Nuprl Lemma : system-type-properties
∀Gamma:j⊢. ∀sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List.
(compatible-system{i:l}(Gamma;sys)
⇒ (Gamma, \/(map(λp.(fst(p));sys)) ⊢ system-type(sys)
∧ (∀i:ℕ||sys||. (system-type(sys) = (snd(sys[i])) ∈ {Gamma, fst(sys[i]) ⊢ _}))))
Proof
Definitions occuring in Statement :
system-type: system-type(sys),
compatible-system: compatible-system{i:l}(Gamma;sys),
context-subset: Gamma, phi,
face-or-list: \/(L),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
int_seg: {i..j-},
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
lambda: λx.A[x],
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
system-type: system-type(sys),
face-or-list: \/(L),
select: L[n],
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
compatible-system: compatible-system{i:l}(Gamma;sys),
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
pi1: fst(t),
bdd-distributive-lattice: BoundedDistributiveLattice,
l_exists: (∃x∈L. P[x]),
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
same-cubical-type: Gamma ⊢ A = B,
compose: f o g,
face-term-iff: Gamma ⊢ (phi ⇐⇒ psi),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-0: 0(𝔽),
cubical-term-at: u(a),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
pi2: snd(t),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
cubical-term_wf,
face-type_wf,
cubical-type_wf,
context-subset_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
list_wf,
cubical_set_wf,
reduce_nil_lemma,
map_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
discrete-cubical-type_wf,
top_wf,
face-0_wf,
int_seg_properties,
int_seg_wf,
compatible-system_wf,
nil_wf,
pairwise-cons,
equal_wf,
face-and_wf,
subset-cubical-type,
face-term-implies-subset,
face-term-and-implies1,
face-term-and-implies2,
cons_wf,
length_wf,
pair-eta,
select_wf,
decidable__lt,
context-subset-subtype-and2,
context-subset-subtype,
face-or-list_wf,
map_wf,
pi1_wf_top,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
face-or-list-eq-1,
map-length,
select-map,
subtype_rel_list,
face-and-eq-1,
cubical-term-at_wf,
subtype_rel_self,
length-map,
face-type-at,
respects-equality_weakening,
same-cubical-type-by-list-cases,
map-map,
reduce_cons_lemma,
map_cons_lemma,
face-lattice-0-not-1,
face-term-iff_wf,
face-or-eq-1,
face-or_wf,
context-subset_functionality,
equal_functionality_wrt_subtype_rel2,
length_of_cons_lemma,
case-type_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
select-cons,
case-type-same1,
add-is-int-iff,
false_wf,
case-type-same2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
productEquality,
cumulativity,
instantiate,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
dependent_pairEquality_alt,
spreadEquality,
productIsType,
universeEquality,
isectEquality,
inlFormation_alt,
inrFormation_alt,
unionIsType,
addEquality,
equalityElimination,
pointwiseFunctionality
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}sys:(phi:\{Gamma \mvdash{} \_:\mBbbF{}\} \mtimes{} \{Gamma, phi \mvdash{} \_\}) List.
(compatible-system\{i:l\}(Gamma;sys)
{}\mRightarrow{} (Gamma, \mbackslash{}/(map(\mlambda{}p.(fst(p));sys)) \mvdash{} system-type(sys)
\mwedge{} (\mforall{}i:\mBbbN{}||sys||. (system-type(sys) = (snd(sys[i]))))))
Date html generated:
2020_05_20-PM-03_09_35
Last ObjectModification:
2020_04_07-AM-08_18_58
Theory : cubical!type!theory
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