Nuprl Lemma : universe-path-type-lemma-1
∀G:j⊢. ∀A,B:{G ⊢ _:c𝕌}. ∀p:{G ⊢ _:(Path_c𝕌 A B)}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀v:G(I+new-name(I)).
(universe-type(B;I+new-name(I);v)((new-name(I)1) ⋅ f)
= fst((p(v) I+new-name(I) 1 <new-name(I)>))((new-name(I)1) ⋅ f)
∈ Type)
Proof
Definitions occuring in Statement :
universe-type: universe-type(t;I;a),
cubical-universe: c𝕌,
path-type: (Path_A a b),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
new-name: new-name(I),
add-name: I+i,
nh-comp: g ⋅ f,
nh-id: 1,
names-hom: I ⟶ J,
dM_inc: <x>,
fset: fset(T),
nat: ℕ,
pi1: fst(t),
all: ∀x:A. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
cubical-universe: c𝕌,
closed-cubical-universe: cc𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
closed-type-to-type: closed-type-to-type(T),
and: P ∧ Q,
names: names(I),
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
fibrant-type: FibrantType(X),
pi1: fst(t),
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
formal-cube: formal-cube(I),
implies: P ⇒ Q,
true: True,
universe-type: universe-type(t;I;a),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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-1: (i1),
fset-singleton: {x},
cons: [a / b],
lattice-1: 1,
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
cubical-term-at_wf,
add-name_wf,
new-name_wf,
path-type-at,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
I_cube_wf,
names-hom_wf,
fset_wf,
nat_wf,
istype-cubical-term,
path-type_wf,
cubical-universe_wf,
istype-cubical-universe-term,
cubical_set_wf,
nh-id_wf,
nh-comp_wf,
nc-1_wf,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
pi1_wf_top,
cubical-type_wf,
formal-cube_wf1,
lifting-strict-spread,
strict4-spread,
cubical-type-at_wf,
I_cube_pair_redex_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube_set_restriction_pair_lemma,
csm-ap-type-at,
csm-ap-context-map,
nh-id-left,
dM1_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,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM-lift-inc,
subtype_rel_self,
iff_weakening_equal,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
dM1-sq-1,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
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,
dM-lift-1-sq,
interval-type-ap-morph
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
Error :memTop,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
productElimination,
universeIsType,
dependent_set_memberEquality_alt,
intEquality,
independent_isectElimination,
natural_numberEquality,
independent_pairEquality,
equalityIstype,
independent_functionElimination,
hyp_replacement,
universeEquality,
productEquality,
cumulativity,
isectEquality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
approximateComputation,
int_eqEquality
Latex:
\mforall{}G:j\mvdash{}. \mforall{}A,B:\{G \mvdash{} \_:c\mBbbU{}\}. \mforall{}p:\{G \mvdash{} \_:(Path\_c\mBbbU{} A B)\}. \mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}v:G(I+new-name(I)).
(universe-type(B;I+new-name(I);v)((new-name(I)1) \mcdot{} f)
= fst((p(v) I+new-name(I) 1 <new-name(I)>))((new-name(I)1) \mcdot{} f))
Date html generated:
2020_05_20-PM-07_36_15
Last ObjectModification:
2020_04_28-PM-01_19_27
Theory : cubical!type!theory
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