redefine maxm2 in strunc
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2 changed files with 35 additions and 26 deletions
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@ -5,19 +5,18 @@ open trunc_index nat
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namespace int
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definition maxm2 : ℤ → ℕ₋₂ :=
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λ n, int.cases_on n trunc_index.of_nat
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(λ m, nat.cases_on m -1 (λ a, -2))
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attribute maxm2 [unfold 1]
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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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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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{ cases n with n,
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{ exact le.step (le.tr_refl -1) },
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{ exact minus_two_le 0 } }
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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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@ -43,14 +42,11 @@ definition loop_ptrunc_maxm2_pequiv (k : ℤ) (X : Type*) :
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begin
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induction k with k k,
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{ exact loop_ptrunc_pequiv k X },
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{ cases k with k,
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{ exact loop_ptrunc_pequiv -1 X },
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{ cases k with k,
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{ exact loop_ptrunc_pequiv -2 X },
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{ exact loop_pequiv_punit_of_is_set (pType.mk (trunc -2 X) (tr pt))
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⬝e* (pequiv_punit_of_is_contr
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(pType.mk (trunc -2 (Point X = Point X)) (tr idp))
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(is_trunc_trunc -2 (Point X = Point X)))⁻¹ᵉ* } } }
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{ refine _ ⬝e* (pequiv_punit_of_is_contr _ !is_trunc_trunc)⁻¹ᵉ*,
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apply @loop_pequiv_punit_of_is_set,
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cases k with k,
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{ change is_set (trunc 0 X), apply _ },
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{ change is_set (trunc -2 X), apply _ }}
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end
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definition is_trunc_of_is_trunc_maxm2 (k : ℤ) (X : Type)
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@ -73,14 +69,12 @@ definition is_trunc_maxm2_loop (A : pType) (k : ℤ)
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begin
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intro H, induction k with k k,
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{ apply is_trunc_loop, exact H },
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{ cases k with k,
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{ apply is_trunc_loop, exact H},
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{ apply is_trunc_loop, cases k with k,
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{ apply is_contr_loop, cases k with k,
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{ exact H },
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{ apply is_trunc_succ, exact H } } }
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{ have H2 : is_contr A, from H, apply _ } }
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end
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definition is_strunc (k : ℤ) (E : spectrum) : Type :=
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definition is_strunc [reducible] (k : ℤ) (E : spectrum) : Type :=
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Π (n : ℤ), is_trunc (maxm2 (k + n)) (E n)
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definition is_strunc_change_int {k l : ℤ} (E : spectrum) (p : k = l)
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@ -95,6 +89,10 @@ definition is_trunc_maxm2_change_int {k l : ℤ} (X : pType) (p : k = l)
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: is_trunc (maxm2 k) X → is_trunc (maxm2 l) X :=
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by induction p; exact id
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definition strunc_functor [constructor] (k : ℤ) {E F : spectrum} (f : E →ₛ F) :
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strunc k E →ₛ strunc k F :=
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smap.mk (λn, ptrunc_functor (maxm2 (k + n)) (f n)) sorry
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definition is_strunc_EM_spectrum (G : AbGroup)
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: is_strunc 0 (EM_spectrum G) :=
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begin
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@ -102,13 +100,19 @@ begin
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{ -- case ≥ 0
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apply is_trunc_maxm2_change_int (EM G n) (zero_add n)⁻¹,
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apply is_trunc_EM },
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{ induction n with n IH,
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{ change is_contr (EM_spectrum G (-[1+n])),
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induction n with n IH,
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{ -- case = -1
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apply is_trunc_loop, exact ab_group.is_set_carrier G },
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apply is_contr_loop, exact is_trunc_EM G 0 },
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{ -- case < -1
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apply is_trunc_maxm2_loop, exact IH }}
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apply is_trunc_loop, apply is_trunc_succ, exact IH }}
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end
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definition strunc_elim [constructor] {k : ℤ} {E F : spectrum} (f : E →ₛ F)
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(H : is_strunc k F) : strunc k E →ₛ F :=
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smap.mk (λn, ptrunc.elim (maxm2 (k + n)) (f n))
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(λn, sorry)
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definition trivial_shomotopy_group_of_is_strunc (E : spectrum)
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{n k : ℤ} (K : is_strunc n E) (H : n < k)
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: is_contr (πₛ[k] E) :=
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@ -192,8 +192,11 @@ namespace pointed
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psquare (pequiv_ap B p) (pequiv_ap C p) (f a) (f a') :=
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begin induction p, exact phrfl end
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definition is_contr_loop (A : Type*) [is_set A] : is_contr (Ω A) :=
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is_contr.mk idp (λa, !is_prop.elim)
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definition loop_pequiv_punit_of_is_set (X : Type*) [is_set X] : Ω X ≃* punit :=
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pequiv_punit_of_is_contr _ (is_contr_of_inhabited_prop pt)
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pequiv_punit_of_is_contr _ (is_contr_loop X)
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definition loop_punit : Ω punit ≃* punit :=
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loop_pequiv_punit_of_is_set punit
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@ -202,4 +205,6 @@ namespace pointed
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f ~* g :=
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phomotopy.mk (λa, !eq_of_is_contr) !eq_of_is_contr
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end pointed
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