
names_wf,
add-name_wf,
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
names-hom_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
nh-comp-sq,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
nc-e'_wf,
trivial-member-add-name1,
nc-r_wf,
eq_int_eq_true,
btrue_wf,
iff_weakening_equal,
dma-neg-dM_inc,
dM_opp_wf,
squash_wf,
true_wf,
dM-lift-opp,
dM-lift-inc,
dM-point-subtype,
f-subset-add-name,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
f-subset_wf,
dM-lift-is-id2,
dM_inc_wf,
names-subtype,
fset_wf,
deq_wf,
sq_stable__fset-member,
sq_stable__not,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
instantiate,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
productEquality,
universeEquality,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
int_eqEquality,
independent_pairFormation,
computeAll,
addLevel,
hyp_replacement,
levelHypothesis
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[j:\{i:\mBbbN{}| \mneg{}i \mmember{} J\} ].
(r\_i \mcdot{} f,i=j = f,i=j \mcdot{} r\_j)
Date html generated:
2017_10_05-AM-01_06_13
Last ObjectModification:
2017_07_28-AM-09_27_48
Theory : cubical!type!theory
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