Nuprl Lemma : nc-e'-comp-m-2
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
(s ⋅ g,i=j ⋅ m(j;l) = g ⋅ s ⋅ s ∈ J+j+l ⟶ I)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-m: m(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,
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],
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],
false: False,
and: P ∧ Q,
prop: ℙ,
top: Top,
compose: f o g,
nc-s: s,
names: names(I),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nc-e': g,i=j,
sq_stable: SqStable(P),
nc-m: m(i;j)
Lemmas referenced :
names_wf,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
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,
istype-void,
names-hom_wf,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM-lift_wf2,
nc-m_wf,
trivial-member-add-name1,
nc-e'_wf,
names-subtype,
f-subset-add-name,
nc-s_wf,
squash_wf,
true_wf,
istype-universe,
dM-lift-inc,
dM-lift-s,
subtype_rel_self,
iff_weakening_equal,
neg_assert_of_eq_int,
assert-bnot,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
bool_subtype_base,
int_subtype_base,
not_wf,
set_subtype_base,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
fset_wf,
deq_wf,
sq_stable__not,
dM-point-subtype,
uall_wf,
f-subset_wf,
dM-lift-is-id2,
f-subset-add-name1,
equal-wf-T-base,
dM_inc_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setIsType,
sqequalRule,
functionIsType,
universeIsType,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
natural_numberEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
productEquality,
cumulativity,
isectEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
equalityIsType1,
promote_hyp,
closedConclusion,
baseApply,
equalityIsType4,
equalityElimination,
lambdaFormation_alt,
hyp_replacement,
lambdaEquality,
dependent_set_memberEquality,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
applyLambdaEquality
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{}[l:\{i:\mBbbN{}| \mneg{}i \mmember{} J+j\} ].
(s \mcdot{} g,i=j \mcdot{} m(j;l) = g \mcdot{} s \mcdot{} s)
Date html generated:
2020_05_20-PM-01_37_41
Last ObjectModification:
2019_12_10-PM-01_11_29
Theory : cubical!type!theory
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