Nuprl Lemma : nc-p-s-commute
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ∀[z:Point(dM(I))]. ((i/z) ⋅ s = s ⋅ (i/z) ∈ I+j ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-p: (i/z),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dM: dM(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
names-hom: I ⟶ J,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
nc-p: (i/z),
nc-s: s,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
names: names(I),
nat: ℕ,
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,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
nc-p_wf,
add-name_wf,
dM-point-subtype,
f-subset-add-name,
equal_wf,
squash_wf,
true_wf,
names-hom_wf,
add-name-com,
subtype_rel_self,
iff_weakening_equal,
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,
names_wf,
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,
nat_wf,
dM-lift-is-id2,
f-subset_weakening,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
f-subset_wf,
nc-s_wf,
dM_inc_wf,
names-subtype,
nat_properties,
full-omega-unsat,
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,
dM-lift-inc,
not-added-name
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
dependent_functionElimination,
instantiate,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
hyp_replacement,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
productEquality,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
intEquality,
approximateComputation,
int_eqEquality,
voidEquality,
independent_pairFormation
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i,j:\mBbbN{}]. \mforall{}[z:Point(dM(I))]. ((i/z) \mcdot{} s = s \mcdot{} (i/z))
Date html generated:
2018_05_23-AM-08_30_34
Last ObjectModification:
2018_05_20-PM-05_44_06
Theory : cubical!type!theory
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