Nuprl Lemma : nc-m-nc-1
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (m(i;j) ⋅ (j1) = 1 ∈ I+i ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-m: m(i;j),
nc-1: (i1),
add-name: I+i,
nh-comp: g ⋅ f,
nh-id: 1,
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],
names-hom: I ⟶ J,
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
names: names(I),
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
squash: ↓T,
DeMorgan-algebra: DeMorganAlgebra,
nc-1: (i1),
nh-id: 1,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_stable: SqStable(P),
true: True,
dM1: 1,
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-m: m(i;j),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
names_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
fset-member_wf,
nat_wf,
int-deq_wf,
istype-void,
istype-nat,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
nc-1_wf,
trivial-member-add-name1,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
not-added-name,
nh-comp-nc-m,
equal_wf,
squash_wf,
true_wf,
istype-universe,
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,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM1-sq-singleton-empty,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
lattice-meet-1,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
dM_inc_wf,
dM-lift-inc,
dM1_wf,
names-subtype,
f-subset-add-name,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionExtensionality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
setIsType,
because_Cache,
functionIsType,
applyEquality,
intEquality,
instantiate,
cumulativity,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
productEquality,
isectEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
productElimination,
equalityIstype,
promote_hyp,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I+i\} ]. (m(i;j) \mcdot{} (j1) = 1)
Date html generated:
2020_05_20-PM-01_36_53
Last ObjectModification:
2020_01_15-PM-02_19_15
Theory : cubical!type!theory
Home
Index