Nuprl Lemma : nc-s-comp-e'
∀I:fset(ℕ). ∀i,j:ℕ. ((¬i ∈ I) ⇒ (s ⋅ 1,i=j = s ∈ I+j ⟶ I))
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
nh-id: 1,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
nat-deq: NatDeq,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
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-s: s,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-e': g,i=j,
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,
nh-id: 1,
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
not: ¬A,
label: ...$L... t,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat-deq: NatDeq,
int-deq: IntDeq
Lemmas referenced :
equal_wf,
lattice-point_wf,
dM_wf,
add-name_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,
dM-lift-inc,
nc-e'_wf,
nh-id_wf,
names-subtype,
f-subset-add-name,
dM_inc_wf,
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,
not_wf,
fset-member_wf,
nat_wf,
nat-deq_wf,
fset_wf,
squash_wf,
true_wf,
deq_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
sqequalRule,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
setElimination,
rename,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}i,j:\mBbbN{}. ((\mneg{}i \mmember{} I) {}\mRightarrow{} (s \mcdot{} 1,i=j = s))
Date html generated:
2017_10_05-AM-01_05_00
Last ObjectModification:
2017_07_28-AM-09_27_17
Theory : cubical!type!theory
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