Nuprl Lemma : nc-e'-lemma5
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[k:{i1:ℕ| ¬i1 ∈ I+i} ]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
(s ⋅ g,i=j,k=l = g ⋅ s ∈ J+j+l ⟶ I)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-s: s,
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-s: s,
all: ∀x:A. B[x],
implies: P ⇒ Q,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-e': g,i=j,
names: names(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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,
not: ¬A,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
names_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,
add-name_wf,
names-hom_wf,
f-subset-add-name1,
f-subset-add-name,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
nc-e'_wf,
names-subtype,
nh-comp-sq,
squash_wf,
true_wf,
dM-lift-inc,
dM-lift-s,
iff_weakening_equal,
dM-point-subtype,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
fset-member-add-name,
trivial-member-add-name1,
nat_properties,
sq_stable__not,
decidable__le,
satisfiable-full-omega-tt,
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,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
instantiate,
productEquality,
cumulativity,
universeEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
inrFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
hyp_replacement,
dependent_set_memberEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ]. \mforall{}[g:J {}\mrightarrow{} I]. \mforall{}[k:\{i1:\mBbbN{}| \mneg{}i1 \mmember{} I+i\} ].
\mforall{}[l:\{i:\mBbbN{}| \mneg{}i \mmember{} J+j\} ].
(s \mcdot{} g,i=j,k=l = g \mcdot{} s)
Date html generated:
2017_10_05-AM-01_05_31
Last ObjectModification:
2017_07_28-AM-09_27_27
Theory : cubical!type!theory
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