Nuprl Lemma : nc-r-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 ⋅ r_j = f,i=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,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
false: False,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
istype-void,
names-hom_wf,
fset_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
istype-le,
nc-r_wf,
trivial-member-add-name1,
nc-e'_wf,
equal_wf,
nh-comp-assoc,
iff_weakening_equal,
nh-comp_wf,
r-comp-r,
nh-id-right,
squash_wf,
true_wf,
istype-universe,
nc-e'-r,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
setIsType,
extract_by_obid,
sqequalRule,
functionIsType,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
natural_numberEquality,
hypothesisEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
instantiate,
universeEquality
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 \mcdot{} r\_j = f,i=j)
Date html generated:
2020_05_20-PM-01_38_09
Last ObjectModification:
2020_01_03-AM-00_25_56
Theory : cubical!type!theory
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