Nuprl Lemma : nc-e'-s-lemma1
∀[I,J:fset(ℕ)]. ∀[i,z:ℕ]. ∀[g:J ⟶ I]. ∀[j,k:ℕ]. g,i=z ⋅ s = s ⋅ g,j=k,i=z ∈ J+z+k ⟶ I+i supposing ¬j ∈ 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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
names-hom: I ⟶ J,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
compose: f o g,
nc-e': g,i=j,
nc-s: s,
names: names(I),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
DeMorgan-algebra: DeMorganAlgebra,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
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),
sq_stable: SqStable(P)
Lemmas referenced :
names_wf,
add-name_wf,
not_wf,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
names-hom_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_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,
dM-lift-inc,
nc-s_wf,
f-subset-add-name,
trivial-member-add-name1,
trivial-member-add-name2,
add-name-com,
subtype_rel_self,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
dM-lift_wf2,
dM-point-subtype,
names-subtype,
nh-comp-sq,
nc-e'_wf,
dM_inc_wf,
nat_properties,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformand_wf,
int_formula_prop_and_lemma,
dM-lift-is-id2,
f-subset_wf,
f-subset-add-name1,
fset-member-add-name
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
instantiate,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
universeEquality,
productEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
dependent_pairFormation,
promote_hyp,
approximateComputation,
int_eqEquality,
independent_pairFormation,
applyLambdaEquality,
inrFormation
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[i,z:\mBbbN{}]. \mforall{}[g:J {}\mrightarrow{} I]. \mforall{}[j,k:\mBbbN{}]. g,i=z \mcdot{} s = s \mcdot{} g,j=k,i=z supposing \mneg{}j \mmember{} I
Date html generated:
2018_05_23-AM-08_30_40
Last ObjectModification:
2018_05_20-PM-05_44_54
Theory : cubical!type!theory
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