add is_strunc_spi
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Ulrik Buchholtz
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2 changed files with 185 additions and 23 deletions
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@ -5,26 +5,32 @@ open trunc_index nat
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namespace int
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/-
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The function from integers to truncation indices which sends positive numbers to themselves, and negative
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numbers to negative 2. In particular -1 is sent to -2, but since we only work with pointed types, that
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doesn't matter for us -/
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definition maxm2 [unfold 1] : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n trunc_index.of_nat (λk, -2)
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section
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private definition maxm2_le.lemma₁ {n k : ℕ} : n+(1:int) + -[1+ k] ≤ n :=
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le.intro (
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calc n + 1 + -[1+ k] + k = n + 1 - (k + 1) + k : by reflexivity
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... = n : sorry)
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definition maxm2_le_maxm0 (n : ℤ) : maxm2 n ≤ max0 n :=
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private definition maxm2_le.lemma₂ {n : ℕ} {k : ℤ} : -[1+ n] + 1 + k ≤ k :=
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le.intro (
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calc -[1+ n] + 1 + k + n = - (n + 1) + 1 + k + n : by reflexivity
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... = k : sorry)
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definition maxm2_le (n k : ℤ) : maxm2 (n+1+k) ≤ (maxm1m1 n).+1+2+(maxm1m1 k) :=
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begin
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rewrite [-(maxm1_eq_succ n)],
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induction n with n n,
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{ exact le.tr_refl n },
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{ exact minus_two_le 0 }
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{ induction k with k k,
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{ induction k with k IH,
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{ apply le.tr_refl },
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{ exact succ_le_succ IH } },
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{ exact trunc_index.le_trans (maxm2_monotone maxm2_le.lemma₁)
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(maxm2_le_maxm1 n) } },
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{ krewrite (add_plus_two_comm -1 (maxm1m1 k)),
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rewrite [-(maxm1_eq_succ k)],
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exact trunc_index.le_trans (maxm2_monotone maxm2_le.lemma₂)
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(maxm2_le_maxm1 k) }
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end
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definition max0_le_of_le {n : ℤ} {m : ℕ} (H : n ≤ of_nat m)
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: nat.le (max0 n) m :=
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begin
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induction n with n n,
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{ exact le_of_of_nat_le_of_nat H },
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{ exact nat.zero_le m }
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end
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end int
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@ -130,4 +136,63 @@ definition str [constructor] (k : ℤ) (E : spectrum) : E →ₛ strunc k E :=
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smap.mk (λ n, ptr (maxm2 (k + n)) (E n))
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(λ n, sorry)
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structure truncspectrum (n : ℤ) :=
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(carrier : spectrum)
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(struct : is_strunc n carrier)
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notation n `-spectrum` := truncspectrum n
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attribute truncspectrum.carrier [coercion]
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definition genspectrum_of_truncspectrum (n : ℤ)
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: n-spectrum → gen_spectrum +ℤ :=
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λ E, truncspectrum.carrier E
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attribute genspectrum_of_truncspectrum [coercion]
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section
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open is_conn
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definition is_conn_maxm1_of_maxm2 (A : Type*) (n : ℤ)
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: is_conn (maxm2 n) A → is_conn (maxm1m1 n).+1 A :=
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begin
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intro H, induction n with n n,
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{ exact H },
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{ exact is_conn_minus_one A (tr pt) }
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end
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definition is_trunc_maxm2_of_maxm1 (A : Type*) (n : ℤ)
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: is_trunc (maxm1m1 n).+1 A → is_trunc (maxm2 n) A :=
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begin
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intro H, induction n with n n,
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{ exact H},
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{ apply is_contr_of_merely_prop,
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{ exact H },
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{ exact tr pt } }
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end
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variables (A : Type*) (n : ℤ) [H : is_conn (maxm2 n) A]
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include H
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definition is_trunc_maxm2_ppi (k : ℤ) (P : A → (maxm2 (n+1+k))-Type*)
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: is_trunc (maxm2 k) (Π*(a : A), P a) :=
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is_trunc_maxm2_of_maxm1 (Π*(a : A), P a) k
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(@is_trunc_ppi A (maxm1m1 n)
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(is_conn_maxm1_of_maxm2 A n H) (maxm1m1 k)
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(λ a, ptrunctype.mk (P a) (is_trunc_of_le (P a) (maxm2_le n k)) pt))
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definition is_strunc_spi (k : ℤ) (P : A → (n+1+k)-spectrum)
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: is_strunc k (spi A P) :=
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begin
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intro m, unfold spi,
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exact is_trunc_maxm2_ppi A n (k+m)
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(λ a, ptrunctype.mk (P a m)
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(is_trunc_maxm2_change_int (P a m) (add.assoc (n+1) k m)
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(truncspectrum.struct (P a) m)) pt)
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end
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end
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end spectrum
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@ -139,6 +139,110 @@ namespace nat
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end nat
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namespace trunc_index
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open is_conn nat trunc is_trunc
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lemma minus_two_add_plus_two (n : ℕ₋₂) : -2+2+n = n :=
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by induction n with n p; reflexivity; exact ap succ p
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protected definition of_nat_monotone {n k : ℕ} : n ≤ k → of_nat n ≤ of_nat k :=
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begin
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intro H, induction H with k H K,
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{ apply le.tr_refl },
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{ apply le.step K }
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end
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lemma add_plus_two_comm (n k : ℕ₋₂) : n +2+ k = k +2+ n :=
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begin
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induction n with n IH,
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{ exact minus_two_add_plus_two k },
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{ exact !succ_add_plus_two ⬝ ap succ IH}
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end
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end trunc_index
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namespace int
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open trunc_index
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/-
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The function from integers to truncation indices which sends
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positive numbers to themselves, and negative numbers to negative
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2. In particular -1 is sent to -2, but since we only work with
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pointed types, that doesn't matter for us -/
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definition maxm2 [unfold 1] : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n trunc_index.of_nat (λk, -2)
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-- we also need the max -1 - function
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definition maxm1 [unfold 1] : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n trunc_index.of_nat (λk, -1)
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definition maxm2_le_maxm1 (n : ℤ) : maxm2 n ≤ maxm1 n :=
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begin
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induction n with n n,
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{ exact le.tr_refl n },
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{ exact minus_two_le -1 }
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end
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-- the is maxm1 minus 1
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definition maxm1m1 [unfold 1] : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n (λ k, k.-1) (λ k, -2)
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definition maxm1_eq_succ (n : ℤ) : maxm1 n = (maxm1m1 n).+1 :=
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begin
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induction n with n n,
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{ reflexivity },
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{ reflexivity }
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end
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definition maxm2_le_maxm0 (n : ℤ) : maxm2 n ≤ max0 n :=
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begin
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induction n with n n,
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{ exact le.tr_refl n },
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{ exact minus_two_le 0 }
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end
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definition max0_le_of_le {n : ℤ} {m : ℕ} (H : n ≤ of_nat m)
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: nat.le (max0 n) m :=
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begin
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induction n with n n,
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{ exact le_of_of_nat_le_of_nat H },
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{ exact nat.zero_le m }
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end
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section
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-- is there a way to get this from int.add_assoc?
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private definition maxm2_monotone.lemma₁ {n k : ℕ}
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: k + n + (1:int) = k + (1:int) + n :=
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begin
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induction n with n IH,
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{ reflexivity },
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{ exact ap (λ z, z + 1) IH }
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end
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private definition maxm2_monotone.lemma₂ {n k : ℕ} : ¬ n ≤ -[1+ k] :=
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int.not_le_of_gt (lt.intro
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(calc -[1+ k] + (succ (k + n))
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= -(k+1) + (k + n + 1) : by reflexivity
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... = -(k+1) + (k + 1 + n) : maxm2_monotone.lemma₁
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... = (-(k+1) + (k+1)) + n : int.add_assoc
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... = (0:int) + n : by rewrite int.add_left_inv
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... = n : int.zero_add))
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definition maxm2_monotone {n k : ℤ} : n ≤ k → maxm2 n ≤ maxm2 k :=
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begin
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intro H,
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induction n with n n,
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{ induction k with k k,
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{ exact trunc_index.of_nat_monotone (le_of_of_nat_le_of_nat H) },
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{ exact empty.elim (maxm2_monotone.lemma₂ H) } },
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{ induction k with k k,
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{ apply minus_two_le },
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{ apply le.tr_refl } }
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end
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end
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end int
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namespace pmap
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definition eta {A B : Type*} (f : A →* B) : pmap.mk f (respect_pt f) = f :=
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@ -184,13 +288,6 @@ namespace trunc
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end trunc
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namespace trunc_index
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open is_conn nat trunc is_trunc
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lemma minus_two_add_plus_two (n : ℕ₋₂) : -2+2+n = n :=
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by induction n with n p; reflexivity; exact ap succ p
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end trunc_index
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namespace sigma
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-- definition sigma_pathover_equiv_of_is_prop {A : Type} {B : A → Type} {C : Πa, B a → Type}
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