Nuprl Lemma : nc-e'-lemma1
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ((i1) ⋅ g = g,i=j ⋅ (j1) ∈ J ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-1: (i1),
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],
prop: ℙ,
false: False,
nc-e': g,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),
names: names(I),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
top: Top,
nc-1: (i1),
true: True,
DeMorgan-algebra: DeMorganAlgebra,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
names_wf,
add-name_wf,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
istype-void,
names-hom_wf,
istype-nat,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
nat_properties,
equal_wf,
dM1-sq-singleton-empty,
nc-1_wf,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
trivial-member-add-name1,
squash_wf,
true_wf,
dM-lift-1,
dM-lift-inc,
subtype_rel_self,
iff_weakening_equal,
dM1_wf,
intformand_wf,
int_formula_prop_and_lemma,
not-added-name,
istype-universe,
dM-lift_wf2,
dM-point-subtype,
f-subset-add-name,
dM-lift-nc-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setIsType,
inhabitedIsType,
sqequalRule,
functionIsType,
universeIsType,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
natural_numberEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
int_eqEquality,
lambdaEquality,
approximateComputation,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
lambdaFormation,
productEquality,
dependent_set_memberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
Error :memTop,
independent_pairFormation,
isectEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[g:J {}\mrightarrow{} I]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ]. ((i1) \mcdot{} g = g,i=j \mcdot{} (j1))
Date html generated:
2020_05_20-PM-01_37_12
Last ObjectModification:
2020_01_06-PM-02_27_57
Theory : cubical!type!theory
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