merged

--- a/NEWS	Tue Apr 09 11:24:47 2019 +0200
+++ b/NEWS	Tue Apr 09 12:36:53 2019 +0100
@@ -233,7 +233,10 @@
 * Session HOL-Number_Theory: More material on residue rings in
 Carmichael's function, primitive roots, more properties for "ord".
 
-* Session HOL-Analysis: More material and better organization.
+* Session HOL-Analysis: Better organization and much more material,
+including algebraic topology.
+
+* Session HOL-Algebra: Much more material on group theory.
 
 * Session HOL-SPARK: .prv files are no longer written to the
 file-system, but exported to the session database. Results may be

--- a/src/HOL/Analysis/Abstract_Euclidean_Space.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Abstract_Euclidean_Space.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -89,6 +89,51 @@
   unfolding Euclidean_space_def continuous_map_in_subtopology
   by (fastforce simp add: continuous_map_componentwise_UNIV continuous_map_diff)
 
+lemma continuous_map_Euclidean_space_iff:
+  "continuous_map (Euclidean_space m) euclideanreal g
+   = continuous_on (topspace (Euclidean_space m)) g"
+proof
+  assume "continuous_map (Euclidean_space m) euclideanreal g"
+  then have "continuous_map (top_of_set {f. \<forall>n\<ge>m. f n = 0}) euclideanreal g"
+    by (simp add: Euclidean_space_def euclidean_product_topology)
+  then show "continuous_on (topspace (Euclidean_space m)) g"
+    by (metis continuous_map_subtopology_eu subtopology_topspace topspace_Euclidean_space)
+next
+  assume "continuous_on (topspace (Euclidean_space m)) g"
+  then have "continuous_map (top_of_set {f. \<forall>n\<ge>m. f n = 0}) euclideanreal g"
+    by (metis (lifting) continuous_map_into_fulltopology continuous_map_subtopology_eu order_refl topspace_Euclidean_space)
+  then show "continuous_map (Euclidean_space m) euclideanreal g"
+    by (simp add: Euclidean_space_def euclidean_product_topology)
+qed
+
+lemma cm_Euclidean_space_iff_continuous_on:
+  "continuous_map (subtopology (Euclidean_space m) S) (Euclidean_space n) f
+   \<longleftrightarrow> continuous_on (topspace (subtopology (Euclidean_space m) S)) f \<and>
+       f ` (topspace (subtopology (Euclidean_space m) S)) \<subseteq> topspace (Euclidean_space n)"
+  (is "?P \<longleftrightarrow> ?Q \<and> ?R")
+proof -
+  have ?Q if ?P
+  proof -
+    have "\<And>n. Euclidean_space n = top_of_set {f. \<forall>m\<ge>n. f m = 0}"
+      by (simp add: Euclidean_space_def euclidean_product_topology)
+    with that show ?thesis
+      by (simp add: subtopology_subtopology)
+  qed
+  moreover
+  have ?R if ?P
+    using that by (simp add: image_subset_iff continuous_map_def)
+  moreover
+  have ?P if ?Q ?R
+  proof -
+    have "continuous_map (top_of_set (topspace (subtopology (subtopology (powertop_real UNIV) {f. \<forall>n\<ge>m. f n = 0}) S))) (top_of_set (topspace (subtopology (powertop_real UNIV) {f. \<forall>na\<ge>n. f na = 0}))) f"
+      using Euclidean_space_def that by auto
+    then show ?thesis
+      by (simp add: Euclidean_space_def euclidean_product_topology subtopology_subtopology)
+  qed
+  ultimately show ?thesis
+    by auto
+qed
+
 lemma homeomorphic_Euclidean_space_product_topology:
   "Euclidean_space n homeomorphic_space product_topology (\<lambda>i. euclideanreal) {..<n}"
 proof -
@@ -125,6 +170,7 @@
    "compact_space (Euclidean_space n) \<longleftrightarrow> n = 0"
   by (auto simp: homeomorphic_compact_space [OF homeomorphic_Euclidean_space_product_topology] compact_space_product_topology)
 
+
 subsection\<open>n-dimensional spheres\<close>
 
 definition nsphere where

--- a/src/HOL/Analysis/Abstract_Topology.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Abstract_Topology.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -2728,7 +2728,7 @@
    "X homeomorphic_space Y \<longleftrightarrow> Y homeomorphic_space X"
   unfolding homeomorphic_space_def by (metis homeomorphic_maps_sym)
 
-lemma homeomorphic_space_trans:
+lemma homeomorphic_space_trans [trans]:
      "\<lbrakk>X1 homeomorphic_space X2; X2 homeomorphic_space X3\<rbrakk> \<Longrightarrow> X1 homeomorphic_space X3"
   unfolding homeomorphic_space_def by (metis homeomorphic_maps_compose)
 

--- a/src/HOL/Analysis/Analysis.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Analysis.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -7,6 +7,8 @@
   Connected
   Abstract_Limits
   Abstract_Euclidean_Space
+  (*Homology*)
+  Simplices
   (* Functional Analysis *)
   Elementary_Normed_Spaces
   Norm_Arith

--- a/src/HOL/Analysis/Convex.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Convex.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -1,4 +1,4 @@
-(* Title:      HOL/Analysis/Convex_Euclidean_Space.thy
+(* Title:      HOL/Analysis/Convex.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh

--- a/src/HOL/Analysis/Function_Topology.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Function_Topology.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -642,6 +642,17 @@
   shows "continuous_on (A \<inter> S) f \<longleftrightarrow> (\<forall>i. continuous_on (A \<inter> S) (\<lambda>x. f x i))"
   by (auto simp: continuous_on_product_then_coordinatewise continuous_on_coordinatewise_then_product)
 
+lemma continuous_map_span_sum:
+  fixes B :: "'a::real_inner set"
+  assumes biB: "\<And>i. i \<in> I \<Longrightarrow> b i \<in> B"
+  shows "continuous_map euclidean (top_of_set (span B)) (\<lambda>x. \<Sum>i\<in>I. x i *\<^sub>R b i)"
+proof (rule continuous_map_euclidean_top_of_set)
+  show "(\<lambda>x. \<Sum>i\<in>I. x i *\<^sub>R b i) -` span B = UNIV"
+    by auto (meson biB lessThan_iff span_base span_scale span_sum)
+  show "continuous_on UNIV (\<lambda>x. \<Sum>i\<in> I. x i *\<^sub>R b i)"
+    by (intro continuous_intros) auto
+qed
+
 subsubsection%important \<open>Topological countability for product spaces\<close>
 
 text \<open>The next two lemmas are useful to prove first or second countability
@@ -867,7 +878,6 @@
   apply standard
   using product_topology_countable_basis topological_basis_imp_subbasis by auto
 
-
 subsection \<open>Metrics on product spaces\<close>
 
 
@@ -1242,7 +1252,7 @@
 qed
 
 instance "fun" :: (countable, polish_space) polish_space
-by standard
+  by standard
 
 
 subsection\<open>The Alexander subbase theorem\<close>

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Simplices.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -0,0 +1,2785 @@
+section\<open>Homology, I: Simplices\<close>
+
+theory "Simplices"
+  imports "Abstract_Euclidean_Space" "HOL-Algebra.Free_Abelian_Groups"
+
+begin
+subsection\<open>Standard simplices, all of which are topological subspaces of @{text"R^n"}.      \<close>
+
+type_synonym 'a chain = "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int"
+
+definition standard_simplex :: "nat \<Rightarrow> (nat \<Rightarrow> real) set" where
+  "standard_simplex p \<equiv>
+    {x. (\<forall>i. 0 \<le> x i \<and> x i \<le> 1) \<and> (\<forall>i>p. x i = 0) \<and> (\<Sum>i\<le>p. x i) = 1}"
+
+lemma topspace_standard_simplex:
+  "topspace(subtopology (powertop_real UNIV) (standard_simplex p))
+    = standard_simplex p"
+  by simp
+
+lemma basis_in_standard_simplex [simp]:
+   "(\<lambda>j. if j = i then 1 else 0) \<in> standard_simplex p \<longleftrightarrow> i \<le> p"
+  by (auto simp: standard_simplex_def)
+
+lemma nonempty_standard_simplex: "standard_simplex p \<noteq> {}"
+  using basis_in_standard_simplex by blast
+
+lemma standard_simplex_0: "standard_simplex 0 = {(\<lambda>j. if j = 0 then 1 else 0)}"
+  by (auto simp: standard_simplex_def)
+
+lemma standard_simplex_mono:
+  assumes "p \<le> q"
+  shows "standard_simplex p \<subseteq> standard_simplex q"
+  using assms
+proof (clarsimp simp: standard_simplex_def)
+  fix x :: "nat \<Rightarrow> real"
+  assume "\<forall>i. 0 \<le> x i \<and> x i \<le> 1" and "\<forall>i>p. x i = 0" and "sum x {..p} = 1"
+  then show "sum x {..q} = 1"
+    using sum.mono_neutral_left [of "{..q}" "{..p}" x] assms by auto
+qed
+
+lemma closedin_standard_simplex:
+   "closedin (powertop_real UNIV) (standard_simplex p)"
+    (is "closedin ?X ?S")
+proof -
+  have eq: "standard_simplex p =
+              (\<Inter>i. {x. x \<in> topspace ?X \<and> x i \<in> {0..1}}) \<inter>
+              (\<Inter>i \<in> {p<..}. {x \<in> topspace ?X. x i \<in> {0}}) \<inter>
+              {x \<in> topspace ?X. (\<Sum>i\<le>p. x i) \<in> {1}}"
+    by (auto simp: standard_simplex_def topspace_product_topology)
+  show ?thesis
+    unfolding eq
+    by (rule closedin_Int closedin_Inter continuous_map_sum
+             continuous_map_product_projection closedin_continuous_map_preimage | force | clarify)+
+qed
+
+lemma standard_simplex_01: "standard_simplex p \<subseteq> UNIV \<rightarrow>\<^sub>E {0..1}"
+  using standard_simplex_def by auto
+
+lemma compactin_standard_simplex:
+   "compactin (powertop_real UNIV) (standard_simplex p)"
+  apply (rule closed_compactin [where K = "PiE UNIV (\<lambda>i. {0..1})"])
+    apply (simp_all add: compactin_PiE standard_simplex_01 closedin_standard_simplex)
+  done
+
+lemma convex_standard_simplex:
+   "\<lbrakk>x \<in> standard_simplex p; y \<in> standard_simplex p;
+     0 \<le> u; u \<le> 1\<rbrakk>
+    \<Longrightarrow> (\<lambda>i. (1 - u) * x i + u * y i) \<in> standard_simplex p"
+  by (simp add: standard_simplex_def sum.distrib convex_bound_le flip: sum_distrib_left)
+
+lemma path_connectedin_standard_simplex:
+   "path_connectedin (powertop_real UNIV) (standard_simplex p)"
+proof -
+  define g where "g \<equiv> \<lambda>x y::nat\<Rightarrow>real. \<lambda>u i. (1 - u) * x i + u * y i"
+  have 1: "continuous_map
+                (subtopology euclideanreal {0..1}) (powertop_real UNIV)
+                (g x y)"
+    if "x \<in> standard_simplex p" "y \<in> standard_simplex p" for x y
+    unfolding g_def continuous_map_componentwise
+    by (force intro: continuous_intros)
+  have 2: "g x y ` {0..1} \<subseteq> standard_simplex p" "g x y 0 = x" "g x y 1 = y"
+    if "x \<in> standard_simplex p" "y \<in> standard_simplex p" for x y
+    using that by (auto simp: convex_standard_simplex g_def)
+  show ?thesis
+    unfolding path_connectedin_def path_connected_space_def pathin_def
+    apply (simp add: topspace_standard_simplex topspace_product_topology continuous_map_in_subtopology)
+    by (metis 1 2)
+qed
+
+lemma connectedin_standard_simplex:
+   "connectedin (powertop_real UNIV) (standard_simplex p)"
+  by (simp add: path_connectedin_imp_connectedin path_connectedin_standard_simplex)
+
+subsection\<open>Face map\<close>
+
+definition simplical_face :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" where
+   "simplical_face k x \<equiv> \<lambda>i. if i < k then x i else if i = k then 0 else x(i -1)"
+
+lemma simplical_face_in_standard_simplex:
+  assumes "1 \<le> p" "k \<le> p" "x \<in> standard_simplex (p - Suc 0)"
+  shows "(simplical_face k x) \<in> standard_simplex p"
+proof -
+  have x01: "\<And>i. 0 \<le> x i \<and> x i \<le> 1" and sumx: "sum x {..p - Suc 0} = 1"
+    using assms by (auto simp: standard_simplex_def simplical_face_def)
+  have gg: "\<And>g. sum g {..p} = sum g {..<k} + sum g {k..p}"
+    using \<open>k \<le> p\<close> sum.union_disjoint [of "{..<k}" "{k..p}"]
+    by (force simp: ivl_disj_un ivl_disj_int)
+  have eq: "(\<Sum>i\<le>p. if i < k then x i else if i = k then 0 else x (i -1))
+         = (\<Sum>i < k. x i) + (\<Sum>i \<in> {k..p}. if i = k then 0 else x (i -1))"
+    by (simp add: gg)
+  consider "k \<le> p - Suc 0" | "k = p"
+    using \<open>k \<le> p\<close> by linarith
+  then have "(\<Sum>i\<le>p. if i < k then x i else if i = k then 0 else x (i -1)) = 1"
+  proof cases
+    case 1
+    have [simp]: "Suc (p - Suc 0) = p"
+      using \<open>1 \<le> p\<close> by auto
+    have "(\<Sum>i = k..p. if i = k then 0 else x (i -1)) = (\<Sum>i = k+1..p. if i = k then 0 else x (i -1))"
+      by (rule sum.mono_neutral_right) auto
+    also have "\<dots> = (\<Sum>i = k+1..p. x (i -1))"
+      by simp
+    also have "\<dots> = (\<Sum>i = k..p-1. x i)"
+      using sum.atLeastAtMost_reindex [of Suc k "p-1" "\<lambda>i. x (i - Suc 0)"] 1 by simp
+    finally have eq2: "(\<Sum>i = k..p. if i = k then 0 else x (i -1)) = (\<Sum>i = k..p-1. x i)" .
+    with 1 show ?thesis
+      apply (simp add: eq eq2)
+      by (metis (mono_tags, lifting) One_nat_def assms(3) finite_atLeastAtMost finite_lessThan ivl_disj_int(4) ivl_disj_un(10) mem_Collect_eq standard_simplex_def sum.union_disjoint)
+  next
+    case 2
+    have [simp]: "({..p} \<inter> {x. x < p}) = {..p - Suc 0}"
+      using assms by auto
+    have "(\<Sum>i\<le>p. if i < p then x i else if i = k then 0 else x (i -1)) = (\<Sum>i\<le>p. if i < p then x i else 0)"
+      by (rule sum.cong) (auto simp: 2)
+    also have "\<dots> = sum x {..p-1}"
+      by (simp add: sum.If_cases)
+    also have "\<dots> = 1"
+      by (simp add: sumx)
+    finally show ?thesis
+      using 2 by simp
+  qed
+  then show ?thesis
+    using assms by (auto simp: standard_simplex_def simplical_face_def)
+qed
+
+subsection\<open>Singular simplices, forcing canonicity outside the intended domain\<close>
+
+definition singular_simplex :: "nat \<Rightarrow> 'a topology \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "singular_simplex p X f \<equiv>
+      continuous_map(subtopology (powertop_real UNIV) (standard_simplex p)) X f
+    \<and> f \<in> extensional (standard_simplex p)"
+
+abbreviation singular_simplex_set :: "nat \<Rightarrow> 'a topology \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'a) set" where
+ "singular_simplex_set p X \<equiv> Collect (singular_simplex p X)"
+
+lemma singular_simplex_empty:
+   "topspace X = {} \<Longrightarrow> \<not> singular_simplex p X f"
+  by (simp add: singular_simplex_def continuous_map nonempty_standard_simplex)
+
+lemma singular_simplex_mono:
+   "\<lbrakk>singular_simplex p (subtopology X T) f; T \<subseteq> S\<rbrakk> \<Longrightarrow> singular_simplex p (subtopology X S) f"
+  by (auto simp: singular_simplex_def continuous_map_in_subtopology)
+
+lemma singular_simplex_subtopology:
+   "singular_simplex p (subtopology X S) f \<longleftrightarrow>
+        singular_simplex p X f \<and> f ` (standard_simplex p) \<subseteq> S"
+  by (auto simp: singular_simplex_def continuous_map_in_subtopology)
+
+subsubsection\<open>Singular face\<close>
+
+definition singular_face :: "nat \<Rightarrow> nat \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> 'a"
+  where "singular_face p k f \<equiv> restrict (f \<circ> simplical_face k) (standard_simplex (p - Suc 0))"
+
+lemma singular_simplex_singular_face:
+  assumes f: "singular_simplex p X f" and "1 \<le> p" "k \<le> p"
+  shows "singular_simplex (p - Suc 0) X (singular_face p k f)"
+proof -
+  let ?PT = "(powertop_real UNIV)"
+  have 0: "simplical_face k ` standard_simplex (p - Suc 0) \<subseteq> standard_simplex p"
+    using assms simplical_face_in_standard_simplex by auto
+  have 1: "continuous_map (subtopology ?PT (standard_simplex (p - Suc 0)))
+                          (subtopology ?PT (standard_simplex p))
+                          (simplical_face k)"
+  proof (clarsimp simp add: continuous_map_in_subtopology simplical_face_in_standard_simplex continuous_map_componentwise 0)
+    fix i
+    have "continuous_map ?PT euclideanreal (\<lambda>x. if i < k then x i else if i = k then 0 else x (i -1))"
+      by (auto intro: continuous_map_product_projection)
+    then show "continuous_map (subtopology ?PT (standard_simplex (p - Suc 0))) euclideanreal
+                              (\<lambda>x. simplical_face k x i)"
+      by (simp add: simplical_face_def continuous_map_from_subtopology)
+  qed
+  have 2: "continuous_map (subtopology ?PT (standard_simplex p)) X f"
+    using assms(1) singular_simplex_def by blast
+  show ?thesis
+    by (simp add: singular_simplex_def singular_face_def continuous_map_compose [OF 1 2])
+qed
+
+
+subsection\<open>Singular chains\<close>
+
+definition singular_chain :: "[nat, 'a topology, 'a chain] \<Rightarrow> bool"
+  where "singular_chain p X c \<equiv> Poly_Mapping.keys c \<subseteq> singular_simplex_set p X"
+
+abbreviation singular_chain_set :: "[nat, 'a topology] \<Rightarrow> ('a chain) set"
+  where "singular_chain_set p X \<equiv> Collect (singular_chain p X)"
+
+lemma singular_chain_empty:
+   "topspace X = {} \<Longrightarrow> singular_chain p X c \<longleftrightarrow> c = 0"
+  by (auto simp: singular_chain_def singular_simplex_empty subset_eq poly_mapping_eqI)
+
+lemma singular_chain_mono:
+   "\<lbrakk>singular_chain p (subtopology X T) c;  T \<subseteq> S\<rbrakk>
+        \<Longrightarrow> singular_chain p (subtopology X S) c"
+  unfolding singular_chain_def using singular_simplex_mono by blast
+
+lemma singular_chain_subtopology:
+   "singular_chain p (subtopology X S) c \<longleftrightarrow>
+        singular_chain p X c \<and> (\<forall>f \<in> Poly_Mapping.keys c. f ` (standard_simplex p) \<subseteq> S)"
+  unfolding singular_chain_def
+  by (fastforce simp add: singular_simplex_subtopology subset_eq)
+
+lemma singular_chain_0 [iff]: "singular_chain p X 0"
+  by (auto simp: singular_chain_def)
+
+lemma singular_chain_of:
+   "singular_chain p X (frag_of c) \<longleftrightarrow> singular_simplex p X c"
+  by (auto simp: singular_chain_def)
+
+lemma singular_chain_cmul:
+   "singular_chain p X c \<Longrightarrow> singular_chain p X (frag_cmul a c)"
+  by (auto simp: singular_chain_def)
+
+lemma singular_chain_minus:
+   "singular_chain p X (-c) \<longleftrightarrow> singular_chain p X c"
+  by (auto simp: singular_chain_def)
+
+lemma singular_chain_add:
+   "\<lbrakk>singular_chain p X a; singular_chain p X b\<rbrakk> \<Longrightarrow> singular_chain p X (a+b)"
+  unfolding singular_chain_def
+  using keys_add [of a b] by blast
+
+lemma singular_chain_diff:
+   "\<lbrakk>singular_chain p X a; singular_chain p X b\<rbrakk> \<Longrightarrow> singular_chain p X (a-b)"
+  unfolding singular_chain_def
+  using keys_diff [of a b] by blast
+
+lemma singular_chain_sum:
+   "(\<And>i. i \<in> I \<Longrightarrow> singular_chain p X (f i)) \<Longrightarrow> singular_chain p X (\<Sum>i\<in>I. f i)"
+  unfolding singular_chain_def
+  using keys_sum [of f I] by blast
+
+lemma singular_chain_extend:
+   "(\<And>c. c \<in> Poly_Mapping.keys x \<Longrightarrow> singular_chain p X (f c))
+        \<Longrightarrow> singular_chain p X (frag_extend f x)"
+  by (simp add: frag_extend_def singular_chain_cmul singular_chain_sum)
+
+subsection\<open>Boundary homomorphism for singular chains\<close>
+
+definition chain_boundary :: "nat \<Rightarrow> ('a chain) \<Rightarrow> 'a chain"
+  where "chain_boundary p c \<equiv>
+          (if p = 0 then 0 else
+           frag_extend (\<lambda>f. (\<Sum>k\<le>p. frag_cmul ((-1) ^ k) (frag_of(singular_face p k f)))) c)"
+
+lemma singular_chain_boundary:
+  "singular_chain p X c
+        \<Longrightarrow> singular_chain (p - Suc 0) X (chain_boundary p c)"
+  unfolding chain_boundary_def
+  apply (clarsimp intro!: singular_chain_extend singular_chain_sum singular_chain_cmul)
+  apply (auto simp: singular_chain_def intro: singular_simplex_singular_face)
+  done
+
+lemma singular_chain_boundary_alt:
+   "singular_chain (Suc p) X c \<Longrightarrow> singular_chain p X (chain_boundary (Suc p) c)"
+  using singular_chain_boundary by force
+
+lemma chain_boundary_0 [simp]: "chain_boundary p 0 = 0"
+  by (simp add: chain_boundary_def)
+
+lemma chain_boundary_cmul:
+   "chain_boundary p (frag_cmul k c) = frag_cmul k (chain_boundary p c)"
+  by (auto simp: chain_boundary_def frag_extend_cmul)
+
+lemma chain_boundary_minus:
+   "chain_boundary p (- c) = - (chain_boundary p c)"
+  by (metis chain_boundary_cmul frag_cmul_minus_one)
+
+lemma chain_boundary_add:
+   "chain_boundary p (a+b) = chain_boundary p a + chain_boundary p b"
+  by (simp add: chain_boundary_def frag_extend_add)
+
+lemma chain_boundary_diff:
+   "chain_boundary p (a-b) = chain_boundary p a - chain_boundary p b"
+  using chain_boundary_add [of p a "-b"]
+  by (simp add: chain_boundary_minus)
+
+lemma chain_boundary_sum:
+   "chain_boundary p (sum g I) = sum (chain_boundary p \<circ> g) I"
+  by (induction I rule: infinite_finite_induct) (simp_all add: chain_boundary_add)
+
+lemma chain_boundary_sum':
+   "finite I \<Longrightarrow> chain_boundary p (sum' g I) = sum' (chain_boundary p \<circ> g) I"
+  by (induction I rule: finite_induct) (simp_all add: chain_boundary_add)
+
+lemma chain_boundary_of:
+   "chain_boundary p (frag_of f) =
+        (if p = 0 then 0
+         else (\<Sum>k\<le>p. frag_cmul ((-1) ^ k) (frag_of(singular_face p k f))))"
+  by (simp add: chain_boundary_def)
+
+subsection\<open>Factoring out chains in a subtopology for relative homology\<close>
+
+definition mod_subset
+  where "mod_subset p X \<equiv> {(a,b). singular_chain p X (a - b)}"
+
+lemma mod_subset_empty [simp]:
+   "(a,b) \<in> (mod_subset p (subtopology X {})) \<longleftrightarrow> a = b"
+  by (simp add: mod_subset_def singular_chain_empty topspace_subtopology)
+
+lemma mod_subset_refl [simp]: "(c,c) \<in> mod_subset p X"
+  by (auto simp: mod_subset_def)
+
+lemma mod_subset_cmul:
+  "(a,b) \<in> (mod_subset p X) \<Longrightarrow> (frag_cmul k a, frag_cmul k b) \<in> (mod_subset p X)"
+  apply (simp add: mod_subset_def)
+  by (metis add_diff_cancel diff_add_cancel frag_cmul_distrib2 singular_chain_cmul)
+
+lemma mod_subset_add:
+   "\<lbrakk>(c1,c2) \<in> (mod_subset p X); (d1,d2) \<in> (mod_subset p X)\<rbrakk>
+    \<Longrightarrow> (c1+d1, c2+d2) \<in> (mod_subset p X)"
+  apply (simp add: mod_subset_def)
+  by (simp add: add_diff_add singular_chain_add)
+
+subsection\<open>Relative cycles $Z_pX (S)$ where $X$ is a topology and $S$ a subset \<close>
+
+definition singular_relcycle :: "nat \<Rightarrow> 'a topology \<Rightarrow> 'a set \<Rightarrow> ('a chain) \<Rightarrow> bool"
+  where "singular_relcycle p X S \<equiv>
+        \<lambda>c. singular_chain p X c \<and> (chain_boundary p c, 0) \<in> mod_subset (p-1) (subtopology X S)"
+
+abbreviation singular_relcycle_set
+  where "singular_relcycle_set p X S \<equiv> Collect (singular_relcycle p X S)"
+
+lemma singular_relcycle_restrict [simp]:
+   "singular_relcycle p X (topspace X \<inter> S) = singular_relcycle p X S"
+proof -
+  have eq: "subtopology X (topspace X \<inter> S) = subtopology X S"
+    by (metis subtopology_subtopology subtopology_topspace)
+  show ?thesis
+    by (force simp: singular_relcycle_def eq)
+qed
+
+lemma singular_relcycle:
+   "singular_relcycle p X S c \<longleftrightarrow>
+    singular_chain p X c \<and> singular_chain (p-1) (subtopology X S) (chain_boundary p c)"
+  by (simp add: singular_relcycle_def mod_subset_def)
+
+lemma singular_relcycle_0 [simp]: "singular_relcycle p X S 0"
+  by (auto simp: singular_relcycle_def)
+
+lemma singular_relcycle_cmul:
+   "singular_relcycle p X S c \<Longrightarrow> singular_relcycle p X S (frag_cmul k c)"
+  by (auto simp: singular_relcycle_def chain_boundary_cmul dest: singular_chain_cmul mod_subset_cmul)
+
+lemma singular_relcycle_minus:
+   "singular_relcycle p X S (-c) \<longleftrightarrow> singular_relcycle p X S c"
+  by (simp add: chain_boundary_minus singular_chain_minus singular_relcycle)
+
+lemma singular_relcycle_add:
+   "\<lbrakk>singular_relcycle p X S a; singular_relcycle p X S b\<rbrakk>
+        \<Longrightarrow> singular_relcycle p X S (a+b)"
+  by (simp add: singular_relcycle_def chain_boundary_add mod_subset_def singular_chain_add)
+
+lemma singular_relcycle_sum:
+   "\<lbrakk>\<And>i. i \<in> I \<Longrightarrow> singular_relcycle p X S (f i)\<rbrakk>
+        \<Longrightarrow> singular_relcycle p X S (sum f I)"
+  by (induction I rule: infinite_finite_induct) (auto simp: singular_relcycle_add)
+
+lemma singular_relcycle_diff:
+   "\<lbrakk>singular_relcycle p X S a; singular_relcycle p X S b\<rbrakk>
+        \<Longrightarrow> singular_relcycle p X S (a-b)"
+  by (metis singular_relcycle_add singular_relcycle_minus uminus_add_conv_diff)
+
+lemma singular_cycle:
+   "singular_relcycle p X {} c \<longleftrightarrow> singular_chain p X c \<and> chain_boundary p c = 0"
+  by (simp add: singular_relcycle_def)
+
+lemma singular_cycle_mono:
+   "\<lbrakk>singular_relcycle p (subtopology X T) {} c; T \<subseteq> S\<rbrakk>
+        \<Longrightarrow> singular_relcycle p (subtopology X S) {} c"
+  by (auto simp: singular_cycle elim: singular_chain_mono)
+
+
+subsection\<open>Relative boundaries $B_p X S$, where $X$ is a topology and $S$ a subset.\<close>
+
+definition singular_relboundary :: "nat \<Rightarrow> 'a topology \<Rightarrow> 'a set \<Rightarrow> ('a chain) \<Rightarrow> bool"
+  where
+  "singular_relboundary p X S \<equiv>
+    \<lambda>c. \<exists>d. singular_chain (Suc p) X d \<and> (chain_boundary (Suc p) d, c) \<in> (mod_subset p (subtopology X S))"
+
+abbreviation singular_relboundary_set :: "nat \<Rightarrow> 'a topology \<Rightarrow> 'a set \<Rightarrow> ('a chain) set"
+  where "singular_relboundary_set p X S \<equiv> Collect (singular_relboundary p X S)"
+
+lemma singular_relboundary_restrict [simp]:
+   "singular_relboundary p X (topspace X \<inter> S) = singular_relboundary p X S"
+  unfolding singular_relboundary_def
+  by (metis (no_types, hide_lams) subtopology_subtopology subtopology_topspace)
+
+lemma singular_relboundary_alt:
+   "singular_relboundary p X S c \<longleftrightarrow>
+    (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and>
+           chain_boundary (Suc p) d = c + e)"
+  unfolding singular_relboundary_def mod_subset_def by fastforce
+
+lemma singular_relboundary:
+   "singular_relboundary p X S c \<longleftrightarrow>
+    (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and>
+              (chain_boundary (Suc p) d) + e = c)"
+  using singular_chain_minus
+  by (fastforce simp add: singular_relboundary_alt)
+
+lemma singular_boundary:
+   "singular_relboundary p X {} c \<longleftrightarrow>
+    (\<exists>d. singular_chain (Suc p) X d \<and> chain_boundary (Suc p) d = c)"
+  by (simp add: singular_relboundary_def)
+
+lemma singular_boundary_imp_chain:
+   "singular_relboundary p X {} c \<Longrightarrow> singular_chain p X c"
+  by (auto simp: singular_relboundary singular_chain_boundary_alt singular_chain_empty topspace_subtopology)
+
+lemma singular_boundary_mono:
+   "\<lbrakk>T \<subseteq> S; singular_relboundary p (subtopology X T) {} c\<rbrakk>
+        \<Longrightarrow> singular_relboundary p (subtopology X S) {} c"
+  by (metis mod_subset_empty singular_chain_mono singular_relboundary_def)
+
+lemma singular_relboundary_imp_chain:
+   "singular_relboundary p X S c \<Longrightarrow> singular_chain p X c"
+  unfolding singular_relboundary singular_chain_subtopology
+  by (blast intro: singular_chain_add singular_chain_boundary_alt)
+
+lemma singular_chain_imp_relboundary:
+   "singular_chain p (subtopology X S) c \<Longrightarrow> singular_relboundary p X S c"
+  unfolding singular_relboundary_def
+  apply (rule_tac x=0 in exI)
+  using mod_subset_def singular_chain_diff by fastforce
+
+lemma singular_relboundary_0 [simp]: "singular_relboundary p X S 0"
+  unfolding singular_relboundary_def
+  by (rule_tac x=0 in exI) auto
+
+lemma singular_relboundary_cmul:
+   "singular_relboundary p X S c \<Longrightarrow> singular_relboundary p X S (frag_cmul a c)"
+  unfolding singular_relboundary_def
+  by (metis chain_boundary_cmul mod_subset_cmul singular_chain_cmul)
+
+lemma singular_relboundary_minus:
+   "singular_relboundary p X S (-c) \<longleftrightarrow> singular_relboundary p X S c"
+  using singular_relboundary_cmul
+  by (metis add.inverse_inverse frag_cmul_minus_one)
+
+lemma singular_relboundary_add:
+   "\<lbrakk>singular_relboundary p X S a; singular_relboundary p X S b\<rbrakk> \<Longrightarrow> singular_relboundary p X S (a+b)"
+  unfolding singular_relboundary_def
+  by (metis chain_boundary_add mod_subset_add singular_chain_add)
+
+lemma singular_relboundary_diff:
+   "\<lbrakk>singular_relboundary p X S a; singular_relboundary p X S b\<rbrakk> \<Longrightarrow> singular_relboundary p X S (a-b)"
+  by (metis uminus_add_conv_diff singular_relboundary_minus singular_relboundary_add)
+
+subsection\<open>The (relative) homology relation\<close>
+
+definition homologous_rel :: "[nat,'a topology,'a set,'a chain,'a chain] \<Rightarrow> bool"
+  where "homologous_rel p X S \<equiv> \<lambda>a b. singular_relboundary p X S (a-b)"
+
+abbreviation homologous_rel_set
+  where "homologous_rel_set p X S a \<equiv> Collect (homologous_rel p X S a)"
+
+lemma homologous_rel_restrict [simp]:
+   "homologous_rel p X (topspace X \<inter> S) = homologous_rel p X S"
+  unfolding homologous_rel_def by (metis singular_relboundary_restrict)
+
+lemma homologous_rel_refl [simp]: "homologous_rel p X S c c"
+  unfolding homologous_rel_def by auto
+
+lemma homologous_rel_sym:
+   "homologous_rel p X S a b = homologous_rel p X S b a"
+  unfolding homologous_rel_def
+  using singular_relboundary_minus by fastforce
+
+lemma homologous_rel_trans:
+  assumes "homologous_rel p X S b c" "homologous_rel p X S a b"
+  shows "homologous_rel p X S a c"
+  using homologous_rel_def
+proof -
+  have "singular_relboundary p X S (b - c)"
+    using assms unfolding homologous_rel_def by blast
+  moreover have "singular_relboundary p X S (b - a)"
+    using assms by (meson homologous_rel_def homologous_rel_sym)
+  ultimately have "singular_relboundary p X S (c - a)"
+    using singular_relboundary_diff by fastforce
+  then show ?thesis
+    by (meson homologous_rel_def homologous_rel_sym)
+qed
+
+lemma homologous_rel_eq:
+   "homologous_rel p X S a = homologous_rel p X S b \<longleftrightarrow>
+    homologous_rel p X S a b"
+  using homologous_rel_sym homologous_rel_trans by fastforce
+
+lemma homologous_rel_set_eq:
+   "homologous_rel_set p X S a = homologous_rel_set p X S b \<longleftrightarrow>
+    homologous_rel p X S a b"
+  by (metis homologous_rel_eq mem_Collect_eq)
+
+lemma homologous_rel_singular_chain:
+  "homologous_rel p X S a b \<Longrightarrow> (singular_chain p X a \<longleftrightarrow> singular_chain p X b)"
+  unfolding homologous_rel_def
+  using singular_chain_diff singular_chain_add
+  by (fastforce dest: singular_relboundary_imp_chain)
+
+lemma homologous_rel_add:
+   "\<lbrakk>homologous_rel p X S a a'; homologous_rel p X S b b'\<rbrakk>
+        \<Longrightarrow> homologous_rel p X S (a+b) (a'+b')"
+  unfolding homologous_rel_def
+  by (simp add: add_diff_add singular_relboundary_add)
+
+lemma homologous_rel_diff:
+  assumes "homologous_rel p X S a a'" "homologous_rel p X S b b'"
+  shows "homologous_rel p X S (a - b) (a' - b')"
+proof -
+  have "singular_relboundary p X S ((a - a') - (b - b'))"
+    using assms singular_relboundary_diff unfolding homologous_rel_def by blast
+  then show ?thesis
+    by (simp add: homologous_rel_def algebra_simps)
+qed
+
+lemma homologous_rel_sum:
+  assumes f: "finite {i \<in> I. f i \<noteq> 0}" and g: "finite {i \<in> I. g i \<noteq> 0}"
+    and h: "\<And>i. i \<in> I \<Longrightarrow> homologous_rel p X S (f i) (g i)"
+  shows "homologous_rel p X S (sum f I) (sum g I)"
+proof (cases "finite I")
+  case True
+  let ?L = "{i \<in> I. f i \<noteq> 0} \<union> {i \<in> I. g i \<noteq> 0}"
+  have L: "finite ?L" "?L \<subseteq> I"
+    using f g by blast+
+  have "sum f I = sum f ?L"
+    by (rule comm_monoid_add_class.sum.mono_neutral_right [OF True]) auto
+  moreover have "sum g I = sum g ?L"
+    by (rule comm_monoid_add_class.sum.mono_neutral_right [OF True]) auto
+  moreover have *: "homologous_rel p X S (f i) (g i)" if "i \<in> ?L" for i
+    using h that by auto
+  have "homologous_rel p X S (sum f ?L) (sum g ?L)"
+    using L
+  proof induction
+    case (insert j J)
+    then show ?case
+      by (simp add: h homologous_rel_add)
+  qed auto
+  ultimately show ?thesis
+    by simp
+qed auto
+
+
+lemma chain_homotopic_imp_homologous_rel:
+  assumes
+   "\<And>c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X' (h c)"
+   "\<And>c. singular_chain (p -1) (subtopology X S) c \<Longrightarrow> singular_chain p (subtopology X' T) (h' c)"
+   "\<And>c. singular_chain p X c
+             \<Longrightarrow> (chain_boundary (Suc p) (h c)) + (h'(chain_boundary p c)) = f c - g c"
+    "singular_relcycle p X S c"
+  shows "homologous_rel p X' T (f c) (g c)"
+  using assms
+  unfolding singular_relcycle_def mod_subset_def homologous_rel_def singular_relboundary_def
+  apply (rule_tac x="h c" in exI, simp)
+  by (metis (no_types, lifting) add_diff_cancel_left' minus_diff_eq singular_chain_minus)
+
+
+subsection\<open>Show that all boundaries are cycles, the key "chain complex" property.\<close>
+
+lemma sum_Int_Diff: "finite A \<Longrightarrow> sum f A = sum f (A \<inter> B) + sum f (A - B)"
+  by (metis Diff_Diff_Int Diff_subset sum.subset_diff)
+
+lemma chain_boundary_boundary:
+  assumes "singular_chain p X c"
+  shows "chain_boundary (p - Suc 0) (chain_boundary p c) = 0"
+proof (cases "p -1 = 0")
+  case False
+  then have "2 \<le> p"
+    by auto
+  show ?thesis
+    using assms
+    unfolding singular_chain_def
+  proof (induction rule: frag_induction)
+    case (one g)
+    then have ss: "singular_simplex p X g"
+      by simp
+    have eql: "{..p} \<times> {..p - Suc 0} \<inter> {(x, y). y < x} = (\<lambda>(j,i). (Suc i, j)) ` {(i,j). i \<le> j \<and> j \<le> p -1}"
+      using False
+      by (auto simp: image_def) (metis One_nat_def diff_Suc_1 diff_le_mono le_refl lessE less_imp_le_nat)
+    have eqr: "{..p} \<times> {..p - Suc 0} - {(x, y). y < x} = {(i,j). i \<le> j \<and> j \<le> p -1}"
+      by auto
+    have eqf: "singular_face (p - Suc 0) i (singular_face p (Suc j) g) =
+               singular_face (p - Suc 0) j (singular_face p i g)" if "i \<le> j" "j \<le> p - Suc 0" for i j
+    proof (rule ext)
+      fix t
+      show "singular_face (p - Suc 0) i (singular_face p (Suc j) g) t =
+            singular_face (p - Suc 0) j (singular_face p i g) t"
+      proof (cases "t \<in> standard_simplex (p -1 -1)")
+        case True
+        have fi: "simplical_face i t \<in> standard_simplex (p - Suc 0)"
+          using False True simplical_face_in_standard_simplex that by force
+        have fj: "simplical_face j t \<in> standard_simplex (p - Suc 0)"
+          by (metis False One_nat_def True simplical_face_in_standard_simplex less_one not_less that(2))
+        have eq: "simplical_face (Suc j) (simplical_face i t) = simplical_face i (simplical_face j t)"
+          using True that ss
+          unfolding standard_simplex_def simplical_face_def by fastforce
+        show ?thesis by (simp add: singular_face_def fi fj eq)
+      qed (simp add: singular_face_def)
+    qed
+    show ?case
+    proof (cases "p = 1")
+      case False
+      have eq0: "frag_cmul (-1) a = b \<Longrightarrow> a + b = 0" for a b
+        by (simp add: neg_eq_iff_add_eq_0)
+      have *: "(\<Sum>x\<le>p. \<Sum>i\<le>p - Suc 0.
+                 frag_cmul ((-1) ^ (x + i)) (frag_of (singular_face (p - Suc 0) i (singular_face p x g))))
+              = 0"
+        apply (simp add: sum.cartesian_product sum_Int_Diff [of "_ \<times> _" _ "{(x,y). y < x}"])
+        apply (rule eq0)
+        apply (simp only: frag_cmul_sum prod.case_distrib [of "frag_cmul (-1)"] frag_cmul_cmul eql eqr flip: power_Suc)
+        apply (force simp: simp add: inj_on_def sum.reindex add.commute eqf intro: sum.cong)
+        done
+      show ?thesis
+        using False by (simp add: chain_boundary_of chain_boundary_sum chain_boundary_cmul frag_cmul_sum * flip: power_add)
+    qed (simp add: chain_boundary_def)
+  next
+    case (diff a b)
+    then show ?case
+      by (simp add: chain_boundary_diff)
+  qed auto
+qed (simp add: chain_boundary_def)
+
+
+lemma chain_boundary_boundary_alt:
+   "singular_chain (Suc p) X c \<Longrightarrow> chain_boundary p (chain_boundary (Suc p) c) = 0"
+  using chain_boundary_boundary by force
+
+lemma singular_relboundary_imp_relcycle:
+  assumes "singular_relboundary p X S c"
+  shows "singular_relcycle p X S c"
+proof -
+  obtain d e where d: "singular_chain (Suc p) X d"
+               and e: "singular_chain p (subtopology X S) e"
+               and c: "c = chain_boundary (Suc p) d + e"
+    using assms by (auto simp: singular_relboundary singular_relcycle)
+  have 1: "singular_chain (p - Suc 0) (subtopology X S) (chain_boundary p (chain_boundary (Suc p) d))"
+    using d chain_boundary_boundary_alt by fastforce
+  have 2: "singular_chain (p - Suc 0) (subtopology X S) (chain_boundary p e)"
+    using \<open>singular_chain p (subtopology X S) e\<close> singular_chain_boundary by auto
+  have "singular_chain p X c"
+    using assms singular_relboundary_imp_chain by auto
+  moreover have "singular_chain (p - Suc 0) (subtopology X S) (chain_boundary p c)"
+    by (simp add: c chain_boundary_add singular_chain_add 1 2)
+  ultimately show ?thesis
+    by (simp add: singular_relcycle)
+qed
+
+lemma homologous_rel_singular_relcycle_1:
+  assumes "homologous_rel p X S c1 c2" "singular_relcycle p X S c1"
+  shows "singular_relcycle p X S c2"
+  using assms
+  by (metis diff_add_cancel homologous_rel_def homologous_rel_sym singular_relboundary_imp_relcycle singular_relcycle_add)
+
+lemma homologous_rel_singular_relcycle:
+  assumes "homologous_rel p X S c1 c2"
+  shows "singular_relcycle p X S c1 = singular_relcycle p X S c2"
+  using assms homologous_rel_singular_relcycle_1
+  using homologous_rel_sym by blast
+
+
+subsection\<open>Operations induced by a continuous map g between topological spaces\<close>
+
+definition simplex_map :: "nat \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> 'a"
+  where "simplex_map p g c \<equiv> restrict (g \<circ> c) (standard_simplex p)"
+
+lemma singular_simplex_simplex_map:
+   "\<lbrakk>singular_simplex p X f; continuous_map X X' g\<rbrakk>
+        \<Longrightarrow> singular_simplex p X' (simplex_map p g f)"
+  unfolding singular_simplex_def simplex_map_def
+  by (auto simp: continuous_map_compose)
+
+lemma simplex_map_eq:
+   "\<lbrakk>singular_simplex p X c;
+     \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk>
+    \<Longrightarrow> simplex_map p f c = simplex_map p g c"
+  by (auto simp: singular_simplex_def simplex_map_def continuous_map_def)
+
+lemma simplex_map_id_gen:
+   "\<lbrakk>singular_simplex p X c;
+     \<And>x. x \<in> topspace X \<Longrightarrow> f x = x\<rbrakk>
+    \<Longrightarrow> simplex_map p f c = c"
+  unfolding singular_simplex_def simplex_map_def continuous_map_def
+  using extensional_arb by fastforce
+
+lemma simplex_map_id [simp]:
+   "simplex_map p id = (\<lambda>c. restrict c (standard_simplex p))"
+  by (auto simp: simplex_map_def)
+
+lemma simplex_map_compose:
+   "simplex_map p (h \<circ> g) = simplex_map p h \<circ> simplex_map p g"
+  unfolding simplex_map_def by force
+
+lemma singular_face_simplex_map:
+   "\<lbrakk>1 \<le> p; k \<le> p\<rbrakk>
+        \<Longrightarrow> singular_face p k (simplex_map p f c) = simplex_map (p - Suc 0) f (c \<circ> simplical_face k)"
+  unfolding simplex_map_def singular_face_def
+  by (force simp: simplical_face_in_standard_simplex)
+
+lemma singular_face_restrict [simp]:
+  assumes "p > 0" "i \<le> p"
+  shows "singular_face p i (restrict f (standard_simplex p)) = singular_face p i f"
+  by (metis assms One_nat_def Suc_leI simplex_map_id singular_face_def singular_face_simplex_map)
+
+
+definition chain_map :: "nat \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a chain"
+  where "chain_map p g c \<equiv> frag_extend (frag_of \<circ> simplex_map p g) c"
+
+lemma singular_chain_chain_map:
+   "\<lbrakk>singular_chain p X c; continuous_map X X' g\<rbrakk> \<Longrightarrow> singular_chain p X' (chain_map p g c)"
+  unfolding chain_map_def
+  apply (rule singular_chain_extend)
+  by (metis comp_apply subsetD mem_Collect_eq singular_chain_def singular_chain_of singular_simplex_simplex_map)
+
+lemma chain_map_0 [simp]: "chain_map p g 0 = 0"
+  by (auto simp: chain_map_def)
+
+lemma chain_map_of [simp]: "chain_map p g (frag_of f) = frag_of (simplex_map p g f)"
+  by (simp add: chain_map_def)
+
+lemma chain_map_cmul [simp]:
+   "chain_map p g (frag_cmul a c) = frag_cmul a (chain_map p g c)"
+  by (simp add: frag_extend_cmul chain_map_def)
+
+lemma chain_map_minus: "chain_map p g (-c) = - (chain_map p g c)"
+  by (simp add: frag_extend_minus chain_map_def)
+
+lemma chain_map_add:
+   "chain_map p g (a+b) = chain_map p g a + chain_map p g b"
+  by (simp add: frag_extend_add chain_map_def)
+
+lemma chain_map_diff:
+   "chain_map p g (a-b) = chain_map p g a - chain_map p g b"
+  by (simp add: frag_extend_diff chain_map_def)
+
+lemma chain_map_sum:
+   "finite I \<Longrightarrow> chain_map p g (sum f I) = sum (chain_map p g \<circ> f) I"
+  by (simp add: frag_extend_sum chain_map_def)
+
+lemma chain_map_eq:
+   "\<lbrakk>singular_chain p X c; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk>
+    \<Longrightarrow> chain_map p f c = chain_map p g c"
+  unfolding singular_chain_def
+  apply (erule frag_induction)
+    apply (auto simp: chain_map_diff)
+  apply (metis simplex_map_eq)
+  done
+
+lemma chain_map_id_gen:
+   "\<lbrakk>singular_chain p X c; \<And>x. x \<in> topspace X \<Longrightarrow> f x = x\<rbrakk>
+    \<Longrightarrow>  chain_map p f c = c"
+  unfolding singular_chain_def
+  by (erule frag_induction) (auto simp: chain_map_diff simplex_map_id_gen)
+
+lemma chain_map_ident:
+   "singular_chain p X c \<Longrightarrow> chain_map p id c = c"
+  by (simp add: chain_map_id_gen)
+
+lemma chain_map_id:
+   "chain_map p id = frag_extend (frag_of \<circ> (\<lambda>f. restrict f (standard_simplex p)))"
+  by (auto simp: chain_map_def)
+
+lemma chain_map_compose:
+   "chain_map p (h \<circ> g) = chain_map p h \<circ> chain_map p g"
+proof
+  show "chain_map p (h \<circ> g) c = (chain_map p h \<circ> chain_map p g) c" for c
+    using subset_UNIV
+  proof (induction c rule: frag_induction)
+    case (one x)
+    then show ?case
+      by simp (metis (mono_tags, lifting) comp_eq_dest_lhs restrict_apply simplex_map_def)
+  next
+    case (diff a b)
+    then show ?case
+      by (simp add: chain_map_diff)
+  qed auto
+qed
+
+lemma singular_simplex_chain_map_id:
+  assumes "singular_simplex p X f"
+  shows "chain_map p f (frag_of (restrict id (standard_simplex p))) = frag_of f"
+proof -
+  have "(restrict (f \<circ> restrict id (standard_simplex p)) (standard_simplex p)) = f"
+    by (rule ext) (metis assms comp_apply extensional_arb id_apply restrict_apply singular_simplex_def)
+  then show ?thesis
+    by (simp add: simplex_map_def)
+qed
+
+lemma chain_boundary_chain_map:
+  assumes "singular_chain p X c"
+  shows "chain_boundary p (chain_map p g c) = chain_map (p - Suc 0) g (chain_boundary p c)"
+  using assms unfolding singular_chain_def
+proof (induction c rule: frag_induction)
+  case (one x)
+  then have "singular_face p i (simplex_map p g x) = simplex_map (p - Suc 0) g (singular_face p i x)"
+    if "0 \<le> i" "i \<le> p" "p \<noteq> 0" for i
+    using that
+    by (fastforce simp add: singular_face_def simplex_map_def simplical_face_in_standard_simplex)
+  then show ?case
+    by (auto simp: chain_boundary_of chain_map_sum)
+next
+  case (diff a b)
+  then show ?case
+    by (simp add: chain_boundary_diff chain_map_diff)
+qed auto
+
+lemma singular_relcycle_chain_map:
+  assumes "singular_relcycle p X S c" "continuous_map X X' g" "g ` S \<subseteq> T"
+  shows "singular_relcycle p X' T (chain_map p g c)"
+proof -
+  have "continuous_map (subtopology X S) (subtopology X' T) g"
+    using assms
+    using continuous_map_from_subtopology continuous_map_in_subtopology topspace_subtopology by fastforce
+  then show ?thesis
+    using chain_boundary_chain_map [of p X c g]
+    by (metis One_nat_def assms(1) assms(2) singular_chain_chain_map singular_relcycle)
+qed
+
+lemma singular_relboundary_chain_map:
+  assumes "singular_relboundary p X S c" "continuous_map X X' g" "g ` S \<subseteq> T"
+  shows "singular_relboundary p X' T (chain_map p g c)"
+proof -
+  obtain d e where d: "singular_chain (Suc p) X d"
+    and e: "singular_chain p (subtopology X S) e" and c: "c = chain_boundary (Suc p) d + e"
+    using assms by (auto simp: singular_relboundary)
+  have "singular_chain (Suc p) X' (chain_map (Suc p) g d)"
+    using assms(2) d singular_chain_chain_map by blast
+  moreover have "singular_chain p (subtopology X' T) (chain_map p g e)"
+  proof -
+    have "\<forall>t. g ` topspace (subtopology t S) \<subseteq> T"
+      by (metis assms(3) closure_of_subset_subtopology closure_of_topspace dual_order.trans image_mono)
+    then show ?thesis
+      by (meson assms(2) continuous_map_from_subtopology continuous_map_in_subtopology e singular_chain_chain_map)
+  qed
+  moreover have "chain_boundary (Suc p) (chain_map (Suc p) g d) + chain_map p g e =
+                 chain_map p g (chain_boundary (Suc p) d + e)"
+    by (metis One_nat_def chain_boundary_chain_map chain_map_add d diff_Suc_1)
+  ultimately show ?thesis
+    unfolding singular_relboundary
+    using c by blast
+qed
+
+
+subsection\<open>Homology of one-point spaces degenerates except for $p = 0$.\<close>
+
+lemma singular_simplex_singleton:
+  assumes "topspace X = {a}"
+  shows "singular_simplex p X f \<longleftrightarrow> f = restrict (\<lambda>x. a) (standard_simplex p)" (is "?lhs = ?rhs")
+proof
+  assume L: ?lhs
+  then show ?rhs
+  proof -
+    have "continuous_map (subtopology (product_topology (\<lambda>n. euclideanreal) UNIV) (standard_simplex p)) X f"
+      using \<open>singular_simplex p X f\<close> singular_simplex_def by blast
+    then have "\<And>c. c \<notin> standard_simplex p \<or> f c = a"
+      by (simp add: assms continuous_map_def)
+    then show ?thesis
+      by (metis (no_types) L extensional_restrict restrict_ext singular_simplex_def)
+  qed
+next
+  assume ?rhs
+  with assms show ?lhs
+    by (auto simp: singular_simplex_def topspace_subtopology)
+qed
+
+lemma singular_chain_singleton:
+  assumes "topspace X = {a}"
+  shows "singular_chain p X c \<longleftrightarrow>
+         (\<exists>b. c = frag_cmul b (frag_of(restrict (\<lambda>x. a) (standard_simplex p))))"
+    (is "?lhs = ?rhs")
+proof
+  let ?f = "restrict (\<lambda>x. a) (standard_simplex p)"
+  assume L: ?lhs
+  with assms have "Poly_Mapping.keys c \<subseteq> {?f}"
+    by (auto simp: singular_chain_def singular_simplex_singleton)
+  then consider "Poly_Mapping.keys c = {}" | "Poly_Mapping.keys c = {?f}"
+    by blast
+  then show ?rhs
+  proof cases
+    case 1
+    with L show ?thesis
+      by (metis frag_cmul_zero keys_eq_empty)
+  next
+    case 2
+    then have "\<exists>b. frag_extend frag_of c = frag_cmul b (frag_of (\<lambda>x\<in>standard_simplex p. a))"
+      by (force simp: frag_extend_def)
+    then show ?thesis
+      by (metis frag_expansion)
+  qed
+next
+  assume ?rhs
+  with assms show ?lhs
+    by (auto simp: singular_chain_def singular_simplex_singleton)
+qed
+
+lemma chain_boundary_of_singleton:
+  assumes tX: "topspace X = {a}" and sc: "singular_chain p X c"
+  shows "chain_boundary p c =
+         (if p = 0 \<or> odd p then 0
+          else frag_extend (\<lambda>f. frag_of(restrict (\<lambda>x. a) (standard_simplex (p -1)))) c)"
+    (is "?lhs = ?rhs")
+proof (cases "p = 0")
+  case False
+  have "?lhs = frag_extend (\<lambda>f. if odd p then 0 else frag_of(restrict (\<lambda>x. a) (standard_simplex (p -1)))) c"
+  proof (simp only: chain_boundary_def False if_False, rule frag_extend_eq)
+    fix f
+    assume "f \<in> Poly_Mapping.keys c"
+    with assms have "singular_simplex p X f"
+      by (auto simp: singular_chain_def)
+    then have *: "\<And>k. k \<le> p \<Longrightarrow> singular_face p k f = (\<lambda>x\<in>standard_simplex (p -1). a)"
+      apply (subst singular_simplex_singleton [OF tX, symmetric])
+      using False singular_simplex_singular_face by fastforce
+    define c where "c \<equiv> frag_of (\<lambda>x\<in>standard_simplex (p -1). a)"
+    have "(\<Sum>k\<le>p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k f)))
+        = (\<Sum>k\<le>p. frag_cmul ((-1) ^ k) c)"
+      by (auto simp: c_def * intro: sum.cong)
+    also have "\<dots> = (if odd p then 0 else c)"
+      by (induction p) (auto simp: c_def restrict_def)
+    finally show "(\<Sum>k\<le>p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k f)))
+                = (if odd p then 0 else frag_of (\<lambda>x\<in>standard_simplex (p -1). a))"
+      unfolding c_def .
+  qed
+  also have "\<dots> = ?rhs"
+    by (auto simp: False frag_extend_eq_0)
+  finally show ?thesis .
+qed (simp add: chain_boundary_def)
+
+
+lemma singular_cycle_singleton:
+  assumes "topspace X = {a}"
+  shows "singular_relcycle p X {} c \<longleftrightarrow> singular_chain p X c \<and> (p = 0 \<or> odd p \<or> c = 0)"
+proof -
+  have "c = 0" if "singular_chain p X c" and "chain_boundary p c = 0" and "even p" and "p \<noteq> 0"
+    using that assms singular_chain_singleton [of X a p c] chain_boundary_of_singleton [OF assms]
+    by (auto simp: frag_extend_cmul)
+  moreover
+  have "chain_boundary p c = 0" if sc: "singular_chain p X c" and "odd p"
+    by (simp add: chain_boundary_of_singleton [OF assms sc] that)
+  moreover have "chain_boundary 0 c = 0" if "singular_chain 0 X c" and "p = 0"
+    by (simp add: chain_boundary_def)
+  ultimately show ?thesis
+  using assms by (auto simp: singular_cycle)
+qed
+
+
+lemma singular_boundary_singleton:
+  assumes "topspace X = {a}"
+  shows "singular_relboundary p X {} c \<longleftrightarrow> singular_chain p X c \<and> (odd p \<or> c = 0)"
+proof (cases "singular_chain p X c")
+  case True
+  have eq: "frag_extend (\<lambda>f. frag_of (\<lambda>x\<in>standard_simplex p. a)) (frag_of (\<lambda>x\<in>standard_simplex (Suc p). a))
+          = frag_of (\<lambda>x\<in>standard_simplex p. a)"
+    by (simp add: singular_chain_singleton frag_extend_cmul assms)
+  have "\<exists>d. singular_chain (Suc p) X d \<and> chain_boundary (Suc p) d = c"
+    if "singular_chain p X c" and "odd p"
+    using assms that
+    apply (clarsimp simp: singular_chain_singleton)
+    apply (rule_tac x = "frag_cmul b (frag_of (\<lambda>x\<in>standard_simplex (Suc p). a))" in exI)
+    apply (subst chain_boundary_of_singleton [of X a "Suc p"])
+    apply (auto simp: singular_chain_singleton frag_extend_cmul eq)
+    done
+  with True assms show ?thesis
+    by (auto simp: singular_boundary chain_boundary_of_singleton)
+next
+  case False
+  with assms singular_boundary_imp_chain show ?thesis
+    by metis
+qed
+
+
+lemma singular_boundary_eq_cycle_singleton:
+  assumes "topspace X = {a}" "1 \<le> p"
+  shows "singular_relboundary p X {} c \<longleftrightarrow> singular_relcycle p X {} c"
+  using assms
+  apply (auto simp: singular_boundary chain_boundary_boundary_alt singular_chain_boundary_alt singular_cycle)
+  by (metis Suc_neq_Zero le_zero_eq singular_boundary singular_boundary_singleton singular_chain_0 singular_cycle_singleton singular_relcycle)
+
+lemma singular_boundary_set_eq_cycle_singleton:
+  assumes "topspace X = {a}" "1 \<le> p"
+  shows "singular_relboundary_set p X {} = singular_relcycle_set p X {}"
+  using singular_boundary_eq_cycle_singleton [OF assms]
+  by blast
+
+subsection\<open>Simplicial chains\<close>
+
+text\<open>Simplicial chains, effectively those resulting from linear maps.
+ We still allow the map to be singular, so the name is questionable.
+These are intended as building-blocks for singular subdivision, rather  than as a axis
+for 1 simplicial homology.\<close>
+
+definition oriented_simplex
+  where "oriented_simplex p l \<equiv> (\<lambda>x\<in>standard_simplex p. \<lambda>i. (\<Sum>j\<le>p. l j i * x j))"
+
+definition simplicial_simplex
+  where
+ "simplicial_simplex p S f \<equiv>
+        singular_simplex p (subtopology (powertop_real UNIV) S) f \<and>
+        (\<exists>l. f = oriented_simplex p l)"
+
+lemma simplicial_simplex:
+  "simplicial_simplex p S f \<longleftrightarrow> f ` (standard_simplex p) \<subseteq> S \<and> (\<exists>l. f = oriented_simplex p l)"
+  (is "?lhs = ?rhs")
+proof
+  assume R: ?rhs
+  show ?lhs
+    using R
+    apply (clarsimp simp: simplicial_simplex_def singular_simplex_subtopology)
+    apply (simp add: singular_simplex_def oriented_simplex_def)
+    apply (clarsimp simp: continuous_map_componentwise)
+    apply (intro continuous_intros continuous_map_from_subtopology continuous_map_product_projection, auto)
+    done
+qed (simp add: simplicial_simplex_def singular_simplex_subtopology)
+
+lemma simplicial_simplex_empty [simp]: "\<not> simplicial_simplex p {} f"
+  by (simp add: nonempty_standard_simplex simplicial_simplex)
+
+definition simplicial_chain
+  where "simplicial_chain p S c \<equiv> Poly_Mapping.keys c \<subseteq> Collect (simplicial_simplex p S)"
+
+lemma simplicial_chain_0 [simp]: "simplicial_chain p S 0"
+  by (simp add: simplicial_chain_def)
+
+lemma simplicial_chain_of [simp]:
+   "simplicial_chain p S (frag_of c) \<longleftrightarrow> simplicial_simplex p S c"
+  by (simp add: simplicial_chain_def)
+
+lemma simplicial_chain_cmul:
+   "simplicial_chain p S c \<Longrightarrow> simplicial_chain p S (frag_cmul a c)"
+  by (auto simp: simplicial_chain_def)
+
+lemma simplicial_chain_diff:
+   "\<lbrakk>simplicial_chain p S c1; simplicial_chain p S c2\<rbrakk> \<Longrightarrow> simplicial_chain p S (c1 - c2)"
+  unfolding simplicial_chain_def  by (meson UnE keys_diff subset_iff)
+
+lemma simplicial_chain_sum:
+   "(\<And>i. i \<in> I \<Longrightarrow> simplicial_chain p S (f i)) \<Longrightarrow> simplicial_chain p S (sum f I)"
+  unfolding simplicial_chain_def
+  using order_trans [OF keys_sum [of f I]]
+  by (simp add: UN_least)
+
+lemma simplicial_simplex_oriented_simplex:
+   "simplicial_simplex p S (oriented_simplex p l)
+    \<longleftrightarrow> ((\<lambda>x i. \<Sum>j\<le>p. l j i * x j) ` standard_simplex p \<subseteq> S)"
+  by (auto simp: simplicial_simplex oriented_simplex_def)
+
+lemma simplicial_imp_singular_simplex:
+   "simplicial_simplex p S f
+      \<Longrightarrow> singular_simplex p (subtopology (powertop_real UNIV) S) f"
+  by (simp add: simplicial_simplex_def)
+
+lemma simplicial_imp_singular_chain:
+   "simplicial_chain p S c
+      \<Longrightarrow> singular_chain p (subtopology (powertop_real UNIV) S) c"
+  unfolding simplicial_chain_def singular_chain_def
+  by (auto intro: simplicial_imp_singular_simplex)
+
+lemma oriented_simplex_eq:
+  "oriented_simplex p l = oriented_simplex p l' \<longleftrightarrow> (\<forall>i. i \<le> p \<longrightarrow> l i = l' i)"
+  (is "?lhs = ?rhs")
+proof
+  assume L: ?lhs
+  show ?rhs
+  proof clarify
+    fix i
+    assume "i \<le> p"
+    let ?fi = "(\<lambda>j. if j = i then 1 else 0)"
+    have "(\<Sum>j\<le>p. l j k * ?fi j) = (\<Sum>j\<le>p. l' j k * ?fi j)" for k
+      apply (rule fun_cong [where x=k])
+      using fun_cong [OF L, of ?fi]
+      apply (simp add: \<open>i \<le> p\<close> oriented_simplex_def)
+      done
+    with \<open>i \<le> p\<close> show "l i = l' i"
+      by (simp add: if_distrib ext cong: if_cong)
+  qed
+qed (auto simp: oriented_simplex_def)
+
+lemma sum_zero_middle:
+  fixes g :: "nat \<Rightarrow> 'a::comm_monoid_add"
+  assumes "1 \<le> p" "k \<le> p"
+  shows "(\<Sum>j\<le>p. if j < k then f j else if j = k then 0 else g (j - Suc 0))
+       = (\<Sum>j\<le>p - Suc 0. if j < k then f j else g j)"  (is "?lhs = ?rhs")
+proof -
+  have [simp]: "{..p - Suc 0} \<inter> {j. j < k} = {..<k}" "{..p - Suc 0} \<inter> - {j. j < k} = {k..p - Suc 0}"
+    using assms by auto
+  have "?lhs = (\<Sum>j<k. f j)  + (\<Sum>j = k..p. if j = k then 0 else g (j - Suc 0))"
+    using sum.union_disjoint [of "{..<k}" "{k..p}", where 'a='a] assms
+    by (simp add: ivl_disj_int_one ivl_disj_un_one)
+  also have "\<dots> = (\<Sum>j<k. f j) + (\<Sum>j = Suc k..p. g (j - Suc 0))"
+    by (simp add: sum_head_Suc [of k p] assms)
+  also have "\<dots> = (\<Sum>j<k. f j) + (\<Sum>j = k..p - Suc 0. g j)"
+    using sum.reindex [of Suc "{k..p - Suc 0}", where 'a='a] assms by simp
+  also have "\<dots> = ?rhs"
+    by (simp add: comm_monoid_add_class.sum.If_cases)
+  finally show ?thesis .
+qed
+
+lemma singular_face_oriented_simplex:
+  assumes "1 \<le> p" "k \<le> p"
+  shows "singular_face p k (oriented_simplex p l) =
+         oriented_simplex (p -1) (\<lambda>j. if j < k then l j else l (Suc j))"
+proof -
+  have "(\<Sum>j\<le>p. l j i * simplical_face k x j)
+      = (\<Sum>j\<le>p - Suc 0. (if j < k then l j else l (Suc j)) i * x j)"
+    if "x \<in> standard_simplex (p - Suc 0)" for i x
+  proof -
+    show ?thesis
+      unfolding simplical_face_def
+      using sum_zero_middle [OF assms, where 'a=real, symmetric]
+      apply (simp add: if_distrib [of "\<lambda>x. _ * x"] if_distrib [of "\<lambda>f. f i * _"] atLeast0AtMost cong: if_cong)
+      done
+  qed
+  then show ?thesis
+    using simplical_face_in_standard_simplex assms
+    by (auto simp: singular_face_def oriented_simplex_def restrict_def)
+qed
+
+lemma simplicial_simplex_singular_face:
+  fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"
+  assumes ss: "simplicial_simplex p S f" and p: "1 \<le> p" "k \<le> p"
+  shows "simplicial_simplex (p - Suc 0) S (singular_face p k f)"
+proof -
+  let ?X = "subtopology (powertop_real UNIV) S"
+  obtain m where l: "singular_simplex p ?X (oriented_simplex p m)"
+       and feq: "f = oriented_simplex p m"
+    using assms by (force simp: simplicial_simplex_def)
+  moreover have "\<exists>l. singular_face p k f = oriented_simplex (p - Suc 0) l"
+    apply (simp add: feq singular_face_def oriented_simplex_def)
+    apply (simp add: simplical_face_in_standard_simplex [OF p] restrict_compose_left subset_eq)
+    apply (rule_tac x="\<lambda>i. if i < k then m i else m (Suc i)" in exI)
+    using sum_zero_middle [OF p, where 'a=real, symmetric]  unfolding simplical_face_def o_def
+    apply (simp add: if_distrib [of "\<lambda>x. _ * x"] if_distrib [of "\<lambda>f. f _ * _"] atLeast0AtMost cong: if_cong)
+    done
+  ultimately
+  show ?thesis
+    using assms by (simp add: singular_simplex_singular_face simplicial_simplex_def)
+qed
+
+lemma simplicial_chain_boundary:
+   "simplicial_chain p S c \<Longrightarrow> simplicial_chain (p -1) S (chain_boundary p c)"
+  unfolding simplicial_chain_def
+proof (induction rule: frag_induction)
+  case (one f)
+  then have "simplicial_simplex p S f"
+    by simp
+  have "simplicial_chain (p - Suc 0) S (frag_of (singular_face p i f))"
+    if "0 < p" "i \<le> p" for i
+    using that one
+    apply (simp add: simplicial_simplex_def singular_simplex_singular_face)
+    apply (force simp: singular_face_oriented_simplex)
+    done
+  then have "simplicial_chain (p - Suc 0) S (chain_boundary p (frag_of f))"
+    unfolding chain_boundary_def frag_extend_of
+    by (auto intro!: simplicial_chain_cmul simplicial_chain_sum)
+  then show ?case
+    by (simp add: simplicial_chain_def [symmetric])
+next
+  case (diff a b)
+  then show ?case
+    by (metis chain_boundary_diff simplicial_chain_def simplicial_chain_diff)
+qed auto
+
+
+subsection\<open>The cone construction on simplicial simplices.\<close>
+
+consts simplex_cone :: "[nat, nat \<Rightarrow> real, [nat \<Rightarrow> real, nat] \<Rightarrow> real, nat \<Rightarrow> real, nat] \<Rightarrow> real"
+specification (simplex_cone)
+  simplex_cone:
+    "\<And>p v l. simplex_cone p v (oriented_simplex p l) =
+          oriented_simplex (Suc p) (\<lambda>i. if i = 0 then v else l(i -1))"
+proof -
+  have *: "\<And>x. \<exists>y. \<forall>v. (\<lambda>l. oriented_simplex (Suc x) (\<lambda>i. if i = 0 then v else l (i -1)))
+                  = (y v \<circ> (oriented_simplex x))"
+    apply (subst choice_iff [symmetric])
+    apply (subst function_factors_left [symmetric])
+    by (simp add: oriented_simplex_eq)
+  then show ?thesis
+    apply (subst choice_iff [symmetric])
+    apply (subst fun_eq_iff [symmetric])
+    unfolding o_def
+    apply (blast intro: sym)
+    done
+qed
+
+lemma simplicial_simplex_simplex_cone:
+  assumes f: "simplicial_simplex p S f"
+    and T: "\<And>x u. \<lbrakk>0 \<le> u; u \<le> 1; x \<in> S\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * v i + u * x i) \<in> T"
+  shows "simplicial_simplex (Suc p) T (simplex_cone p v f)"
+proof -
+  obtain l where l: "\<And>x. x \<in> standard_simplex p \<Longrightarrow> oriented_simplex p l x \<in> S"
+    and feq: "f = oriented_simplex p l"
+    using f by (auto simp: simplicial_simplex)
+  have "oriented_simplex p l x \<in> S" if "x \<in> standard_simplex p" for x
+    using f that by (auto simp: simplicial_simplex feq)
+  then have S: "\<And>x. \<lbrakk>\<And>i. 0 \<le> x i \<and> x i \<le> 1; \<And>i. i>p \<Longrightarrow> x i = 0; sum x {..p} = 1\<rbrakk>
+                 \<Longrightarrow> (\<lambda>i. \<Sum>j\<le>p. l j i * x j) \<in> S"
+    by (simp add: oriented_simplex_def standard_simplex_def)
+  have "oriented_simplex (Suc p) (\<lambda>i. if i = 0 then v else l (i -1)) x \<in> T"
+    if "x \<in> standard_simplex (Suc p)" for x
+  proof (simp add: that oriented_simplex_def sum_atMost_Suc_shift del: sum_atMost_Suc)
+    have x01: "\<And>i. 0 \<le> x i \<and> x i \<le> 1" and x0: "\<And>i. i > Suc p \<Longrightarrow> x i = 0" and x1: "sum x {..Suc p} = 1"
+      using that by (auto simp: oriented_simplex_def standard_simplex_def)
+    obtain a where "a \<in> S"
+      using f by force
+    show "(\<lambda>i. v i * x 0 + (\<Sum>j\<le>p. l j i * x (Suc j))) \<in> T"
+    proof (cases "x 0 = 1")
+      case True
+      then have "sum x {Suc 0..Suc p} = 0"
+        using x1 by (simp add: atMost_atLeast0 sum_head_Suc)
+      then have [simp]: "x (Suc j) = 0" if "j\<le>p" for j
+        unfolding sum.atLeast_Suc_atMost_Suc_shift
+        using x01 that by (simp add: sum_nonneg_eq_0_iff)
+      then show ?thesis
+        using T [of 0 a] \<open>a \<in> S\<close> by (auto simp: True)
+    next
+      case False
+      then have "(\<lambda>i. v i * x 0 + (\<Sum>j\<le>p. l j i * x (Suc j))) = (\<lambda>i. (1 - (1 - x 0)) * v i + (1 - x 0) * (inverse (1 - x 0) * (\<Sum>j\<le>p. l j i * x (Suc j))))"
+        by (force simp: field_simps)
+      also have "\<dots> \<in> T"
+      proof (rule T)
+        have "x 0 < 1"
+          by (simp add: False less_le x01)
+        have xle: "x (Suc i) \<le> (1 - x 0)" for i
+        proof (cases "i \<le> p")
+          case True
+          have "sum x {0, Suc i} \<le> sum x {..Suc p}"
+            by (rule sum_mono2) (auto simp: True x01)
+          then show ?thesis
+           using x1 x01 by (simp add: algebra_simps not_less)
+        qed (simp add: x0 x01)
+        have "(\<lambda>i. (\<Sum>j\<le>p. l j i * (x (Suc j) * inverse (1 - x 0)))) \<in> S"
+        proof (rule S)
+          have "x 0 + (\<Sum>j\<le>p. x (Suc j)) = sum x {..Suc p}"
+            by (metis sum_atMost_Suc_shift)
+          with x1 have "(\<Sum>j\<le>p. x (Suc j)) = 1 - x 0"
+            by simp
+          with False show "(\<Sum>j\<le>p. x (Suc j) * inverse (1 - x 0)) = 1"
+            by (metis add_diff_cancel_left' diff_diff_eq2 diff_zero right_inverse sum_distrib_right)
+      qed (use x01 x0 xle \<open>x 0 < 1\<close> in \<open>auto simp: divide_simps\<close>)
+      then show "(\<lambda>i. inverse (1 - x 0) * (\<Sum>j\<le>p. l j i * x (Suc j))) \<in> S"
+        by (simp add: field_simps sum_divide_distrib)
+    qed (use x01 in auto)
+    finally show ?thesis .
+  qed
+qed
+  then show ?thesis
+    by (auto simp: simplicial_simplex feq  simplex_cone)
+qed
+
+definition simplicial_cone
+  where "simplicial_cone p v \<equiv> frag_extend (frag_of \<circ> simplex_cone p v)"
+
+lemma simplicial_chain_simplicial_cone:
+  assumes c: "simplicial_chain p S c"
+    and T: "\<And>x u. \<lbrakk>0 \<le> u; u \<le> 1; x \<in> S\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * v i + u * x i) \<in> T"
+  shows "simplicial_chain (Suc p) T (simplicial_cone p v c)"
+  using c unfolding simplicial_chain_def simplicial_cone_def
+proof (induction rule: frag_induction)
+  case (one x)
+  then show ?case
+    by (simp add: T simplicial_simplex_simplex_cone)
+next
+  case (diff a b)
+  then show ?case
+    by (metis frag_extend_diff simplicial_chain_def simplicial_chain_diff)
+qed auto
+
+
+lemma chain_boundary_simplicial_cone_of':
+  assumes "f = oriented_simplex p l"
+  shows "chain_boundary (Suc p) (simplicial_cone p v (frag_of f)) =
+         frag_of f
+         - (if p = 0 then frag_of (\<lambda>u\<in>standard_simplex p. v)
+            else simplicial_cone (p -1) v (chain_boundary p (frag_of f)))"
+proof (simp, intro impI conjI)
+  assume "p = 0"
+  have eq: "(oriented_simplex 0 (\<lambda>j. if j = 0 then v else l j)) = (\<lambda>u\<in>standard_simplex 0. v)"
+    by (force simp: oriented_simplex_def standard_simplex_def)
+  show "chain_boundary (Suc 0) (simplicial_cone 0 v (frag_of f))
+        = frag_of f - frag_of (\<lambda>u\<in>standard_simplex 0. v)"
+    by (simp add: assms simplicial_cone_def chain_boundary_of \<open>p = 0\<close> simplex_cone singular_face_oriented_simplex eq cong: if_cong)
+next
+  assume "0 < p"
+  have 0: "simplex_cone (p - Suc 0) v (singular_face p x (oriented_simplex p l))
+         = oriented_simplex p
+              (\<lambda>j. if j < Suc x
+                   then if j = 0 then v else l (j -1)
+                   else if Suc j = 0 then v else l (Suc j -1))" if "x \<le> p" for x
+    using \<open>0 < p\<close> that
+    by (auto simp: Suc_leI singular_face_oriented_simplex simplex_cone oriented_simplex_eq)
+  have 1: "frag_extend (frag_of \<circ> simplex_cone (p - Suc 0) v)
+                     (\<Sum>k = 0..p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k (oriented_simplex p l))))
+         = - (\<Sum>k = Suc 0..Suc p. frag_cmul ((-1) ^ k)
+               (frag_of (singular_face (Suc p) k (simplex_cone p v (oriented_simplex p l)))))"
+    apply (subst sum.atLeast_Suc_atMost_Suc_shift)
+    apply (simp add: frag_extend_sum frag_extend_cmul flip: sum_negf)
+    apply (auto simp: simplex_cone singular_face_oriented_simplex 0 intro: sum.cong)
+    done
+  moreover have 2: "singular_face (Suc p) 0 (simplex_cone p v (oriented_simplex p l))
+                    = oriented_simplex p l"
+    by (simp add: simplex_cone singular_face_oriented_simplex)
+  show "chain_boundary (Suc p) (simplicial_cone p v (frag_of f))
+        = frag_of f - simplicial_cone (p - Suc 0) v (chain_boundary p (frag_of f))"
+    using \<open>p > 0\<close>
+    apply (simp add: assms simplicial_cone_def chain_boundary_of atMost_atLeast0 del: sum_atMost_Suc)
+    apply (subst sum_head_Suc [of 0])
+     apply (simp_all add: 1 2 del: sum_atMost_Suc)
+    done
+qed
+
+lemma chain_boundary_simplicial_cone_of:
+  assumes "simplicial_simplex p S f"
+  shows "chain_boundary (Suc p) (simplicial_cone p v (frag_of f)) =
+         frag_of f
+         - (if p = 0 then frag_of (\<lambda>u\<in>standard_simplex p. v)
+            else simplicial_cone (p -1) v (chain_boundary p (frag_of f)))"
+  using chain_boundary_simplicial_cone_of' assms unfolding simplicial_simplex_def
+  by blast
+
+lemma chain_boundary_simplicial_cone:
+  "simplicial_chain p S c
+   \<Longrightarrow> chain_boundary (Suc p) (simplicial_cone p v c) =
+       c - (if p = 0 then frag_extend (\<lambda>f. frag_of (\<lambda>u\<in>standard_simplex p. v)) c
+            else simplicial_cone (p -1) v (chain_boundary p c))"
+  unfolding simplicial_chain_def
+proof (induction rule: frag_induction)
+  case (one x)
+  then show ?case
+    by (auto simp: chain_boundary_simplicial_cone_of)
+qed (auto simp: chain_boundary_diff simplicial_cone_def frag_extend_diff)
+
+lemma simplex_map_oriented_simplex:
+  assumes l: "simplicial_simplex p (standard_simplex q) (oriented_simplex p l)"
+    and g: "simplicial_simplex r S g" and "q \<le> r"
+  shows "simplex_map p g (oriented_simplex p l) = oriented_simplex p (g \<circ> l)"
+proof -
+  obtain m where geq: "g = oriented_simplex r m"
+    using g by (auto simp: simplicial_simplex_def)
+  have "g (\<lambda>i. \<Sum>j\<le>p. l j i * x j) i = (\<Sum>j\<le>p. g (l j) i * x j)"
+    if "x \<in> standard_simplex p" for x i
+  proof -
+    have ssr: "(\<lambda>i. \<Sum>j\<le>p. l j i * x j) \<in> standard_simplex r"
+      using l that standard_simplex_mono [OF \<open>q \<le> r\<close>]
+      unfolding simplicial_simplex_oriented_simplex by auto
+    have lss: "l j \<in> standard_simplex r" if "j\<le>p" for j
+    proof -
+      have q: "(\<lambda>x i. \<Sum>j\<le>p. l j i * x j) ` standard_simplex p \<subseteq> standard_simplex q"
+        using l by (simp add: simplicial_simplex_oriented_simplex)
+      have p: "l j \<in> (\<lambda>x i. \<Sum>j\<le>p. l j i * x j) ` standard_simplex p"
+        apply (rule_tac x="(\<lambda>i. if i = j then 1 else 0)" in rev_image_eqI)
+        using \<open>j\<le>p\<close> by (force simp: basis_in_standard_simplex if_distrib cong: if_cong)+
+      show ?thesis
+        apply (rule subsetD [OF standard_simplex_mono [OF \<open>q \<le> r\<close>]])
+        apply (rule subsetD [OF q p])
+        done
+    qed
+    show ?thesis
+      apply (simp add: geq oriented_simplex_def sum_distrib_left sum_distrib_right mult.assoc ssr lss)
+      by (rule sum.swap)
+  qed
+  then show ?thesis
+    by (force simp: oriented_simplex_def simplex_map_def o_def)
+qed
+
+
+lemma chain_map_simplicial_cone:
+  assumes g: "simplicial_simplex r S g"
+      and c: "simplicial_chain p (standard_simplex q) c"
+      and v: "v \<in> standard_simplex q" and "q \<le> r"
+  shows "chain_map (Suc p) g (simplicial_cone p v c) = simplicial_cone p (g v) (chain_map p g c)"
+proof -
+  have *: "simplex_map (Suc p) g (simplex_cone p v f) = simplex_cone p (g v) (simplex_map p g f)"
+    if "f \<in> Poly_Mapping.keys c" for f
+  proof -
+    have "simplicial_simplex p (standard_simplex q) f"
+      using c that by (auto simp: simplicial_chain_def)
+    then obtain m where feq: "f = oriented_simplex p m"
+      by (auto simp: simplicial_simplex)
+    have 0: "simplicial_simplex p (standard_simplex q) (oriented_simplex p m)"
+      using \<open>simplicial_simplex p (standard_simplex q) f\<close> feq by blast
+    then have 1: "simplicial_simplex (Suc p) (standard_simplex q)
+                      (oriented_simplex (Suc p) (\<lambda>i. if i = 0 then v else m (i -1)))"
+      using convex_standard_simplex v
+      by (simp flip: simplex_cone add: simplicial_simplex_simplex_cone)
+    show ?thesis
+      using simplex_map_oriented_simplex [OF 1 g \<open>q \<le> r\<close>]
+            simplex_map_oriented_simplex [of p q m r S g, OF 0 g \<open>q \<le> r\<close>]
+      by (simp add: feq oriented_simplex_eq simplex_cone)
+  qed
+  show ?thesis
+    by (auto simp: chain_map_def simplicial_cone_def frag_extend_compose * intro: frag_extend_eq)
+qed
+
+
+subsection\<open>Barycentric subdivision of a linear ("simplicial") simplex's image\<close>
+
+definition simplicial_vertex
+  where "simplicial_vertex i f = f(\<lambda>j. if j = i then 1 else 0)"
+
+lemma simplicial_vertex_oriented_simplex:
+   "simplicial_vertex i (oriented_simplex p l) = (if i \<le> p then l i else undefined)"
+  by (simp add: simplicial_vertex_def oriented_simplex_def if_distrib cong: if_cong)
+
+
+primrec simplicial_subdivision
+where
+  "simplicial_subdivision 0 = id"
+| "simplicial_subdivision (Suc p) =
+     frag_extend
+      (\<lambda>f. simplicial_cone p
+            (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (p + 2))
+            (simplicial_subdivision p (chain_boundary (Suc p) (frag_of f))))"
+
+
+lemma simplicial_subdivision_0 [simp]:
+   "simplicial_subdivision p 0 = 0"
+  by (induction p) auto
+
+lemma simplicial_subdivision_diff:
+   "simplicial_subdivision p (c1-c2) = simplicial_subdivision p c1 - simplicial_subdivision p c2"
+  by (induction p) (auto simp: frag_extend_diff)
+
+lemma simplicial_subdivision_of:
+   "simplicial_subdivision p (frag_of f) =
+         (if p = 0 then frag_of f
+         else simplicial_cone (p -1)
+               (\<lambda>i. (\<Sum>j\<le>p. simplicial_vertex j f i) / (Suc p))
+               (simplicial_subdivision (p -1) (chain_boundary p (frag_of f))))"
+  by (induction p) (auto simp: add.commute)
+
+
+lemma simplicial_chain_simplicial_subdivision:
+   "simplicial_chain p S c
+           \<Longrightarrow> simplicial_chain p S (simplicial_subdivision p c)"
+proof (induction p arbitrary: S c)
+  case (Suc p)
+  show ?case
+    using Suc.prems [unfolded simplicial_chain_def]
+  proof (induction c rule: frag_induction)
+    case (one f)
+    then have f: "simplicial_simplex (Suc p) S f"
+      by auto
+    then have "simplicial_chain p (f ` standard_simplex (Suc p))
+                 (simplicial_subdivision p (chain_boundary (Suc p) (frag_of f)))"
+      by (metis Suc.IH diff_Suc_1 simplicial_chain_boundary simplicial_chain_of simplicial_simplex subsetI)
+    moreover
+    obtain l where l: "\<And>x. x \<in> standard_simplex (Suc p) \<Longrightarrow> (\<lambda>i. (\<Sum>j\<le>Suc p. l j i * x j)) \<in> S"
+      and feq: "f = oriented_simplex (Suc p) l"
+      using f by (fastforce simp: simplicial_simplex oriented_simplex_def simp del: sum_atMost_Suc)
+    have "(\<lambda>i. (1 - u) * ((\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (real p + 2)) + u * y i) \<in> S"
+      if "0 \<le> u" "u \<le> 1" and y: "y \<in> f ` standard_simplex (Suc p)" for y u
+    proof -
+      obtain x where x: "x \<in> standard_simplex (Suc p)" and yeq: "y = oriented_simplex (Suc p) l x"
+        using y feq by blast
+      have "(\<lambda>i. \<Sum>j\<le>Suc p. l j i * ((if j \<le> Suc p then (1 - u) * inverse (p + 2) + u * x j else 0))) \<in> S"
+      proof (rule l)
+        have i2p: "inverse (2 + real p) \<le> 1"
+          by (simp add: divide_simps)
+        show "(\<lambda>j. if j \<le> Suc p then (1 - u) * inverse (real (p + 2)) + u * x j else 0) \<in> standard_simplex (Suc p)"
+          using x \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>
+          apply (simp add: sum.distrib standard_simplex_def i2p linepath_le_1 flip: sum_distrib_left del: sum_atMost_Suc)
+          apply (simp add: divide_simps)
+          done
+      qed
+      moreover have "(\<lambda>i. \<Sum>j\<le>Suc p. l j i * ((1 - u) * inverse (2 + real p) + u * x j))
+                   = (\<lambda>i. (1 - u) * (\<Sum>j\<le>Suc p. l j i) / (real p + 2) + u * (\<Sum>j\<le>Suc p. l j i * x j))"
+      proof
+        fix i
+        have "(\<Sum>j\<le>Suc p. l j i * ((1 - u) * inverse (2 + real p) + u * x j))
+            = (\<Sum>j\<le>Suc p. (1 - u) * l j i / (real p + 2) + u * l j i * x j)" (is "?lhs = _")
+          by (simp add: field_simps cong: sum.cong)
+        also have "\<dots> = (1 - u) * (\<Sum>j\<le>Suc p. l j i) / (real p + 2) + u * (\<Sum>j\<le>Suc p. l j i * x j)" (is "_ = ?rhs")
+          by (simp add: sum_distrib_left sum.distrib sum_divide_distrib mult.assoc del: sum_atMost_Suc)
+        finally show "?lhs = ?rhs" .
+      qed
+      ultimately show ?thesis
+        using feq x yeq
+        by (simp add: simplicial_vertex_oriented_simplex) (simp add: oriented_simplex_def)
+    qed
+    ultimately show ?case
+      by (simp add: simplicial_chain_simplicial_cone)
+  next
+    case (diff a b)
+    then show ?case
+      by (metis simplicial_chain_diff simplicial_subdivision_diff)
+  qed auto
+qed auto
+
+lemma chain_boundary_simplicial_subdivision:
+   "simplicial_chain p S c
+    \<Longrightarrow> chain_boundary p (simplicial_subdivision p c) = simplicial_subdivision (p -1) (chain_boundary p c)"
+proof (induction p arbitrary: c)
+  case (Suc p)
+  show ?case
+    using Suc.prems [unfolded simplicial_chain_def]
+  proof (induction c rule: frag_induction)
+    case (one f)
+    then have f: "simplicial_simplex (Suc p) S f"
+      by simp
+    then have "simplicial_chain p S (simplicial_subdivision p (chain_boundary (Suc p) (frag_of f)))"
+      by (metis diff_Suc_1 simplicial_chain_boundary simplicial_chain_of simplicial_chain_simplicial_subdivision)
+    moreover have "simplicial_chain p S (chain_boundary (Suc p) (frag_of f))"
+      using one simplicial_chain_boundary simplicial_chain_of by fastforce
+    moreover have "simplicial_subdivision (p - Suc 0) (chain_boundary p (chain_boundary (Suc p) (frag_of f))) = 0"
+      by (metis f chain_boundary_boundary_alt simplicial_simplex_def simplicial_subdivision_0 singular_chain_of)
+    ultimately show ?case
+      apply (simp add: chain_boundary_simplicial_cone Suc)
+       apply (auto simp: chain_boundary_of frag_extend_diff simplicial_cone_def)
+      done
+  next
+    case (diff a b)
+    then show ?case
+      by (simp add: simplicial_subdivision_diff chain_boundary_diff frag_extend_diff)
+  qed auto
+qed auto
+
+
+(*A MESS AND USED ONLY ONCE*)
+lemma simplicial_subdivision_shrinks:
+   "\<lbrakk>simplicial_chain p S c;
+     \<And>f x y. \<lbrakk>f \<in> Poly_Mapping.keys c; x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk> \<Longrightarrow> \<bar>f x k - f y k\<bar> \<le> d;
+     f \<in> Poly_Mapping.keys(simplicial_subdivision p c);
+     x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk>
+    \<Longrightarrow> \<bar>f x k - f y k\<bar> \<le> (p / (Suc p)) * d"
+proof (induction p arbitrary: d c f x y)
+  case (Suc p)
+  define Sigp where "Sigp \<equiv> \<lambda>f:: (nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real. \<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / real (p + 2)"
+  let ?CB = "\<lambda>f. chain_boundary (Suc p) (frag_of f)"
+  have *: "Poly_Mapping.keys
+             (simplicial_cone p (Sigp f)
+               (simplicial_subdivision p (?CB f)))
+           \<subseteq> {f. \<forall>x\<in>standard_simplex (Suc p). \<forall>y\<in>standard_simplex (Suc p).
+                      \<bar>f x k - f y k\<bar> \<le> real (Suc p) / real (Suc p + 1) * d}" (is "?lhs \<subseteq> ?rhs")
+    if f: "f \<in> Poly_Mapping.keys c" for f
+  proof -
+    have ssf: "simplicial_simplex (Suc p) S f"
+      using Suc.prems(1) simplicial_chain_def that by auto
+    have 2: "\<And>x y. \<lbrakk>x \<in> standard_simplex (Suc p); y \<in> standard_simplex (Suc p)\<rbrakk> \<Longrightarrow> \<bar>f x k - f y k\<bar> \<le> d"
+      by (meson Suc.prems(2) f subsetD le_Suc_eq order_refl standard_simplex_mono)
+    have sub: "Poly_Mapping.keys ((frag_of \<circ> simplex_cone p (Sigp f)) g) \<subseteq> ?rhs"
+      if "g \<in> Poly_Mapping.keys (simplicial_subdivision p (?CB f))" for g
+    proof -
+      have 1: "simplicial_chain p S (?CB f)"
+        using ssf simplicial_chain_boundary simplicial_chain_of by fastforce
+      have "simplicial_chain (Suc p) (f ` standard_simplex(Suc p)) (frag_of f)"
+        by (metis simplicial_chain_of simplicial_simplex ssf subset_refl)
+      then have sc_sub: "Poly_Mapping.keys (?CB f)
+                         \<subseteq> Collect (simplicial_simplex p (f ` standard_simplex (Suc p)))"
+        by (metis diff_Suc_1 simplicial_chain_boundary simplicial_chain_def)
+      have led: "\<And>h x y. \<lbrakk>h \<in> Poly_Mapping.keys (chain_boundary (Suc p) (frag_of f));
+                          x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk> \<Longrightarrow> \<bar>h x k - h y k\<bar> \<le> d"
+        using Suc.prems(2) f sc_sub
+        by (simp add: simplicial_simplex subset_iff image_iff) metis
+      have "\<And>f' x y. \<lbrakk>f' \<in> Poly_Mapping.keys (simplicial_subdivision p (?CB f)); x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk>
+            \<Longrightarrow> \<bar>f' x k - f' y k\<bar> \<le> (p / (Suc p)) * d"
+        by (blast intro: led Suc.IH [of "chain_boundary (Suc p) (frag_of f)", OF 1])
+      then have g: "\<And>x y. \<lbrakk>x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk> \<Longrightarrow> \<bar>g x k - g y k\<bar> \<le> (p / (Suc p)) * d"
+        using that by blast
+      have "d \<ge> 0"
+        using Suc.prems(2)[OF f] \<open>x \<in> standard_simplex (Suc p)\<close> by force
+      have 3: "simplex_cone p (Sigp f) g \<in> ?rhs"
+      proof -
+        have "simplicial_simplex p (f ` standard_simplex(Suc p)) g"
+          by (metis (mono_tags, hide_lams) sc_sub mem_Collect_eq simplicial_chain_def simplicial_chain_simplicial_subdivision subsetD that)
+        then obtain m where m: "g ` standard_simplex p \<subseteq> f ` standard_simplex (Suc p)"
+          and geq: "g = oriented_simplex p m"
+          using ssf by (auto simp: simplicial_simplex)
+        have m_in_gim: "\<And>i. i \<le> p \<Longrightarrow> m i \<in> g ` standard_simplex p"
+          apply (rule_tac x = "\<lambda>j. if j = i then 1 else 0" in image_eqI)
+           apply (simp_all add: geq oriented_simplex_def if_distrib cong: if_cong)
+          done
+        obtain l where l: "f ` standard_simplex (Suc p) \<subseteq> S"
+          and feq: "f = oriented_simplex (Suc p) l"
+          using ssf by (auto simp: simplicial_simplex)
+        show ?thesis
+        proof (clarsimp simp add: geq simp del: sum_atMost_Suc)
+          fix x y
+          assume x: "x \<in> standard_simplex (Suc p)" and y: "y \<in> standard_simplex (Suc p)"
+          then have x': "(\<forall>i. 0 \<le> x i \<and> x i \<le> 1) \<and> (\<forall>i>Suc p. x i = 0) \<and> (\<Sum>i\<le>Suc p. x i) = 1"
+            and y': "(\<forall>i. 0 \<le> y i \<and> y i \<le> 1) \<and> (\<forall>i>Suc p. y i = 0) \<and> (\<Sum>i\<le>Suc p. y i) = 1"
+            by (auto simp: standard_simplex_def)
+          have "\<bar>(\<Sum>j\<le>Suc p. (if j = 0 then \<lambda>i. (\<Sum>j\<le>Suc p. l j i) / (2 + real p) else m (j -1)) k * x j) -
+                 (\<Sum>j\<le>Suc p. (if j = 0 then \<lambda>i. (\<Sum>j\<le>Suc p. l j i) / (2 + real p) else m (j -1)) k * y j)\<bar>
+                \<le> (1 + real p) * d / (2 + real p)"
+          proof -
+            have zero: "\<bar>m (s - Suc 0) k - (\<Sum>j\<le>Suc p. l j k) / (2 + real p)\<bar> \<le> (1 + real p) * d / (2 + real p)"
+              if "0 < s" and "s \<le> Suc p" for s
+            proof -
+              have "m (s - Suc 0) \<in> f ` standard_simplex (Suc p)"
+                using m m_in_gim that(2) by auto
+              then obtain z where eq: "m (s - Suc 0) = (\<lambda>i. \<Sum>j\<le>Suc p. l j i * z j)" and z: "z \<in> standard_simplex (Suc p)"
+                using feq unfolding oriented_simplex_def by auto
+              show ?thesis
+                unfolding eq
+              proof (rule convex_sum_bound_le)
+                fix i
+                assume i: "i \<in> {..Suc p}"
+                then have [simp]: "card ({..Suc p} - {i}) = Suc p"
+                  by (simp add: card_Suc_Diff1)
+                have "(\<Sum>j\<le>Suc p. \<bar>l i k / (p + 2) - l j k / (p + 2)\<bar>) = (\<Sum>j\<le>Suc p. \<bar>l i k - l j k\<bar> / (p + 2))"
+                  by (rule sum.cong) (simp_all add: flip: diff_divide_distrib)
+                also have "\<dots> = (\<Sum>j \<in> {..Suc p} - {i}. \<bar>l i k - l j k\<bar> / (p + 2))"
+                  by (rule sum.mono_neutral_right) auto
+                also have "\<dots> \<le> (1 + real p) * d / (p + 2)"
+                proof (rule sum_bounded_above_divide)
+                  fix i' :: "nat"
+                  assume i': "i' \<in> {..Suc p} - {i}"
+                  have lf: "\<And>r. r \<le> Suc p \<Longrightarrow> l r \<in> f ` standard_simplex(Suc p)"
+                    apply (rule_tac x="\<lambda>j. if j = r then 1 else 0" in image_eqI)
+                     apply (auto simp: feq oriented_simplex_def if_distrib [of "\<lambda>x. _ * x"] cong: if_cong)
+                    done
+                  show "\<bar>l i k - l i' k\<bar> / real (p + 2) \<le> (1 + real p) * d / real (p + 2) / real (card ({..Suc p} - {i}))"
+                    using i i' lf [of i] lf [of i'] 2
+                    by (auto simp: divide_simps image_iff)
+                qed auto
+                finally have "(\<Sum>j\<le>Suc p. \<bar>l i k / (p + 2) - l j k / (p + 2)\<bar>) \<le> (1 + real p) * d / (p + 2)" .
+                then have "\<bar>\<Sum>j\<le>Suc p. l i k / (p + 2) - l j k / (p + 2)\<bar> \<le> (1 + real p) * d / (p + 2)"
+                  by (rule order_trans [OF sum_abs])
+                then show "\<bar>l i k - (\<Sum>j\<le>Suc p. l j k) / (2 + real p)\<bar> \<le> (1 + real p) * d / (2 + real p)"
+                  by (simp add: sum_subtractf sum_divide_distrib del: sum_atMost_Suc)
+              qed (use standard_simplex_def z in auto)
+            qed
+            have nonz: "\<bar>m (s - Suc 0) k - m (r - Suc 0) k\<bar> \<le> (1 + real p) * d / (2 + real p)" (is "?lhs \<le> ?rhs")
+              if "r < s" and "0 < r" and "r \<le> Suc p" and "s \<le> Suc p" for r s
+            proof -
+              have "?lhs \<le> (p / (Suc p)) * d"
+                using m_in_gim [of "r - Suc 0"] m_in_gim [of "s - Suc 0"] that g by fastforce
+              also have "\<dots> \<le> ?rhs"
+                by (simp add: field_simps \<open>0 \<le> d\<close>)
+              finally show ?thesis .
+            qed
+            have jj: "j \<le> Suc p \<and> j' \<le> Suc p
+                \<longrightarrow> \<bar>(if j' = 0 then \<lambda>i. (\<Sum>j\<le>Suc p. l j i) / (2 + real p) else m (j' -1)) k -
+                     (if j = 0 then \<lambda>i. (\<Sum>j\<le>Suc p. l j i) / (2 + real p) else m (j -1)) k\<bar>
+                     \<le> (1 + real p) * d / (2 + real p)" for j j'
+              apply (rule_tac a=j and b = "j'" in linorder_less_wlog)
+                apply (force simp: zero nonz \<open>0 \<le> d\<close> simp del: sum_atMost_Suc)+
+              done
+            show ?thesis
+              apply (rule convex_sum_bound_le)
+              using x' apply blast
+              using x' apply blast
+              apply (subst abs_minus_commute)
+              apply (rule convex_sum_bound_le)
+              using y' apply blast
+              using y' apply blast
+              using jj by blast
+          qed
+          then show "\<bar>simplex_cone p (Sigp f) (oriented_simplex p m) x k - simplex_cone p (Sigp f) (oriented_simplex p m) y k\<bar>
+                \<le> (1 + real p) * d / (2 + real p)"
+            apply (simp add: feq Sigp_def simplicial_vertex_oriented_simplex simplex_cone del: sum_atMost_Suc)
+            apply (simp add: oriented_simplex_def x y del: sum_atMost_Suc)
+            done
+        qed
+      qed
+      show ?thesis
+        using Suc.IH [OF 1, where f=g] 2 3 by simp
+    qed
+    then show ?thesis
+      unfolding simplicial_chain_def simplicial_cone_def
+      by (simp add: order_trans [OF keys_frag_extend] sub UN_subset_iff)
+  qed
+  show ?case
+    using Suc
+    apply (simp del: sum_atMost_Suc)
+    apply (drule subsetD [OF keys_frag_extend])
+    apply (simp del: sum_atMost_Suc)
+    apply clarify (*OBTAIN?*)
+    apply (rename_tac FFF)
+    using *
+    apply (simp add: add.commute Sigp_def subset_iff)
+    done
+qed (auto simp: standard_simplex_0)
+
+
+subsection\<open>Singular subdivision\<close>
+
+definition singular_subdivision
+  where "singular_subdivision p \<equiv>
+        frag_extend
+           (\<lambda>f. chain_map p f
+                  (simplicial_subdivision p
+                         (frag_of(restrict id (standard_simplex p)))))"
+
+lemma singular_subdivision_0 [simp]: "singular_subdivision p 0 = 0"
+  by (simp add: singular_subdivision_def)
+
+lemma singular_subdivision_add:
+   "singular_subdivision p (a + b) = singular_subdivision p a + singular_subdivision p b"
+  by (simp add: singular_subdivision_def frag_extend_add)
+
+lemma singular_subdivision_diff:
+   "singular_subdivision p (a - b) = singular_subdivision p a - singular_subdivision p b"
+  by (simp add: singular_subdivision_def frag_extend_diff)
+
+lemma simplicial_simplex_id [simp]:
+   "simplicial_simplex p S (restrict id (standard_simplex p)) \<longleftrightarrow> standard_simplex p \<subseteq> S"
+    (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    by (simp add: simplicial_simplex_def singular_simplex_def continuous_map_in_subtopology set_mp)
+next
+  assume R: ?rhs
+  then have cm: "continuous_map
+                 (subtopology (powertop_real UNIV) (standard_simplex p))
+                 (subtopology (powertop_real UNIV) S) id"
+    using continuous_map_from_subtopology_mono continuous_map_id by blast
+  moreover have "\<exists>l. restrict id (standard_simplex p) = oriented_simplex p l"
+    apply (rule_tac x="\<lambda>i j. if i = j then 1 else 0" in exI)
+    apply (force simp: oriented_simplex_def standard_simplex_def if_distrib [of "\<lambda>u. u * _"] cong: if_cong)
+    done
+  ultimately show ?lhs
+    by (simp add: simplicial_simplex_def singular_simplex_def)
+qed
+
+lemma singular_chain_singular_subdivision:
+   "singular_chain p X c
+        \<Longrightarrow> singular_chain p X (singular_subdivision p c)"
+  unfolding singular_subdivision_def
+  apply (rule singular_chain_extend)
+  apply (rule singular_chain_chain_map [where X = "subtopology (powertop_real UNIV)
+                          (standard_simplex p)"])
+  apply (simp add: simplicial_chain_simplicial_subdivision simplicial_imp_singular_chain)
+  by (simp add: singular_chain_def singular_simplex_def subset_iff)
+
+lemma naturality_singular_subdivision:
+   "singular_chain p X c
+    \<Longrightarrow> singular_subdivision p (chain_map p g c) = chain_map p g (singular_subdivision p c)"
+  unfolding singular_chain_def
+proof (induction rule: frag_induction)
+  case (one f)
+  then have "singular_simplex p X f"
+    by auto
+  have "\<lbrakk>simplicial_chain p (standard_simplex p) d\<rbrakk>
+    \<Longrightarrow> chain_map p (simplex_map p g f) d = chain_map p g (chain_map p f d)" for d
+    unfolding simplicial_chain_def
+  proof (induction rule: frag_induction)
+    case (one x)
+    then have "simplex_map p (simplex_map p g f) x = simplex_map p g (simplex_map p f x)"
+      by (force simp: simplex_map_def restrict_compose_left simplicial_simplex)
+    then show ?case
+      by auto
+  qed (auto simp: chain_map_diff)
+  then show ?case
+    using simplicial_chain_simplicial_subdivision [of p "standard_simplex p" "frag_of (restrict id (standard_simplex p))"]
+    by (simp add: singular_subdivision_def)
+next
+  case (diff a b)
+  then show ?case
+    by (simp add: chain_map_diff singular_subdivision_diff)
+qed auto
+
+lemma simplicial_chain_chain_map:
+  assumes f: "simplicial_simplex q X f" and c: "simplicial_chain p (standard_simplex q) c"
+  shows "simplicial_chain p X (chain_map p f c)"
+  using c unfolding simplicial_chain_def
+proof (induction c rule: frag_induction)
+  case (one g)
+  have "\<exists>n. simplex_map p (oriented_simplex q l)
+                 (oriented_simplex p m) = oriented_simplex p n"
+    if m: "singular_simplex p
+                (subtopology (powertop_real UNIV) (standard_simplex q)) (oriented_simplex p m)"
+    for l m
+  proof -
+    have "(\<lambda>i. \<Sum>j\<le>p. m j i * x j) \<in> standard_simplex q"
+      if "x \<in> standard_simplex p" for x
+      using that m unfolding oriented_simplex_def singular_simplex_def
+      by (auto simp: continuous_map_in_subtopology image_subset_iff)
+    then show ?thesis
+      unfolding oriented_simplex_def simplex_map_def
+      apply (rule_tac x="\<lambda>j k. (\<Sum>i\<le>q. l i k * m j i)" in exI)
+      apply (force simp: sum_distrib_left sum_distrib_right mult.assoc intro: sum.swap)
+      done
+  qed
+  then show ?case
+    using f one
+    apply (auto simp: simplicial_simplex_def)
+    apply (rule singular_simplex_simplex_map
+        [where X = "subtopology (powertop_real UNIV) (standard_simplex q)"])
+    unfolding singular_simplex_def apply (fastforce simp add:)+
+    done
+next
+  case (diff a b)
+  then show ?case
+    by (metis chain_map_diff simplicial_chain_def simplicial_chain_diff)
+qed auto
+
+
+lemma singular_subdivision_simplicial_simplex:
+   "simplicial_chain p S c
+           \<Longrightarrow> singular_subdivision p c = simplicial_subdivision p c"
+proof (induction p arbitrary: S c)
+  case 0
+  then show ?case
+    unfolding simplicial_chain_def
+  proof (induction rule: frag_induction)
+    case (one x)
+    then show ?case
+      using singular_simplex_chain_map_id simplicial_imp_singular_simplex
+      by (fastforce simp: singular_subdivision_def simplicial_subdivision_def)
+  qed (auto simp: singular_subdivision_diff)
+next
+  case (Suc p)
+  show ?case
+    using Suc.prems unfolding simplicial_chain_def
+  proof (induction rule: frag_induction)
+    case (one f)
+    then have ssf: "simplicial_simplex (Suc p) S f"
+      by (auto simp: simplicial_simplex)
+    then have 1: "simplicial_chain p (standard_simplex (Suc p))
+                   (simplicial_subdivision p
+                     (chain_boundary (Suc p)
+                       (frag_of (restrict id (standard_simplex (Suc p))))))"
+      by (metis diff_Suc_1 order_refl simplicial_chain_boundary simplicial_chain_of simplicial_chain_simplicial_subdivision simplicial_simplex_id)
+    have 2: "(\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j (restrict id (standard_simplex (Suc p))) i) / (real p + 2))
+                  \<in> standard_simplex (Suc p)"
+      by (simp add: simplicial_vertex_def standard_simplex_def del: sum_atMost_Suc)
+    have ss_Sp: "(\<lambda>i. (if i \<le> Suc p then 1 else 0) / (real p + 2)) \<in> standard_simplex (Suc p)"
+      by (simp add: standard_simplex_def divide_simps)
+    obtain l where feq: "f = oriented_simplex (Suc p) l"
+      using one unfolding simplicial_simplex by blast
+    then have 3: "f (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j (restrict id (standard_simplex (Suc p))) i) / (real p + 2))
+                = (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (real p + 2))"
+      unfolding simplicial_vertex_def oriented_simplex_def
+      by (simp add: ss_Sp if_distrib [of "\<lambda>x. _ * x"] sum_divide_distrib del: sum_atMost_Suc cong: if_cong)
+    have scp: "singular_chain (Suc p)
+                 (subtopology (powertop_real UNIV) (standard_simplex (Suc p)))
+                 (frag_of (restrict id (standard_simplex (Suc p))))"
+      by (simp add: simplicial_imp_singular_chain)
+    have scps: "simplicial_chain p (standard_simplex (Suc p))
+                  (chain_boundary (Suc p) (frag_of (restrict id (standard_simplex (Suc p)))))"
+      by (metis diff_Suc_1 order_refl simplicial_chain_boundary simplicial_chain_of simplicial_simplex_id)
+    have scpf: "simplicial_chain p S
+                 (chain_map p f
+                   (chain_boundary (Suc p) (frag_of (restrict id (standard_simplex (Suc p))))))"
+      using scps simplicial_chain_chain_map ssf by blast
+    have 4: "chain_map p f
+                (simplicial_subdivision p
+                   (chain_boundary (Suc p) (frag_of (restrict id (standard_simplex (Suc p))))))
+           = simplicial_subdivision p (chain_boundary (Suc p) (frag_of f))"
+      apply (simp add: chain_boundary_chain_map [OF scp] del: chain_map_of
+          flip: singular_simplex_chain_map_id [OF simplicial_imp_singular_simplex [OF ssf]])
+      by (metis (no_types) scp singular_chain_boundary_alt Suc.IH [OF scps] Suc.IH [OF scpf] naturality_singular_subdivision)
+    show ?case
+      apply (simp add: singular_subdivision_def del: sum_atMost_Suc)
+      apply (simp only: ssf 1 2 3 4 chain_map_simplicial_cone [of "Suc p" S _ p "Suc p"])
+      done
+  qed (auto simp: frag_extend_diff singular_subdivision_diff)
+qed
+
+
+lemma naturality_simplicial_subdivision:
+   "\<lbrakk>simplicial_chain p (standard_simplex q) c; simplicial_simplex q S g\<rbrakk>
+    \<Longrightarrow> simplicial_subdivision p (chain_map p g c) = chain_map p g (simplicial_subdivision p c)"
+apply (simp flip: singular_subdivision_simplicial_simplex)
+  by (metis naturality_singular_subdivision simplicial_chain_chain_map simplicial_imp_singular_chain singular_subdivision_simplicial_simplex)
+
+lemma chain_boundary_singular_subdivision:
+   "singular_chain p X c
+        \<Longrightarrow> chain_boundary p (singular_subdivision p c) =
+            singular_subdivision (p - Suc 0) (chain_boundary p c)"
+  unfolding singular_chain_def
+proof (induction rule: frag_induction)
+  case (one f)
+    then have ssf: "singular_simplex p X f"
+      by (auto simp: singular_simplex_def)
+    then have scp: "simplicial_chain p (standard_simplex p) (frag_of (restrict id (standard_simplex p)))"
+      by simp
+    have scp1: "simplicial_chain (p - Suc 0) (standard_simplex p)
+                  (chain_boundary p (frag_of (restrict id (standard_simplex p))))"
+      using simplicial_chain_boundary by force
+    have sgp1: "singular_chain (p - Suc 0)
+                   (subtopology (powertop_real UNIV) (standard_simplex p))
+                   (chain_boundary p (frag_of (restrict id (standard_simplex p))))"
+      using scp1 simplicial_imp_singular_chain by blast
+    have scpp: "singular_chain p (subtopology (powertop_real UNIV) (standard_simplex p))
+                  (frag_of (restrict id (standard_simplex p)))"
+      using scp simplicial_imp_singular_chain by blast
+    then show ?case
+      unfolding singular_subdivision_def
+      using chain_boundary_chain_map [of p "subtopology (powertop_real UNIV)
+                              (standard_simplex p)" _ f]
+      apply (simp add: simplicial_chain_simplicial_subdivision
+          simplicial_imp_singular_chain chain_boundary_simplicial_subdivision [OF scp]
+          flip: singular_subdivision_simplicial_simplex [OF scp1] naturality_singular_subdivision [OF sgp1])
+      by (metis (full_types)   singular_subdivision_def  chain_boundary_chain_map [OF scpp] singular_simplex_chain_map_id [OF ssf])
+qed (auto simp: singular_subdivision_def frag_extend_diff chain_boundary_diff)
+
+lemma singular_subdivision_zero:
+  "singular_chain 0 X c \<Longrightarrow> singular_subdivision 0 c = c"
+  unfolding singular_chain_def
+proof (induction rule: frag_induction)
+  case (one f)
+  then have "restrict (f \<circ> restrict id (standard_simplex 0)) (standard_simplex 0) = f"
+    by (simp add: extensional_restrict restrict_compose_right singular_simplex_def)
+  then show ?case
+    by (auto simp: singular_subdivision_def simplex_map_def)
+qed (auto simp: singular_subdivision_def frag_extend_diff)
+
+
+primrec subd where
+  "subd 0 = (\<lambda>x. 0)"
+| "subd (Suc p) =
+      frag_extend
+       (\<lambda>f. simplicial_cone (Suc p) (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / real (Suc p + 1))
+               (simplicial_subdivision (Suc p) (frag_of f) - frag_of f -
+                subd p (chain_boundary (Suc p) (frag_of f))))"
+
+lemma subd_0 [simp]: "subd p 0 = 0"
+  by (induction p) auto
+
+lemma subd_diff [simp]: "subd p (c1 - c2) = subd p c1 - subd p c2"
+  by (induction p) (auto simp: frag_extend_diff)
+
+lemma subd_uminus [simp]: "subd p (-c) = - subd p c"
+  by (metis diff_0 subd_0 subd_diff)
+
+lemma subd_power_uminus: "subd p (frag_cmul ((-1) ^ k) c) = frag_cmul ((-1) ^ k) (subd p c)"
+  apply (induction k, simp_all)
+  by (metis minus_frag_cmul subd_uminus)
+
+lemma subd_power_sum: "subd p (sum f I) = sum (subd p \<circ> f) I"
+  apply (induction I rule: infinite_finite_induct)
+  by auto (metis add_diff_cancel_left' diff_add_cancel subd_diff)
+
+lemma subd: "simplicial_chain p (standard_simplex s) c
+     \<Longrightarrow> (\<forall>r g. simplicial_simplex s (standard_simplex r) g \<longrightarrow> chain_map (Suc p) g (subd p c) = subd p (chain_map p g c))
+         \<and> simplicial_chain (Suc p) (standard_simplex s) (subd p c)
+         \<and> (chain_boundary (Suc p) (subd p c)) + (subd (p - Suc 0) (chain_boundary p c)) = (simplicial_subdivision p c) - c"
+proof (induction p arbitrary: c)
+  case (Suc p)
+  show ?case
+    using Suc.prems [unfolded simplicial_chain_def]
+  proof (induction rule: frag_induction)
+    case (one f)
+    then obtain l where l: "(\<lambda>x i. \<Sum>j\<le>Suc p. l j i * x j) ` standard_simplex (Suc p) \<subseteq> standard_simplex s"
+                  and feq: "f = oriented_simplex (Suc p) l"
+      by (metis (mono_tags) mem_Collect_eq simplicial_simplex simplicial_simplex_oriented_simplex)
+    have scf: "simplicial_chain (Suc p) (standard_simplex s) (frag_of f)"
+      using one by simp
+    have lss: "l i \<in> standard_simplex s" if "i \<le> Suc p" for i
+    proof -
+      have "(\<lambda>i'. \<Sum>j\<le>Suc p. l j i' * (if j = i then 1 else 0)) \<in> standard_simplex s"
+        using subsetD [OF l] basis_in_standard_simplex that by blast
+      moreover have "(\<lambda>i'. \<Sum>j\<le>Suc p. l j i' * (if j = i then 1 else 0)) = l i"
+        using that by (simp add: if_distrib [of "\<lambda>x. _ * x"] del: sum_atMost_Suc cong: if_cong)
+      ultimately show ?thesis
+        by simp
+    qed
+    have *: "(\<And>i. i \<le> n \<Longrightarrow> l i \<in> standard_simplex s)
+     \<Longrightarrow> (\<lambda>i. (\<Sum>j\<le>n. l j i) / (Suc n)) \<in> standard_simplex s" for n
+    proof (induction n)
+      case (Suc n)
+      let ?x = "\<lambda>i. (1 - inverse (n + 2)) * ((\<Sum>j\<le>n. l j i) / (Suc n)) + inverse (n + 2) * l (Suc n) i"
+      have "?x \<in> standard_simplex s"
+      proof (rule convex_standard_simplex)
+        show "(\<lambda>i. (\<Sum>j\<le>n. l j i) / real (Suc n)) \<in> standard_simplex s"
+          using Suc by simp
+      qed (auto simp: lss Suc inverse_le_1_iff)
+      moreover have "?x = (\<lambda>i. (\<Sum>j\<le>Suc n. l j i) / real (Suc (Suc n)))"
+        by (force simp: divide_simps)
+      ultimately show ?case
+        by simp
+    qed auto
+    have **: "(\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (2 + real p)) \<in> standard_simplex s"
+      using * [of "Suc p"] lss by (simp add: simplicial_vertex_oriented_simplex feq)
+    show ?case
+    proof (intro conjI impI allI)
+      fix r g
+      assume g: "simplicial_simplex s (standard_simplex r) g"
+      then obtain m where geq: "g = oriented_simplex s m"
+        using simplicial_simplex by blast
+      have 1: "simplicial_chain (Suc p) (standard_simplex s) (simplicial_subdivision (Suc p) (frag_of f))"
+        by (metis mem_Collect_eq one.hyps simplicial_chain_of simplicial_chain_simplicial_subdivision)
+      have 2: "(\<Sum>j\<le>Suc p. \<Sum>i\<le>s. m i k * simplicial_vertex j f i)
+             = (\<Sum>j\<le>Suc p. simplicial_vertex j
+                                (simplex_map (Suc p) (oriented_simplex s m) f) k)" for k
+      proof (rule sum.cong [OF refl])
+        fix j
+        assume j: "j \<in> {..Suc p}"
+        have eq: "simplex_map (Suc p) (oriented_simplex s m) (oriented_simplex (Suc p) l)
+                = oriented_simplex (Suc p) (oriented_simplex s m \<circ> l)"
+        proof (rule simplex_map_oriented_simplex)
+          show "simplicial_simplex (Suc p) (standard_simplex s) (oriented_simplex (Suc p) l)"
+            using one by (simp add: feq flip: oriented_simplex_def)
+          show "simplicial_simplex s (standard_simplex r) (oriented_simplex s m)"
+            using g by (simp add: geq)
+        qed auto
+        show "(\<Sum>i\<le>s. m i k * simplicial_vertex j f i)
+            = simplicial_vertex j (simplex_map (Suc p) (oriented_simplex s m) f) k"
+          using one j
+          apply (simp add: feq eq simplicial_vertex_oriented_simplex simplicial_simplex_oriented_simplex image_subset_iff)
+          apply (drule_tac x="(\<lambda>i. if i = j then 1 else 0)" in bspec)
+           apply (auto simp: oriented_simplex_def lss)
+          done
+      qed
+      have 4: "chain_map (Suc p) g (subd p (chain_boundary (Suc p) (frag_of f)))
+             = subd p (chain_boundary (Suc p) (frag_of (simplex_map (Suc p) g f)))"
+        by (metis (no_types) One_nat_def scf Suc.IH chain_boundary_chain_map chain_map_of diff_Suc_Suc diff_zero g simplicial_chain_boundary simplicial_imp_singular_chain)
+      show "chain_map (Suc (Suc p)) g (subd (Suc p) (frag_of f)) = subd (Suc p) (chain_map (Suc p) g (frag_of f))"
+        using g
+        apply (simp only: subd.simps frag_extend_of)
+        apply (subst chain_map_simplicial_cone [of s "standard_simplex r" _ "Suc p" s], assumption)
+           apply (intro simplicial_chain_diff)
+        using "1" apply auto[1]
+        using one.hyps apply auto[1]
+        apply (metis Suc.IH diff_Suc_1 mem_Collect_eq one.hyps simplicial_chain_boundary simplicial_chain_of)
+        using "**" apply auto[1]
+         apply (rule order_refl)
+         apply (simp only: chain_map_of frag_extend_of)
+        apply (rule arg_cong2 [where f = "simplicial_cone (Suc p)"])
+         apply (simp add: geq sum_distrib_left oriented_simplex_def ** del: sum_atMost_Suc flip: sum_divide_distrib)
+        using 2  apply (simp only: oriented_simplex_def sum.swap [where A = "{..s}"])
+        using naturality_simplicial_subdivision scf apply (fastforce simp add: 4 chain_map_diff)
+        done
+    next
+      have sc: "simplicial_chain (Suc p) (standard_simplex s)
+               (simplicial_cone p
+                 (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (Suc (Suc p)))
+                 (simplicial_subdivision p
+                   (chain_boundary (Suc p) (frag_of f))))"
+          by (metis diff_Suc_1 nat.simps(3) simplicial_subdivision_of scf simplicial_chain_simplicial_subdivision)
+      have ff: "simplicial_chain (Suc p) (standard_simplex s) (subd p (chain_boundary (Suc p) (frag_of f)))"
+        by (metis (no_types) Suc.IH diff_Suc_1 scf simplicial_chain_boundary)
+      show "simplicial_chain (Suc (Suc p)) (standard_simplex s) (subd (Suc p) (frag_of f))"
+        using one
+        apply (simp only: subd.simps frag_extend_of)
+        apply (rule_tac S="standard_simplex s" in simplicial_chain_simplicial_cone)
+         apply (intro simplicial_chain_diff ff)
+        using sc apply (simp add: algebra_simps)
+        using "**" convex_standard_simplex  apply force+
+        done
+      have "simplicial_chain p (standard_simplex s) (chain_boundary (Suc p) (frag_of f))"
+        using scf simplicial_chain_boundary by fastforce
+      then have "chain_boundary (Suc p) (simplicial_subdivision (Suc p) (frag_of f) - frag_of f
+                                         - subd p (chain_boundary (Suc p) (frag_of f))) = 0"
+        apply (simp only: chain_boundary_diff)
+        using Suc.IH chain_boundary_boundary [of "Suc p" "subtopology (powertop_real UNIV)
+                                (standard_simplex s)" "frag_of f"]
+        by (metis One_nat_def add_diff_cancel_left' subd_0 chain_boundary_simplicial_subdivision plus_1_eq_Suc scf simplicial_imp_singular_chain)
+      then show "chain_boundary (Suc (Suc p)) (subd (Suc p) (frag_of f))
+          + subd (Suc p - Suc 0) (chain_boundary (Suc p) (frag_of f))
+          = simplicial_subdivision (Suc p) (frag_of f) - frag_of f"
+        apply (simp only: subd.simps frag_extend_of)
+        apply (subst chain_boundary_simplicial_cone [of "Suc p" "standard_simplex s"])
+         apply (meson ff scf simplicial_chain_diff simplicial_chain_simplicial_subdivision)
+        apply (simp add: simplicial_cone_def del: sum_atMost_Suc simplicial_subdivision.simps)
+        done
+    qed
+  next
+    case (diff a b)
+    then show ?case
+      apply safe
+        apply (metis chain_map_diff subd_diff)
+       apply (metis simplicial_chain_diff subd_diff)
+      apply (auto simp:  simplicial_subdivision_diff chain_boundary_diff
+          simp del: simplicial_subdivision.simps subd.simps)
+      by (metis (no_types, lifting) add_diff_add add_uminus_conv_diff diff_0 diff_diff_add)
+  qed auto
+qed simp
+
+lemma chain_homotopic_simplicial_subdivision1:
+  "\<lbrakk>simplicial_chain p (standard_simplex q) c; simplicial_simplex q (standard_simplex r) g\<rbrakk>
+       \<Longrightarrow> chain_map (Suc p) g (subd p c) = subd p (chain_map p g c)"
+  by (simp add: subd)
+
+lemma chain_homotopic_simplicial_subdivision2:
+  "simplicial_chain p (standard_simplex q) c
+       \<Longrightarrow> simplicial_chain (Suc p) (standard_simplex q) (subd p c)"
+  by (simp add: subd)
+
+lemma chain_homotopic_simplicial_subdivision3:
+  "simplicial_chain p (standard_simplex q) c
+   \<Longrightarrow> chain_boundary (Suc p) (subd p c) = (simplicial_subdivision p c) - c - subd (p - Suc 0) (chain_boundary p c)"
+  by (simp add: subd algebra_simps)
+
+lemma chain_homotopic_simplicial_subdivision:
+  "\<exists>h. (\<forall>p. h p 0 = 0) \<and>
+       (\<forall>p c1 c2. h p (c1-c2) = h p c1 - h p c2) \<and>
+       (\<forall>p q r g c.
+                simplicial_chain p (standard_simplex q) c
+                \<longrightarrow> simplicial_simplex q (standard_simplex r) g
+                \<longrightarrow> chain_map (Suc p) g (h p c) = h p (chain_map p g c)) \<and>
+       (\<forall>p q c. simplicial_chain p (standard_simplex q) c
+                \<longrightarrow> simplicial_chain (Suc p) (standard_simplex q) (h p c)) \<and>
+       (\<forall>p q c. simplicial_chain p (standard_simplex q) c
+                \<longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c)
+                  = (simplicial_subdivision p c) - c)"
+  by (rule_tac x=subd in exI) (fastforce simp: subd)
+
+lemma chain_homotopic_singular_subdivision:
+  obtains h where
+        "\<And>p. h p 0 = 0"
+        "\<And>p c1 c2. h p (c1-c2) = h p c1 - h p c2"
+        "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"
+        "\<And>p X c. singular_chain p X c
+                 \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
+proof -
+  define k where "k \<equiv> \<lambda>p. frag_extend (\<lambda>f:: (nat \<Rightarrow> real) \<Rightarrow> 'a. chain_map (Suc p) f (subd p (frag_of(restrict id (standard_simplex p)))))"
+  show ?thesis
+  proof
+    fix p X and c :: "'a chain"
+    assume c: "singular_chain p X c"
+    have "singular_chain (Suc p) X (k p c) \<and>
+               chain_boundary (Suc p) (k p c) + k (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
+      using c [unfolded singular_chain_def]
+    proof (induction rule: frag_induction)
+      case (one f)
+      let ?X = "subtopology (powertop_real UNIV) (standard_simplex p)"
+      show ?case
+      proof (simp add: k_def, intro conjI)
+        show "singular_chain (Suc p) X (chain_map (Suc p) f (subd p (frag_of (restrict id (standard_simplex p)))))"
+        proof (rule singular_chain_chain_map)
+          show "singular_chain (Suc p) ?X  (subd p (frag_of (restrict id (standard_simplex p))))"
+            by (simp add: chain_homotopic_simplicial_subdivision2 simplicial_imp_singular_chain)
+          show "continuous_map ?X X f"
+            using one.hyps singular_simplex_def by auto
+        qed
+      next
+        have scp: "singular_chain (Suc p) ?X (subd p (frag_of (restrict id (standard_simplex p))))"
+          by (simp add: chain_homotopic_simplicial_subdivision2 simplicial_imp_singular_chain)
+        have feqf: "frag_of (simplex_map p f (restrict id (standard_simplex p))) = frag_of f"
+          using one.hyps singular_simplex_chain_map_id by auto
+        have *: "chain_map p f
+                   (subd (p - Suc 0)
+                     (\<Sum>k\<le>p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k id))))
+              = (\<Sum>x\<le>p. frag_cmul ((-1) ^ x)
+                         (chain_map p (singular_face p x f)
+                           (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0)))))))"
+                  (is "?lhs = ?rhs")
+                  if "p > 0"
+        proof -
+          have eqc: "subd (p - Suc 0) (frag_of (singular_face p i id))
+                   = chain_map p (singular_face p i id)
+                                 (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0)))))"
+            if "i \<le> p" for i
+          proof -
+            have 1: "simplicial_chain (p - Suc 0) (standard_simplex (p - Suc 0))
+                       (frag_of (restrict id (standard_simplex (p - Suc 0))))"
+              by simp
+            have 2: "simplicial_simplex (p - Suc 0) (standard_simplex p) (singular_face p i id)"
+              by (metis One_nat_def Suc_leI \<open>0 < p\<close> simplicial_simplex_id simplicial_simplex_singular_face singular_face_restrict subsetI that)
+            have 3: "simplex_map (p - Suc 0) (singular_face p i id) (restrict id (standard_simplex (p - Suc 0)))
+                   = singular_face p i id"
+              by (force simp: simplex_map_def singular_face_def)
+            show ?thesis
+              using chain_homotopic_simplicial_subdivision1 [OF 1 2]
+                that \<open>p > 0\<close>  by (simp add: 3)
+          qed
+          have xx: "simplicial_chain p (standard_simplex(p - Suc 0))
+                    (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0)))))"
+            by (metis Suc_pred chain_homotopic_simplicial_subdivision2 order_refl simplicial_chain_of simplicial_simplex_id that)
+          have yy: "\<And>k. k \<le> p \<Longrightarrow>
+                 chain_map p f
+                  (chain_map p (singular_face p k id) h) = chain_map p (singular_face p k f) h"
+            if "simplicial_chain p (standard_simplex(p - Suc 0)) h" for h
+            using that unfolding simplicial_chain_def
+          proof (induction h rule: frag_induction)
+            case (one x)
+            then show ?case
+                using one
+              apply (simp add: chain_map_of singular_simplex_def simplicial_simplex_def, auto)
+                apply (rule_tac f=frag_of in arg_cong, rule)
+                apply (simp add: simplex_map_def)
+                by (simp add: continuous_map_in_subtopology image_subset_iff singular_face_def)
+          qed (auto simp: chain_map_diff)
+          have "?lhs
+                = chain_map p f
+                      (\<Sum>k\<le>p. frag_cmul ((-1) ^ k)
+                          (chain_map p (singular_face p k id)
+                           (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0)))))))"
+            by (simp add: subd_power_sum subd_power_uminus eqc)
+          also have "\<dots> = ?rhs"
+            by (simp add: chain_map_sum xx yy)
+          finally show ?thesis .
+      qed
+        have "chain_map p f
+                   (simplicial_subdivision p (frag_of (restrict id (standard_simplex p)))
+                   - subd (p - Suc 0) (chain_boundary p (frag_of (restrict id (standard_simplex p)))))
+              = singular_subdivision p (frag_of f)
+              - frag_extend
+                   (\<lambda>f. chain_map (Suc (p - Suc 0)) f
+                         (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0))))))
+                   (chain_boundary p (frag_of f))"
+          apply (simp add: singular_subdivision_def chain_map_diff)
+          apply (clarsimp simp add: chain_boundary_def)
+          apply (simp add: frag_extend_sum frag_extend_cmul *)
+          done
+        then show "chain_boundary (Suc p) (chain_map (Suc p) f (subd p (frag_of (restrict id (standard_simplex p)))))
+                 + frag_extend
+                   (\<lambda>f. chain_map (Suc (p - Suc 0)) f
+                         (subd (p - Suc 0) (frag_of (restrict id (standard_simplex (p - Suc 0))))))
+                   (chain_boundary p (frag_of f))
+                 = singular_subdivision p (frag_of f) - frag_of f"
+          by (simp add: chain_boundary_chain_map [OF scp] chain_homotopic_simplicial_subdivision3 [where q=p] chain_map_diff feqf)
+      qed
+    next
+      case (diff a b)
+      then show ?case
+        apply (simp only: k_def singular_chain_diff chain_boundary_diff frag_extend_diff singular_subdivision_diff)
+        by (metis (no_types, lifting) add_diff_add diff_add_cancel)
+    qed (auto simp: k_def)
+    then show "singular_chain (Suc p) X (k p c)" "chain_boundary (Suc p) (k p c) + k (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
+        by auto
+  qed (auto simp: k_def frag_extend_diff)
+qed
+
+
+lemma homologous_rel_singular_subdivision:
+  assumes "singular_relcycle p X T c"
+  shows "homologous_rel p X T (singular_subdivision p c) c"
+proof (cases "p = 0")
+  case True
+  with assms show ?thesis
+    by (auto simp: singular_relcycle_def singular_subdivision_zero)
+next
+  case False
+  with assms show ?thesis
+    unfolding homologous_rel_def singular_relboundary singular_relcycle
+    by (metis One_nat_def Suc_diff_1 chain_homotopic_singular_subdivision gr_zeroI)
+qed
+
+
+subsection\<open>Excision argument that we keep doing singular subdivision\<close>
+
+lemma singular_subdivision_power_0 [simp]: "(singular_subdivision p ^^ n) 0 = 0"
+  by (induction n) auto
+
+lemma singular_subdivision_power_diff:
+  "(singular_subdivision p ^^ n) (a - b) = (singular_subdivision p ^^ n) a - (singular_subdivision p ^^ n) b"
+  by (induction n) (auto simp: singular_subdivision_diff)
+
+lemma iterated_singular_subdivision:
+   "singular_chain p X c
+        \<Longrightarrow> (singular_subdivision p ^^ n) c =
+            frag_extend
+             (\<lambda>f. chain_map p f
+                       ((simplicial_subdivision p ^^ n)
+                         (frag_of(restrict id (standard_simplex p))))) c"
+proof (induction n arbitrary: c)
+  case 0
+  then show ?case
+    unfolding singular_chain_def
+  proof (induction c rule: frag_induction)
+    case (one f)
+    then have "restrict f (standard_simplex p) = f"
+      by (simp add: extensional_restrict singular_simplex_def)
+    then show ?case
+      by (auto simp: simplex_map_def cong: restrict_cong)
+  qed (auto simp: frag_extend_diff)
+next
+  case (Suc n)
+  show ?case
+    using Suc.prems unfolding singular_chain_def
+  proof (induction c rule: frag_induction)
+    case (one f)
+    then have "singular_simplex p X f"
+      by simp
+    have scp: "simplicial_chain p (standard_simplex p)
+                 ((simplicial_subdivision p ^^ n) (frag_of (restrict id (standard_simplex p))))"
+    proof (induction n)
+      case 0
+      then show ?case
+        by (metis funpow_0 order_refl simplicial_chain_of simplicial_simplex_id)
+    next
+      case (Suc n)
+      then show ?case
+        by (simp add: simplicial_chain_simplicial_subdivision)
+    qed
+    have scnp: "simplicial_chain p (standard_simplex p)
+                  ((simplicial_subdivision p ^^ n) (frag_of (\<lambda>x\<in>standard_simplex p. x)))"
+    proof (induction n)
+      case 0
+      then show ?case
+        by (metis eq_id_iff funpow_0 order_refl simplicial_chain_of simplicial_simplex_id)
+    next
+      case (Suc n)
+      then show ?case
+        by (simp add: simplicial_chain_simplicial_subdivision)
+    qed
+    have sff: "singular_chain p X (frag_of f)"
+      by (simp add: \<open>singular_simplex p X f\<close> singular_chain_of)
+    then show ?case
+      using Suc.IH [OF sff] naturality_singular_subdivision [OF simplicial_imp_singular_chain [OF scp], of f] singular_subdivision_simplicial_simplex [OF scnp]
+      by (simp add: singular_chain_of id_def del: restrict_apply)
+  qed (auto simp: singular_subdivision_power_diff singular_subdivision_diff frag_extend_diff)
+qed
+
+
+lemma chain_homotopic_iterated_singular_subdivision:
+  obtains h where
+        "\<And>p. h p 0 = (0 :: 'a chain)"
+        "\<And>p c1 c2. h p (c1-c2) = h p c1 - h p c2"
+        "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"
+        "\<And>p X c. singular_chain p X c
+                 \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c)
+                   = (singular_subdivision p ^^ n) c - c"
+proof (induction n arbitrary: thesis)
+  case 0
+  show ?case
+    by (rule 0 [of "(\<lambda>p x. 0)"]) auto
+next
+  case (Suc n)
+  then obtain k where k:
+        "\<And>p. k p 0 = (0 :: 'a chain)"
+        "\<And>p c1 c2. k p (c1-c2) = k p c1 - k p c2"
+        "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (k p c)"
+        "\<And>p X c. singular_chain p X c
+                 \<Longrightarrow> chain_boundary (Suc p) (k p c) + k (p - Suc 0) (chain_boundary p c)
+                     = (singular_subdivision p ^^ n) c - c"
+    by metis
+  obtain h where h:
+        "\<And>p. h p 0 = (0 :: 'a chain)"
+        "\<And>p c1 c2. h p (c1-c2) = h p c1 - h p c2"
+        "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"
+        "\<And>p X c. singular_chain p X c
+                 \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
+    by (blast intro: chain_homotopic_singular_subdivision)
+  let ?h = "(\<lambda>p c. singular_subdivision (Suc p) (k p c) + h p c)"
+  show ?case
+  proof (rule Suc.prems)
+    fix p X and c :: "'a chain"
+    assume "singular_chain p X c"
+    then show "singular_chain (Suc p) X (?h p c)"
+      by (simp add: h k singular_chain_add singular_chain_singular_subdivision)
+  next
+    fix p :: "nat" and X :: "'a topology" and c :: "'a chain"
+    assume sc: "singular_chain p X c"
+    have f5: "chain_boundary (Suc p) (singular_subdivision (Suc p) (k p c)) = singular_subdivision p (chain_boundary (Suc p) (k p c))"
+      using chain_boundary_singular_subdivision k(3) sc by fastforce
+    have [simp]: "singular_subdivision (Suc (p - Suc 0)) (k (p - Suc 0) (chain_boundary p c)) =
+                  singular_subdivision p (k (p - Suc 0) (chain_boundary p c))"
+    proof (cases p)
+      case 0
+      then show ?thesis
+        by (simp add: k chain_boundary_def)
+    qed auto
+    show "chain_boundary (Suc p) (?h p c) + ?h (p - Suc 0) (chain_boundary p c) = (singular_subdivision p ^^ Suc n) c - c"
+      using chain_boundary_singular_subdivision [of "Suc p" X]
+      apply (simp add: chain_boundary_add f5 h k algebra_simps)
+      apply (erule thin_rl)
+      using h(4) [OF sc] k(4) [OF sc] singular_subdivision_add [of p "chain_boundary (Suc p) (k p c)" "k (p - Suc 0) (chain_boundary p c)"]
+      apply (simp add: algebra_simps)
+      by (smt add.assoc add.left_commute singular_subdivision_add)
+  qed (auto simp: k h singular_subdivision_diff)
+qed
+
+lemma llemma:
+  assumes p: "standard_simplex p \<subseteq> \<Union>\<C>"
+      and \<C>: "\<And>U. U \<in> \<C> \<Longrightarrow> openin (powertop_real UNIV) U"
+  obtains d where "0 < d"
+                  "\<And>K. \<lbrakk>K \<subseteq> standard_simplex p;
+                        \<And>x y i. \<lbrakk>i \<le> p; x \<in> K; y \<in> K\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> d\<rbrakk>
+                       \<Longrightarrow> \<exists>U. U \<in> \<C> \<and> K \<subseteq> U"
+proof -
+  have "\<exists>e U. 0 < e \<and> U \<in> \<C> \<and> x \<in> U \<and>
+                (\<forall>y. (\<forall>i\<le>p. \<bar>y i - x i\<bar> \<le> 2 * e) \<and> (\<forall>i>p. y i = 0) \<longrightarrow> y \<in> U)"
+    if x: "x \<in> standard_simplex p" for x
+  proof-
+    obtain U where U: "U \<in> \<C>" "x \<in> U"
+      using x p by blast
+    then obtain V where finV: "finite {i. V i \<noteq> UNIV}" and openV: "\<And>i. open (V i)"
+                  and xV: "x \<in> Pi\<^sub>E UNIV V" and UV: "Pi\<^sub>E UNIV V \<subseteq> U"
+      using \<C> unfolding openin_product_topology_alt by force
+    have xVi: "x i \<in> V i" for i
+      using PiE_mem [OF xV] by simp
+    have "\<And>i. \<exists>e>0. \<forall>x'. \<bar>x' - x i\<bar> < e \<longrightarrow> x' \<in> V i"
+      by (rule openV [unfolded open_real, rule_format, OF xVi])
+    then obtain d where d: "\<And>i. d i > 0" and dV: "\<And>i x'. \<bar>x' - x i\<bar> < d i \<Longrightarrow> x' \<in> V i"
+      by metis
+    define e where "e \<equiv> Inf (insert 1 (d ` {i. V i \<noteq> UNIV})) / 3"
+    have ed3: "e \<le> d i / 3" if "V i \<noteq> UNIV" for i
+      using that finV by (auto simp: e_def intro: cInf_le_finite)
+    show "\<exists>e U. 0 < e \<and> U \<in> \<C> \<and> x \<in> U \<and>
+                (\<forall>y. (\<forall>i\<le>p. \<bar>y i - x i\<bar> \<le> 2 * e) \<and> (\<forall>i>p. y i = 0) \<longrightarrow> y \<in> U)"
+    proof (intro exI conjI allI impI)
+      show "e > 0"
+        using d finV by (simp add: e_def finite_less_Inf_iff)
+      fix y assume y: "(\<forall>i\<le>p. \<bar>y i - x i\<bar> \<le> 2 * e) \<and> (\<forall>i>p. y i = 0)"
+      have "y \<in> Pi\<^sub>E UNIV V"
+      proof
+        show "y i \<in> V i" for i
+        proof (cases "p < i")
+          case True
+          then show ?thesis
+            by (metis (mono_tags, lifting) y x mem_Collect_eq standard_simplex_def xVi)
+        next
+          case False show ?thesis
+          proof (cases "V i = UNIV")
+            case False show ?thesis
+            proof (rule dV)
+              have "\<bar>y i - x i\<bar> \<le> 2 * e"
+                using y \<open>\<not> p < i\<close> by simp
+              also have "\<dots> < d i"
+                using ed3 [OF False] \<open>e > 0\<close> by simp
+              finally show "\<bar>y i - x i\<bar> < d i" .
+            qed
+          qed auto
+        qed
+      qed auto
+      with UV show "y \<in> U"
+        by blast
+    qed (use U in auto)
+  qed
+  then obtain e U where
+      eU: "\<And>x. x \<in> standard_simplex p \<Longrightarrow>
+                0 < e x \<and> U x \<in> \<C> \<and> x \<in> U x"
+      and  UI: "\<And>x y. \<lbrakk>x \<in> standard_simplex p;  \<And>i. i \<le> p \<Longrightarrow> \<bar>y i - x i\<bar> \<le> 2 * e x; \<And>i. i > p \<Longrightarrow> y i = 0\<rbrakk>
+                       \<Longrightarrow> y \<in> U x"
+    by metis
+  define F where "F \<equiv> \<lambda>x. Pi\<^sub>E UNIV (\<lambda>i. if i \<le> p then {x i - e x<..<x i + e x} else UNIV)"
+  have "\<forall>S \<in> F ` standard_simplex p. openin (powertop_real UNIV) S"
+    by (simp add: F_def openin_PiE_gen)
+  moreover have pF: "standard_simplex p \<subseteq> \<Union>(F ` standard_simplex p)"
+    by (force simp: F_def PiE_iff eU)
+  ultimately have "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> F ` standard_simplex p \<and> standard_simplex p \<subseteq> \<Union>\<F>"
+    using compactin_standard_simplex [of p]
+    unfolding compactin_def by force
+  then obtain S where "finite S" and ssp: "S \<subseteq> standard_simplex p" "standard_simplex p \<subseteq> \<Union>(F ` S)"
+    unfolding exists_finite_subset_image by (auto simp: exists_finite_subset_image)
+  then have "S \<noteq> {}"
+    by (auto simp: nonempty_standard_simplex)
+  show ?thesis
+  proof
+    show "Inf (e ` S) > 0"
+      using \<open>finite S\<close> \<open>S \<noteq> {}\<close> ssp eU by (auto simp: finite_less_Inf_iff)
+    fix k :: "(nat \<Rightarrow> real) set"
+    assume k: "k \<subseteq> standard_simplex p"
+         and kle: "\<And>x y i. \<lbrakk>i \<le> p; x \<in> k; y \<in> k\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> Inf (e ` S)"
+    show "\<exists>U. U \<in> \<C> \<and> k \<subseteq> U"
+    proof (cases "k = {}")
+      case True
+      then show ?thesis
+        using \<open>S \<noteq> {}\<close> eU equals0I ssp(1) subset_eq p by auto
+    next
+      case False
+      with k ssp obtain x a where "x \<in> k" "x \<in> standard_simplex p"
+                            and a: "a \<in> S" and Fa: "x \<in> F a"
+        by blast
+      then have le_ea: "\<And>i. i \<le> p \<Longrightarrow> abs (x i - a i) < e a"
+        by (simp add: F_def PiE_iff if_distrib abs_diff_less_iff cong: if_cong)
+      show ?thesis
+      proof (intro exI conjI)
+        show "U a \<in> \<C>"
+          using a eU ssp(1) by auto
+        show "k \<subseteq> U a"
+        proof clarify
+          fix y assume "y \<in> k"
+          with k have y: "y \<in> standard_simplex p"
+            by blast
+          show "y \<in> U a"
+          proof (rule UI)
+            show "a \<in> standard_simplex p"
+              using a ssp(1) by auto
+            fix i :: "nat"
+            assume "i \<le> p"
+            then have "\<bar>x i - y i\<bar> \<le> e a"
+              by (meson kle [OF \<open>i \<le> p\<close>] a \<open>finite S\<close> \<open>x \<in> k\<close> \<open>y \<in> k\<close> cInf_le_finite finite_imageI imageI order_trans)
+            then show "\<bar>y i - a i\<bar> \<le> 2 * e a"
+              using le_ea [OF \<open>i \<le> p\<close>] by linarith
+          next
+            fix i assume "p < i"
+            then show "y i = 0"
+              using standard_simplex_def y by auto
+          qed
+        qed
+      qed
+    qed
+  qed
+qed
+
+
+proposition sufficient_iterated_singular_subdivision_exists:
+  assumes \<C>: "\<And>U. U \<in> \<C> \<Longrightarrow> openin X U"
+      and X: "topspace X \<subseteq> \<Union>\<C>"
+      and p: "singular_chain p X c"
+  obtains n where "\<And>m f. \<lbrakk>n \<le> m; f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)\<rbrakk>
+                      \<Longrightarrow> \<exists>V \<in> \<C>. f ` (standard_simplex p) \<subseteq> V"
+proof (cases "c = 0")
+  case False
+  then show ?thesis
+  proof (cases "topspace X = {}")
+    case True
+    show ?thesis
+      using p that by (force simp: singular_chain_empty True)
+  next
+    case False
+    show ?thesis
+    proof (cases "\<C> = {}")
+      case True
+      then show ?thesis
+        using False X by blast
+    next
+      case False
+      have "\<exists>e. 0 < e \<and>
+                (\<forall>K. K \<subseteq> standard_simplex p \<longrightarrow> (\<forall>x y i. x \<in> K \<and> y \<in> K \<and> i \<le> p \<longrightarrow> \<bar>x i - y i\<bar> \<le> e)
+                     \<longrightarrow> (\<exists>V. V \<in> \<C> \<and> f ` K \<subseteq> V))"
+        if f: "f \<in> Poly_Mapping.keys c" for f
+      proof -
+        have ssf: "singular_simplex p X f"
+          using f p by (auto simp: singular_chain_def)
+        then have fp: "\<And>x. x \<in> standard_simplex p \<Longrightarrow> f x \<in> topspace X"
+          by (auto simp: singular_simplex_def image_subset_iff dest: continuous_map_image_subset_topspace)
+        have "\<exists>T. openin (powertop_real UNIV) T \<and>
+                    standard_simplex p \<inter> f -` V = T \<inter> standard_simplex p"
+          if V: "V \<in> \<C>" for V
+        proof -
+          have "singular_simplex p X f"
+            using p f unfolding singular_chain_def by blast
+          then have "openin (subtopology (powertop_real UNIV) (standard_simplex p))
+                            {x \<in> standard_simplex p. f x \<in> V}"
+            using \<C> [OF \<open>V \<in> \<C>\<close>] by (simp add: singular_simplex_def continuous_map_def)
+          moreover have "standard_simplex p \<inter> f -` V = {x \<in> standard_simplex p. f x \<in> V}"
+            by blast
+          ultimately show ?thesis
+            by (simp add: openin_subtopology)
+        qed
+        then obtain g where gope: "\<And>V. V \<in> \<C> \<Longrightarrow> openin (powertop_real UNIV) (g V)"
+                and geq: "\<And>V. V \<in> \<C> \<Longrightarrow> standard_simplex p \<inter> f -` V = g V \<inter> standard_simplex p"
+          by metis
+        obtain d where "0 < d"
+              and d: "\<And>K. \<lbrakk>K \<subseteq> standard_simplex p; \<And>x y i. \<lbrakk>i \<le> p; x \<in> K; y \<in> K\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> d\<rbrakk>
+                       \<Longrightarrow> \<exists>U. U \<in> g ` \<C> \<and> K \<subseteq> U"
+        proof (rule llemma [of p "g ` \<C>"])
+          show "standard_simplex p \<subseteq> \<Union>(g ` \<C>)"
+            using geq X fp by (fastforce simp add:)
+          show "openin (powertop_real UNIV) U" if "U \<in> g ` \<C>" for U :: "(nat \<Rightarrow> real) set"
+            using gope that by blast
+        qed auto
+        show ?thesis
+        proof (rule exI, intro allI conjI impI)
+          fix K :: "(nat \<Rightarrow> real) set"
+          assume K: "K \<subseteq> standard_simplex p"
+             and Kd: "\<forall>x y i. x \<in> K \<and> y \<in> K \<and> i \<le> p \<longrightarrow> \<bar>x i - y i\<bar> \<le> d"
+          then have "\<exists>U. U \<in> g ` \<C> \<and> K \<subseteq> U"
+            using d [OF K] by auto
+          then show "\<exists>V. V \<in> \<C> \<and> f ` K \<subseteq> V"
+            using K geq by fastforce
+        qed (rule \<open>d > 0\<close>)
+      qed
+      then obtain \<psi> where epos: "\<forall>f \<in> Poly_Mapping.keys c. 0 < \<psi> f"
+                 and e: "\<And>f K. \<lbrakk>f \<in> Poly_Mapping.keys c; K \<subseteq> standard_simplex p;
+                                \<And>x y i. x \<in> K \<and> y \<in> K \<and> i \<le> p \<Longrightarrow> \<bar>x i - y i\<bar> \<le> \<psi> f\<rbrakk>
+                               \<Longrightarrow> \<exists>V. V \<in> \<C> \<and> f ` K \<subseteq> V"
+        by metis
+      obtain d where "0 < d"
+               and d: "\<And>f K. \<lbrakk>f \<in> Poly_Mapping.keys c; K \<subseteq> standard_simplex p;
+                              \<And>x y i. \<lbrakk>x \<in> K; y \<in> K; i \<le> p\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> d\<rbrakk>
+                              \<Longrightarrow> \<exists>V. V \<in> \<C> \<and> f ` K \<subseteq> V"
+      proof
+        show "Inf (\<psi> ` Poly_Mapping.keys c) > 0"
+          by (simp add: finite_less_Inf_iff \<open>c \<noteq> 0\<close> epos)
+        fix f K
+        assume fK: "f \<in> Poly_Mapping.keys c" "K \<subseteq> standard_simplex p"
+          and le: "\<And>x y i. \<lbrakk>x \<in> K; y \<in> K; i \<le> p\<rbrakk> \<Longrightarrow> \<bar>x i - y i\<bar> \<le> Inf (\<psi> ` Poly_Mapping.keys c)"
+        then have lef: "Inf (\<psi> ` Poly_Mapping.keys c) \<le> \<psi> f"
+          by (auto intro: cInf_le_finite)
+        show "\<exists>V. V \<in> \<C> \<and> f ` K \<subseteq> V"
+          using le lef by (blast intro: dual_order.trans e [OF fK])
+      qed
+      let ?d = "\<lambda>m. (simplicial_subdivision p ^^ m) (frag_of (restrict id (standard_simplex p)))"
+      obtain n where n: "(p / (Suc p)) ^ n < d"
+        using real_arch_pow_inv \<open>0 < d\<close> by fastforce
+      show ?thesis
+      proof
+        fix m h
+        assume "n \<le> m" and "h \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)"
+        then obtain f where "f \<in> Poly_Mapping.keys c" "h \<in> Poly_Mapping.keys (chain_map p f (?d m))"
+          using subsetD [OF keys_frag_extend] iterated_singular_subdivision [OF p, of m] by force
+        then obtain g where g: "g \<in> Poly_Mapping.keys (?d m)" and heq: "h = restrict (f \<circ> g) (standard_simplex p)"
+          using keys_frag_extend by (force simp: chain_map_def simplex_map_def)
+        have xx: "simplicial_chain p (standard_simplex p) (?d n) \<and>
+                  (\<forall>f \<in> Poly_Mapping.keys(?d n). \<forall>x \<in> standard_simplex p. \<forall>y \<in> standard_simplex p.
+                       \<bar>f x i - f y i\<bar> \<le> (p / (Suc p)) ^ n)"
+          for n i
+        proof (induction n)
+          case 0
+          have "simplicial_simplex p (standard_simplex p) (\<lambda>a\<in>standard_simplex p. a)"
+            by (metis eq_id_iff order_refl simplicial_simplex_id)
+          moreover have "(\<forall>x\<in>standard_simplex p. \<forall>y\<in>standard_simplex p. \<bar>x i - y i\<bar> \<le> 1)"
+            unfolding standard_simplex_def
+            by (auto simp: abs_if dest!: spec [where x=i])
+          ultimately show ?case
+            unfolding power_0 funpow_0 by simp
+        next
+          case (Suc n)
+          show ?case
+            unfolding power_Suc funpow.simps o_def
+          proof (intro conjI ballI)
+            show "simplicial_chain p (standard_simplex p) (simplicial_subdivision p (?d n))"
+              by (simp add: Suc simplicial_chain_simplicial_subdivision)
+            show "\<bar>f x i - f y i\<bar> \<le> real p / real (Suc p) * (real p / real (Suc p)) ^ n"
+              if "f \<in> Poly_Mapping.keys (simplicial_subdivision p (?d n))"
+                and "x \<in> standard_simplex p" and "y \<in> standard_simplex p" for f x y
+              using Suc that by (blast intro: simplicial_subdivision_shrinks)
+          qed
+        qed
+        have "g ` standard_simplex p \<subseteq> standard_simplex p"
+          using g xx [of m] unfolding simplicial_chain_def simplicial_simplex by auto
+        moreover
+        have "\<bar>g x i - g y i\<bar> \<le> d" if "i \<le> p" "x \<in> standard_simplex p" "y \<in> standard_simplex p" for x y i
+        proof -
+          have "\<bar>g x i - g y i\<bar> \<le> (p / (Suc p)) ^ m"
+            using g xx [of m] that by blast
+          also have "\<dots> \<le> (p / (Suc p)) ^ n"
+            by (auto intro: power_decreasing [OF \<open>n \<le> m\<close>])
+          finally show ?thesis using n by simp
+        qed
+        then have "\<bar>x i - y i\<bar> \<le> d"
+          if "x \<in> g ` (standard_simplex p)" "y \<in> g ` (standard_simplex p)" "i \<le> p" for i x y
+          using that by blast
+        ultimately show "\<exists>V\<in>\<C>. h ` standard_simplex p \<subseteq> V"
+          using \<open>f \<in> Poly_Mapping.keys c\<close> d [of f "g ` standard_simplex p"]
+          by (simp add: Bex_def heq image_image)
+      qed
+    qed
+  qed
+qed force
+
+
+lemma small_homologous_rel_relcycle_exists:
+  assumes \<C>: "\<And>U. U \<in> \<C> \<Longrightarrow> openin X U"
+      and X: "topspace X \<subseteq> \<Union>\<C>"
+      and p: "singular_relcycle p X S c"
+    obtains c' where "singular_relcycle p X S c'" "homologous_rel p X S c c'"
+                      "\<And>f. f \<in> Poly_Mapping.keys c' \<Longrightarrow> \<exists>V \<in> \<C>. f ` (standard_simplex p) \<subseteq> V"
+proof -
+  have "singular_chain p X c"
+       "(chain_boundary p c, 0) \<in> (mod_subset (p - Suc 0) (subtopology X S))"
+    using p unfolding singular_relcycle_def by auto
+  then obtain n where n: "\<And>m f. \<lbrakk>n \<le> m; f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)\<rbrakk>
+                            \<Longrightarrow> \<exists>V \<in> \<C>. f ` (standard_simplex p) \<subseteq> V"
+    by (blast intro: sufficient_iterated_singular_subdivision_exists [OF \<C> X])
+  let ?c' = "(singular_subdivision p ^^ n) c"
+  show ?thesis
+  proof
+    show "homologous_rel p X S c ?c'"
+      apply (induction n, simp_all)
+      by (metis p homologous_rel_singular_subdivision homologous_rel_singular_relcycle homologous_rel_trans homologous_rel_sym)
+    then show "singular_relcycle p X S ?c'"
+      by (metis homologous_rel_singular_relcycle p)
+  next
+    fix f :: "(nat \<Rightarrow> real) \<Rightarrow> 'a"
+    assume "f \<in> Poly_Mapping.keys ?c'"
+    then show "\<exists>V\<in>\<C>. f ` standard_simplex p \<subseteq> V"
+      by (rule n [OF order_refl])
+  qed
+qed
+
+lemma excised_chain_exists:
+  fixes S :: "'a set"
+  assumes "X closure_of U \<subseteq> X interior_of T" "T \<subseteq> S" "singular_chain p (subtopology X S) c"
+  obtains n d e where "singular_chain p (subtopology X (S - U)) d"
+                      "singular_chain p (subtopology X T) e"
+                      "(singular_subdivision p ^^ n) c = d + e"
+proof -
+  have *: "\<exists>n d e. singular_chain p (subtopology X (S - U)) d \<and>
+                  singular_chain p (subtopology X T) e \<and>
+                  (singular_subdivision p ^^ n) c = d + e"
+    if c: "singular_chain p (subtopology X S) c"
+       and X: "X closure_of U \<subseteq> X interior_of T" "U \<subseteq> topspace X" and S: "T \<subseteq> S" "S \<subseteq> topspace X"
+       for p X c S and T U :: "'a set"
+  proof -
+    obtain n where n: "\<And>m f. \<lbrakk>n \<le> m; f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)\<rbrakk>
+                             \<Longrightarrow> \<exists>V \<in> {S \<inter> X interior_of T, S - X closure_of U}. f ` standard_simplex p \<subseteq> V"
+      apply (rule sufficient_iterated_singular_subdivision_exists
+                   [of "{S \<inter> X interior_of T, S - X closure_of U}"])
+      using X S c
+      by (auto simp: topspace_subtopology openin_subtopology_Int2 openin_subtopology_diff_closed)
+    let ?c' = "\<lambda>n. (singular_subdivision p ^^ n) c"
+    have "singular_chain p (subtopology X S) (?c' m)" for m
+      by (induction m) (auto simp: singular_chain_singular_subdivision c)
+    then have scp: "singular_chain p (subtopology X S) (?c' n)" .
+
+    have SS: "Poly_Mapping.keys (?c' n) \<subseteq> singular_simplex_set p (subtopology X (S - U))
+                              \<union> singular_simplex_set p (subtopology X T)"
+    proof (clarsimp)
+      fix f
+      assume f: "f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ n) c)"
+         and non: "\<not> singular_simplex p (subtopology X T) f"
+      show "singular_simplex p (subtopology X (S - U)) f"
+        using n [OF order_refl f] scp f non closure_of_subset [OF \<open>U \<subseteq> topspace X\<close>] interior_of_subset [of X T]
+        by (fastforce simp: image_subset_iff singular_simplex_subtopology singular_chain_def)
+    qed
+    show ?thesis
+       unfolding singular_chain_def using frag_split [OF SS] by metis
+  qed
+  have "(subtopology X (topspace X \<inter> S)) = (subtopology X S)"
+    by (metis subtopology_subtopology subtopology_topspace)
+  with assms have c: "singular_chain p (subtopology X (topspace X \<inter> S)) c"
+    by simp
+  have Xsub: "X closure_of (topspace X \<inter> U) \<subseteq> X interior_of (topspace X \<inter> T)"
+    using assms closure_of_restrict interior_of_restrict by fastforce
+  obtain n d e where
+    d: "singular_chain p (subtopology X (topspace X \<inter> S - topspace X \<inter> U)) d"
+    and e: "singular_chain p (subtopology X (topspace X \<inter> T)) e"
+    and de: "(singular_subdivision p ^^ n) c = d + e"
+    using *[OF c Xsub, simplified] assms by force
+  show thesis
+  proof
+    show "singular_chain p (subtopology X (S - U)) d"
+      by (metis d Diff_Int_distrib inf.cobounded2 singular_chain_mono)
+    show "singular_chain p (subtopology X T) e"
+      by (metis e inf.cobounded2 singular_chain_mono)
+    show "(singular_subdivision p ^^ n) c = d + e"
+      by (rule de)
+  qed
+qed
+
+
+lemma excised_relcycle_exists:
+  fixes S :: "'a set"
+  assumes X: "X closure_of U \<subseteq> X interior_of T" and "T \<subseteq> S"
+      and c: "singular_relcycle p (subtopology X S) T c"
+  obtains c' where "singular_relcycle p (subtopology X (S - U)) (T - U) c'"
+                   "homologous_rel p (subtopology X S) T c c'"
+proof -
+  have [simp]: "(S - U) \<inter> (T - U) = T - U" "S \<inter> T = T"
+    using \<open>T \<subseteq> S\<close> by auto
+  have scc: "singular_chain p (subtopology X S) c"
+    and scp1: "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p c)"
+    using c by (auto simp: singular_relcycle_def mod_subset_def subtopology_subtopology)
+  obtain n d e where d: "singular_chain p (subtopology X (S - U)) d"
+    and e: "singular_chain p (subtopology X T) e"
+    and de: "(singular_subdivision p ^^ n) c = d + e"
+    using excised_chain_exists [OF X \<open>T \<subseteq> S\<close> scc] .
+  have scSUd: "singular_chain (p - Suc 0) (subtopology X (S - U)) (chain_boundary p d)"
+    by (simp add: singular_chain_boundary d)
+  have sccn: "singular_chain p (subtopology X S) ((singular_subdivision p ^^ n) c)" for n
+    by (induction n) (auto simp: singular_chain_singular_subdivision scc)
+  have "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p ((singular_subdivision p ^^ n) c))"
+  proof (induction n)
+    case (Suc n)
+    then show ?case
+      by (simp add: singular_chain_singular_subdivision chain_boundary_singular_subdivision [OF sccn])
+  qed (auto simp: scp1)
+  then have "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p ((singular_subdivision p ^^ n) c - e))"
+    by (simp add: chain_boundary_diff singular_chain_diff singular_chain_boundary e)
+  with de have scTd: "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p d)"
+    by simp
+  show thesis
+  proof
+    have "singular_chain (p - Suc 0) X (chain_boundary p d)"
+      using scTd singular_chain_subtopology by blast
+    with scSUd scTd have "singular_chain (p - Suc 0) (subtopology X (T - U)) (chain_boundary p d)"
+      by (fastforce simp add: singular_chain_subtopology)
+    then show "singular_relcycle p (subtopology X (S - U)) (T - U) d"
+      by (auto simp: singular_relcycle_def mod_subset_def subtopology_subtopology d)
+    have "homologous_rel p (subtopology X S) T (c-0) ((singular_subdivision p ^^ n) c - e)"
+    proof (rule homologous_rel_diff)
+      show "homologous_rel p (subtopology X S) T c ((singular_subdivision p ^^ n) c)"
+      proof (induction n)
+        case (Suc n)
+        then show ?case
+          apply simp
+          apply (rule homologous_rel_trans)
+          using c homologous_rel_singular_relcycle_1 homologous_rel_singular_subdivision homologous_rel_sym by blast
+      qed auto
+      show "homologous_rel p (subtopology X S) T 0 e"
+        unfolding homologous_rel_def using e
+        by (intro singular_relboundary_diff singular_chain_imp_relboundary; simp add: subtopology_subtopology)
+    qed
+    with de show "homologous_rel p (subtopology X S) T c d"
+      by simp
+  qed
+qed
+
+end
+

--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Apr 09 12:36:53 2019 +0100
@@ -34,6 +34,9 @@
 
 subsection \<open>Continuity of the representation WRT an orthogonal basis\<close>
 
+lemma orthogonal_Basis: "pairwise orthogonal Basis"
+  by (simp add: inner_not_same_Basis orthogonal_def pairwise_def)
+
 lemma representation_bound:
   fixes B :: "'N::real_inner set"
   assumes "finite B" "independent B" "b \<in> B" and orth: "pairwise orthogonal B"

--- a/src/HOL/ROOT	Tue Apr 09 11:24:47 2019 +0200
+++ b/src/HOL/ROOT	Tue Apr 09 12:36:53 2019 +0100
@@ -63,6 +63,7 @@
     document_variants = "document:manual=-proof,-ML,-unimportant"]
   sessions
     "HOL-Library"
+    "HOL-Algebra"
     "HOL-Computational_Algebra"
   theories
     Analysis