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redefine is_trunc_ppi and is_trunc_spi with unbundled families

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Floris van Doorn 2017-07-05 15:42:27 +01:00
parent 58b007c873
commit 9d39f7771f
3 changed files with 36 additions and 40 deletions
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2
homotopy/EM.hlean
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@ -405,7 +405,7 @@ namespace EM
begin
cases n with n, { exact _ },
cases Y with Y H1 H2, cases Y with Y y₀,
exact is_trunc_pmap_of_is_conn X n -1 (ptrunctype.mk Y _ y₀),
exact is_trunc_pmap_of_is_conn X n -1 _ (pointed.MK Y y₀) !le.refl H2,
end
open category functor nat_trans

30
homotopy/strunc.hlean
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@ -251,39 +251,33 @@ section
variables (A : Type*) (n : ℤ) [H : is_conn (maxm2 n) A]
include H
definition is_trunc_maxm2_ppi (k : ℤ) (P : A → (maxm2 (n+1+k))-Type*)
: is_trunc (maxm2 k) (Π*(a : A), P a) :=
definition is_trunc_maxm2_ppi (k l : ℤ) (H3 : l ≤ n+1+k) (P : A → Type*)
(H2 : Πa, is_trunc (maxm2 l) (P a)) : is_trunc (maxm2 k) (Π*(a : A), P a) :=
is_trunc_maxm2_of_maxm1 (Π*(a : A), P a) k
(@is_trunc_ppi_of_is_conn A (maxm1m1 n)
(is_conn_maxm1_of_maxm2 A n H) (maxm1m1 k)
(λ a, ptrunctype.mk (P a) (is_trunc_of_le (P a) (maxm2_le n k)) pt))
(@is_trunc_ppi_of_is_conn A (maxm1m1 n) (is_conn_maxm1_of_maxm2 A n H) (maxm1m1 k) _
(le.trans (maxm2_monotone H3) (maxm2_le n k)) P H2)
definition is_strunc_spi_of_is_conn (k : ℤ) (P : A → (n+1+k)-spectrum)
: is_strunc k (spi A P) :=
definition is_strunc_spi_of_is_conn (k l : ℤ) (H3 : l ≤ n+1+k) (P : A → spectrum)
(H2 : Πa, is_strunc l (P a)) : is_strunc k (spi A P) :=
begin
intro m, unfold spi,
exact is_trunc_maxm2_ppi A n (k+m)
(λ a, ptrunctype.mk (P a m)
(is_trunc_maxm2_change_int (P a m) (add.assoc (n+1) k m)
(truncspectrum.struct (P a) m)) pt)
exact is_trunc_maxm2_ppi A n (k+m) _ (le.trans (add_le_add_right H3 _)
(le_of_eq (add.assoc (n+1) k m))) (λ a, P a m) (λa, H2 a m)
end
end
definition is_strunc_spi_of_le {A : Type*} (k n : ℤ) (H : n ≤ k) (P : A → n-spectrum)
: is_strunc k (spi A P) :=
definition is_strunc_spi_of_le {A : Type*} (k n : ℤ) (H : n ≤ k) (P : A → spectrum)
(H2 : Πa, is_strunc n (P a)) : is_strunc k (spi A P) :=
begin
assert K : n ≤ -[1+ 0] + 1 + k,
{ krewrite (int.zero_add k), exact H },
{ exact @is_strunc_spi_of_is_conn A (-[1+ 0])
(is_conn.is_conn_minus_two A) k
(λ a, truncspectrum.mk (P a) (is_strunc_of_le (P a) K
(truncspectrum.struct (P a)))) }
{ exact @is_strunc_spi_of_is_conn A (-[1+ 0]) (is_conn.is_conn_minus_two A) k _ K P H2 }
end
definition is_strunc_spi {A : Type*} (n : ℤ) (P : A → spectrum) (H : Πa, is_strunc n (P a))
: is_strunc n (spi A P) :=
is_strunc_spi_of_le n n !le.refl (λa, truncspectrum.mk (P a) (H a))
is_strunc_spi_of_le n n !le.refl P H
definition is_strunc_sp_cotensor (n : ℤ) (A : Type*) {Y : spectrum} (H : is_strunc n Y)
: is_strunc n (sp_cotensor A Y) :=

44
pointed_pi.hlean
View file

@ -494,7 +494,7 @@ namespace is_conn
variables (A : Type*) (n : ℕ₋₂) [H : is_conn (n.+1) A]
include H
definition is_contr_ppi_match (P : A → (n.+1)-Type*)
definition is_contr_ppi_match (P : A → Type*) (H : Πa, is_trunc (n.+1) (P a))
: is_contr (Π*(a : A), P a) :=
begin
apply is_contr.mk pt,
@ -504,44 +504,46 @@ namespace is_conn
{ krewrite (is_conn.elim_β n), apply con.left_inv }
end
definition is_trunc_ppi_of_is_conn (k : ℕ₋₂) (P : A → (n.+1+2+k)-Type*)
: is_trunc k.+1 (Π*(a : A), P a) :=
-- definition is_trunc_ppi_of_is_conn (k : ℕ₋₂) (P : A → Type*)
-- : is_trunc k.+1 (Π*(a : A), P a) :=
definition is_trunc_ppi_of_is_conn (k l : ℕ₋₂) (H2 : l ≤ n.+1+2+k)
(P : A → Type*) (H3 : Πa, is_trunc l (P a)) :
is_trunc k.+1 (Π*(a : A), P a) :=
begin
induction k with k IH,
have H4 : Πa, is_trunc (n.+1+2+k) (P a), from λa, is_trunc_of_le (P a) H2,
clear H2 H3, revert P H4,
induction k with k IH: intro P H4,
{ apply is_prop_of_imp_is_contr, intro f,
apply is_contr_ppi_match },
apply is_contr_ppi_match A n P H4 },
{ apply is_trunc_succ_of_is_trunc_loop
(trunc_index.succ_le_succ (trunc_index.minus_two_le k)),
intro f,
apply @is_trunc_equiv_closed_rev _ _ k.+1
(ppi_loop_equiv f),
apply @is_trunc_equiv_closed_rev _ _ k.+1 (ppi_loop_equiv f),
apply IH, intro a,
apply ptrunctype.mk (Ω (pType.mk (P a) (f a))),
{ apply is_trunc_loop, exact is_trunc_ptrunctype (P a) },
{ exact pt } }
apply is_trunc_loop, apply H4 }
end
definition is_trunc_pmap_of_is_conn (k : ℕ₋₂) (B : (n.+1+2+k)-Type*)
: is_trunc k.+1 (A →* B) :=
definition is_trunc_pmap_of_is_conn (k l : ℕ₋₂) (B : Type*) (H2 : l ≤ n.+1+2+k)
(H3 : is_trunc l B) : is_trunc k.+1 (A →* B) :=
@is_trunc_equiv_closed _ _ k.+1 (ppi_equiv_pmap A B)
(is_trunc_ppi_of_is_conn A n k (λ a, B))
(is_trunc_ppi_of_is_conn A n k l H2 (λ a, B) _)
end
-- this is probably much easier to prove directly
definition is_trunc_ppi (A : Type*) (n k : ℕ₋₂) (H : n ≤ k) (P : A → n-Type*)
: is_trunc k (Π*(a : A), P a) :=
definition is_trunc_ppi (A : Type*) (n k : ℕ₋₂) (H : n ≤ k) (P : A → Type*)
(H2 : Πa, is_trunc n (P a)) : is_trunc k (Π*(a : A), P a) :=
begin
cases k with k,
{ apply trunc.is_contr_of_merely_prop,
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) -2
(λ a, ptrunctype.mk (P a) (is_trunc_of_le (P a)
(trunc_index.le.step H)) pt) },
{ apply is_contr_of_merely_prop,
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) -2 _
(trunc_index.le.step H) P H2 },
{ exact tr pt } },
{ assert K : n ≤ -1 +2+ k,
{ rewrite (trunc_index.add_plus_two_comm -1 k), exact H },
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) k
(λ a, ptrunctype.mk (P a) (is_trunc_of_le (P a) K) pt) } }
{ exact @is_trunc_ppi_of_is_conn A -2 (is_conn_minus_one A (tr pt)) k _ K P H2 } }
end
end is_conn