Theory Simplices

```section‹Homology, I: Simplices›

theory "Simplices"
imports
"HOL-Analysis.Function_Metric"
"HOL-Analysis.Abstract_Euclidean_Space"
"HOL-Algebra.Free_Abelian_Groups"
begin

subsection‹Standard simplices, all of which are topological subspaces of @{text"R^n"}.      ›

type_synonym 'a chain = "((nat ⇒ real) ⇒ 'a) ⇒⇩0 int"

definition standard_simplex :: "nat ⇒ (nat ⇒ real) set" where
"standard_simplex p ≡
{x. (∀i. 0 ≤ x i ∧ x i ≤ 1) ∧ (∀i>p. x i = 0) ∧ (∑i≤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]:
"(λj. if j = i then 1 else 0) ∈ standard_simplex p ⟷ i ≤ p"
by (auto simp: standard_simplex_def)

lemma nonempty_standard_simplex: "standard_simplex p ≠ {}"
using basis_in_standard_simplex by blast

lemma standard_simplex_0: "standard_simplex 0 = {(λj. if j = 0 then 1 else 0)}"
by (auto simp: standard_simplex_def)

lemma standard_simplex_mono:
assumes "p ≤ q"
shows "standard_simplex p ⊆ standard_simplex q"
using assms
proof (clarsimp simp: standard_simplex_def)
fix x :: "nat ⇒ real"
assume "∀i. 0 ≤ x i ∧ x i ≤ 1" and "∀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 =
(⋂i. {x. x ∈ topspace ?X ∧ x i ∈ {0..1}}) ∩
(⋂i ∈ {p<..}. {x ∈ topspace ?X. x i ∈ {0}}) ∩
{x ∈ topspace ?X. (∑i≤p. x i) ∈ {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 ⊆ UNIV →⇩E {0..1}"
using standard_simplex_def by auto

lemma compactin_standard_simplex:
"compactin (powertop_real UNIV) (standard_simplex p)"
proof (rule closed_compactin)
show "compactin (powertop_real UNIV) (UNIV →⇩E {0..1})"
show "standard_simplex p ⊆ UNIV →⇩E {0..1}"
show "closedin (powertop_real UNIV) (standard_simplex p)"
qed

lemma convex_standard_simplex:
"⟦x ∈ standard_simplex p; y ∈ standard_simplex p;
0 ≤ u; u ≤ 1⟧
⟹ (λi. (1 - u) * x i + u * y i) ∈ 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 ≡ λx y::nat⇒real. λu i. (1 - u) * x i + u * y i"
have "continuous_map
(subtopology euclideanreal {0..1}) (powertop_real UNIV)
(g x y)"
if "x ∈ standard_simplex p" "y ∈ standard_simplex p" for x y
unfolding g_def continuous_map_componentwise
by (force intro: continuous_intros)
moreover
have "g x y ` {0..1} ⊆ standard_simplex p" "g x y 0 = x" "g x y 1 = y"
if "x ∈ standard_simplex p" "y ∈ standard_simplex p" for x y
using that by (auto simp: convex_standard_simplex g_def)
ultimately
show ?thesis
unfolding path_connectedin_def path_connected_space_def pathin_def
by (metis continuous_map_in_subtopology euclidean_product_topology top_greatest topspace_euclidean topspace_euclidean_subtopology)
qed

lemma connectedin_standard_simplex:
"connectedin (powertop_real UNIV) (standard_simplex p)"

subsection‹Face map›

definition simplical_face :: "nat ⇒ (nat ⇒ 'a) ⇒ nat ⇒ 'a::comm_monoid_add" where
"simplical_face k x ≡ λi. if i < k then x i else if i = k then 0 else x(i -1)"

lemma simplical_face_in_standard_simplex:
assumes "1 ≤ p" "k ≤ p" "x ∈ standard_simplex (p - Suc 0)"
shows "(simplical_face k x) ∈ standard_simplex p"
proof -
have x01: "⋀i. 0 ≤ x i ∧ x i ≤ 1" and sumx: "sum x {..p - Suc 0} = 1"
using assms by (auto simp: standard_simplex_def simplical_face_def)
have gg: "⋀g. sum g {..p} = sum g {..<k} + sum g {k..p}"
using ‹k ≤ p› sum.union_disjoint [of "{..<k}" "{k..p}"]
by (force simp: ivl_disj_un ivl_disj_int)
have eq: "(∑i≤p. if i < k then x i else if i = k then 0 else x (i -1))
= (∑i < k. x i) + (∑i ∈ {k..p}. if i = k then 0 else x (i -1))"
consider "k ≤ p - Suc 0" | "k = p"
using ‹k ≤ p› by linarith
then have "(∑i≤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 ‹1 ≤ p› by auto
have "(∑i = k..p. if i = k then 0 else x (i -1)) = (∑i = k+1..p. if i = k then 0 else x (i -1))"
by (rule sum.mono_neutral_right) auto
also have "… = (∑i = k+1..p. x (i -1))"
by simp
also have "… = (∑i = k..p-1. x i)"
using sum.atLeastAtMost_reindex [of Suc k "p-1" "λi. x (i - Suc 0)"] 1 by simp
finally have eq2: "(∑i = k..p. if i = k then 0 else x (i -1)) = (∑i = k..p-1. x i)" .
with 1 show ?thesis
by (metis (no_types, lifting) One_nat_def eq finite_atLeastAtMost finite_lessThan ivl_disj_int(4) ivl_disj_un(10) sum.union_disjoint sumx)
next
case 2
have [simp]: "({..p} ∩ {x. x < p}) = {..p - Suc 0}"
using assms by auto
have "(∑i≤p. if i < p then x i else if i = k then 0 else x (i -1)) = (∑i≤p. if i < p then x i else 0)"
by (rule sum.cong) (auto simp: 2)
also have "… = sum x {..p-1}"
also have "… = 1"
finally show ?thesis
using 2 by simp
qed
then show ?thesis
using assms by (auto simp: standard_simplex_def simplical_face_def)
qed

subsection‹Singular simplices, forcing canonicity outside the intended domain›

definition singular_simplex :: "nat ⇒ 'a topology ⇒ ((nat ⇒ real) ⇒ 'a) ⇒ bool" where
"singular_simplex p X f ≡
continuous_map(subtopology (powertop_real UNIV) (standard_simplex p)) X f
∧ f ∈ extensional (standard_simplex p)"

abbreviation singular_simplex_set :: "nat ⇒ 'a topology ⇒ ((nat ⇒ real) ⇒ 'a) set" where
"singular_simplex_set p X ≡ Collect (singular_simplex p X)"

lemma singular_simplex_empty:
"topspace X = {} ⟹ ¬ singular_simplex p X f"
by (simp add: singular_simplex_def continuous_map nonempty_standard_simplex)

lemma singular_simplex_mono:
"⟦singular_simplex p (subtopology X T) f; T ⊆ S⟧ ⟹ 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 ⟷
singular_simplex p X f ∧ f ` (standard_simplex p) ⊆ S"
by (auto simp: singular_simplex_def continuous_map_in_subtopology)

subsubsection‹Singular face›

definition singular_face :: "nat ⇒ nat ⇒ ((nat ⇒ real) ⇒ 'a) ⇒ (nat ⇒ real) ⇒ 'a"
where "singular_face p k f ≡ restrict (f ∘ simplical_face k) (standard_simplex (p - Suc 0))"

lemma singular_simplex_singular_face:
assumes f: "singular_simplex p X f" and "1 ≤ p" "k ≤ 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) ⊆ 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 (λx. 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
(λx. simplical_face k x i)"
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‹Singular chains›

definition singular_chain :: "[nat, 'a topology, 'a chain] ⇒ bool"
where "singular_chain p X c ≡ Poly_Mapping.keys c ⊆ singular_simplex_set p X"

abbreviation singular_chain_set :: "[nat, 'a topology] ⇒ ('a chain) set"
where "singular_chain_set p X ≡ Collect (singular_chain p X)"

lemma singular_chain_empty:
"topspace X = {} ⟹ singular_chain p X c ⟷ c = 0"
by (auto simp: singular_chain_def singular_simplex_empty subset_eq poly_mapping_eqI)

lemma singular_chain_mono:
"⟦singular_chain p (subtopology X T) c;  T ⊆ S⟧
⟹ 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 ⟷
singular_chain p X c ∧ (∀f ∈ Poly_Mapping.keys c. f ` (standard_simplex p) ⊆ 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) ⟷ singular_simplex p X c"
by (auto simp: singular_chain_def)

lemma singular_chain_cmul:
"singular_chain p X c ⟹ singular_chain p X (frag_cmul a c)"
by (auto simp: singular_chain_def)

lemma singular_chain_minus:
"singular_chain p X (-c) ⟷ singular_chain p X c"
by (auto simp: singular_chain_def)

"⟦singular_chain p X a; singular_chain p X b⟧ ⟹ singular_chain p X (a+b)"
unfolding singular_chain_def
using keys_add [of a b] by blast

lemma singular_chain_diff:
"⟦singular_chain p X a; singular_chain p X b⟧ ⟹ singular_chain p X (a-b)"
unfolding singular_chain_def
using keys_diff [of a b] by blast

lemma singular_chain_sum:
"(⋀i. i ∈ I ⟹ singular_chain p X (f i)) ⟹ singular_chain p X (∑i∈I. f i)"
unfolding singular_chain_def
using keys_sum [of f I] by blast

lemma singular_chain_extend:
"(⋀c. c ∈ Poly_Mapping.keys x ⟹ singular_chain p X (f c))
⟹ singular_chain p X (frag_extend f x)"
by (simp add: frag_extend_def singular_chain_cmul singular_chain_sum)

subsection‹Boundary homomorphism for singular chains›

definition chain_boundary :: "nat ⇒ ('a chain) ⇒ 'a chain"
where "chain_boundary p c ≡
(if p = 0 then 0 else
frag_extend (λf. (∑k≤p. frag_cmul ((-1) ^ k) (frag_of(singular_face p k f)))) c)"

lemma singular_chain_boundary:
assumes "singular_chain p X c"
shows "singular_chain (p - Suc 0) X (chain_boundary p c)"
unfolding chain_boundary_def
proof (clarsimp intro!: singular_chain_extend singular_chain_sum singular_chain_cmul)
show "⋀d k. ⟦0 < p; d ∈ Poly_Mapping.keys c; k ≤ p⟧
⟹ singular_chain (p - Suc 0) X (frag_of (singular_face p k d))"
using assms by (auto simp: singular_chain_def intro: singular_simplex_singular_face)
qed

lemma singular_chain_boundary_alt:
"singular_chain (Suc p) X c ⟹ 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"

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)

"chain_boundary p (a+b) = chain_boundary p a + chain_boundary p b"

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"]

lemma chain_boundary_sum:
"chain_boundary p (sum g I) = sum (chain_boundary p ∘ g) I"

lemma chain_boundary_sum':
"finite I ⟹ chain_boundary p (sum' g I) = sum' (chain_boundary p ∘ g) I"

lemma chain_boundary_of:
"chain_boundary p (frag_of f) =
(if p = 0 then 0
else (∑k≤p. frag_cmul ((-1) ^ k) (frag_of(singular_face p k f))))"

subsection‹Factoring out chains in a subtopology for relative homology›

definition mod_subset
where "mod_subset p X ≡ {(a,b). singular_chain p X (a - b)}"

lemma mod_subset_empty [simp]:
"(a,b) ∈ (mod_subset p (subtopology X {})) ⟷ a = b"

lemma mod_subset_refl [simp]: "(c,c) ∈ mod_subset p X"
by (auto simp: mod_subset_def)

lemma mod_subset_cmul:
assumes "(a,b) ∈ mod_subset p X"
shows "(frag_cmul k a, frag_cmul k b) ∈ mod_subset p X"
using assms

"⟦(c1,c2) ∈ mod_subset p X; (d1,d2) ∈ mod_subset p X⟧ ⟹ (c1+d1, c2+d2) ∈ mod_subset p X"

subsection‹Relative cycles \$Z_pX (S)\$ where \$X\$ is a topology and \$S\$ a subset ›

definition singular_relcycle :: "nat ⇒ 'a topology ⇒ 'a set ⇒ ('a chain) ⇒ bool"
where "singular_relcycle p X S ≡
λc. singular_chain p X c ∧ (chain_boundary p c, 0) ∈ mod_subset (p-1) (subtopology X S)"

abbreviation singular_relcycle_set
where "singular_relcycle_set p X S ≡ Collect (singular_relcycle p X S)"

lemma singular_relcycle_restrict [simp]:
"singular_relcycle p X (topspace X ∩ S) = singular_relcycle p X S"
proof -
have eq: "subtopology X (topspace X ∩ 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 ⟷
singular_chain p X c ∧ singular_chain (p-1) (subtopology X S) (chain_boundary p c)"

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 ⟹ 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) ⟷ singular_relcycle p X S c"
by (simp add: chain_boundary_minus singular_chain_minus singular_relcycle)

"⟦singular_relcycle p X S a; singular_relcycle p X S b⟧
⟹ singular_relcycle p X S (a+b)"

lemma singular_relcycle_sum:
"⟦⋀i. i ∈ I ⟹ singular_relcycle p X S (f i)⟧
⟹ singular_relcycle p X S (sum f I)"
by (induction I rule: infinite_finite_induct) (auto simp: singular_relcycle_add)

lemma singular_relcycle_diff:
"⟦singular_relcycle p X S a; singular_relcycle p X S b⟧
⟹ singular_relcycle p X S (a-b)"

lemma singular_cycle:
"singular_relcycle p X {} c ⟷ singular_chain p X c ∧ chain_boundary p c = 0"
using mod_subset_empty by (auto simp: singular_relcycle_def)

lemma singular_cycle_mono:
"⟦singular_relcycle p (subtopology X T) {} c; T ⊆ S⟧
⟹ singular_relcycle p (subtopology X S) {} c"
by (auto simp: singular_cycle elim: singular_chain_mono)

subsection‹Relative boundaries \$B_p X S\$, where \$X\$ is a topology and \$S\$ a subset.›

definition singular_relboundary :: "nat ⇒ 'a topology ⇒ 'a set ⇒ ('a chain) ⇒ bool"
where
"singular_relboundary p X S ≡
λc. ∃d. singular_chain (Suc p) X d ∧ (chain_boundary (Suc p) d, c) ∈ (mod_subset p (subtopology X S))"

abbreviation singular_relboundary_set :: "nat ⇒ 'a topology ⇒ 'a set ⇒ ('a chain) set"
where "singular_relboundary_set p X S ≡ Collect (singular_relboundary p X S)"

lemma singular_relboundary_restrict [simp]:
"singular_relboundary p X (topspace X ∩ S) = singular_relboundary p X S"
unfolding singular_relboundary_def
by (metis (no_types, opaque_lifting) subtopology_subtopology subtopology_topspace)

lemma singular_relboundary_alt:
"singular_relboundary p X S c ⟷
(∃d e. singular_chain (Suc p) X d ∧ singular_chain p (subtopology X S) e ∧
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 ⟷
(∃d e. singular_chain (Suc p) X d ∧ singular_chain p (subtopology X S) e ∧
(chain_boundary (Suc p) d) + e = c)"
using singular_chain_minus

lemma singular_boundary:
"singular_relboundary p X {} c ⟷
(∃d. singular_chain (Suc p) X d ∧ chain_boundary (Suc p) d = c)"
by (meson mod_subset_empty singular_relboundary_def)

lemma singular_boundary_imp_chain:
"singular_relboundary p X {} c ⟹ singular_chain p X c"
by (auto simp: singular_relboundary singular_chain_boundary_alt singular_chain_empty)

lemma singular_boundary_mono:
"⟦T ⊆ S; singular_relboundary p (subtopology X T) {} c⟧
⟹ 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 ⟹ singular_chain p X c"
unfolding singular_relboundary singular_chain_subtopology

lemma singular_chain_imp_relboundary:
"singular_chain p (subtopology X S) c ⟹ singular_relboundary p X S c"
unfolding singular_relboundary_def
using mod_subset_def singular_chain_minus 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 ⟹ 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) ⟷ singular_relboundary p X S c"
using singular_relboundary_cmul

"⟦singular_relboundary p X S a; singular_relboundary p X S b⟧ ⟹ singular_relboundary p X S (a+b)"
unfolding singular_relboundary_def

lemma singular_relboundary_diff:
"⟦singular_relboundary p X S a; singular_relboundary p X S b⟧ ⟹ singular_relboundary p X S (a-b)"

subsection‹The (relative) homology relation›

definition homologous_rel :: "[nat,'a topology,'a set,'a chain,'a chain] ⇒ bool"
where "homologous_rel p X S ≡ λa b. singular_relboundary p X S (a-b)"

abbreviation homologous_rel_set
where "homologous_rel_set p X S a ≡ Collect (homologous_rel p X S a)"

lemma homologous_rel_restrict [simp]:
"homologous_rel p X (topspace X ∩ 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 ⟷
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 ⟷
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 ⟹ (singular_chain p X a ⟷ singular_chain p X b)"
unfolding homologous_rel_def
by (fastforce dest: singular_relboundary_imp_chain)

"⟦homologous_rel p X S a a'; homologous_rel p X S b b'⟧
⟹ homologous_rel p X S (a+b) (a'+b')"
unfolding homologous_rel_def

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
qed

lemma homologous_rel_sum:
assumes f: "finite {i ∈ I. f i ≠ 0}" and g: "finite {i ∈ I. g i ≠ 0}"
and h: "⋀i. i ∈ I ⟹ 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 ∈ I. f i ≠ 0} ∪ {i ∈ I. g i ≠ 0}"
have L: "finite ?L" "?L ⊆ 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 ∈ ?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
qed auto
ultimately show ?thesis
by simp
qed auto

lemma chain_homotopic_imp_homologous_rel:
assumes
"⋀c. singular_chain p X c ⟹ singular_chain (Suc p) X' (h c)"
"⋀c. singular_chain (p -1) (subtopology X S) c ⟹ singular_chain p (subtopology X' T) (h' c)"
"⋀c. singular_chain p X c
⟹ (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)"
proof -
have "singular_chain p (subtopology X' T) (chain_boundary (Suc p) (h c) - (f c - g c))"
using assms
by (metis (no_types, lifting) add_diff_cancel_left' minus_diff_eq singular_chain_minus singular_relcycle)
then show ?thesis
using assms
by (metis homologous_rel_def singular_relboundary singular_relcycle)
qed

subsection‹Show that all boundaries are cycles, the key "chain complex" property.›

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 ≤ 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} × {..p - Suc 0} ∩ {(x, y). y < x} = (λ(j,i). (Suc i, j)) ` {(i,j). i ≤ j ∧ j ≤ 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} × {..p - Suc 0} - {(x, y). y < x} = {(i,j). i ≤ j ∧ j ≤ 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 ≤ j" "j ≤ 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 ∈ standard_simplex (p -1 -1)")
case True
have fi: "simplical_face i t ∈ standard_simplex (p - Suc 0)"
using False True simplical_face_in_standard_simplex that by force
have fj: "simplical_face j t ∈ 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
show ?case
proof (cases "p = 1")
case False
have eq0: "frag_cmul (-1) a = b ⟹ a + b = 0" for a b
have *: "(∑x≤p. ∑i≤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 "_ × _" _ "{(x,y). y < x}"])
apply (rule eq0)
unfolding frag_cmul_sum prod.case_distrib [of "frag_cmul (-1)"] frag_cmul_cmul eql eqr
apply (force simp: 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)
next
case (diff a b)
then show ?case
qed auto

lemma chain_boundary_boundary_alt:
"singular_chain (Suc p) X c ⟹ 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 ‹singular_chain p (subtopology X S) e› 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)"
ultimately show ?thesis
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

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‹Operations induced by a continuous map g between topological spaces›

definition simplex_map :: "nat ⇒ ('b ⇒ 'a) ⇒ ((nat ⇒ real) ⇒ 'b) ⇒ (nat ⇒ real) ⇒ 'a"
where "simplex_map p g c ≡ restrict (g ∘ c) (standard_simplex p)"

lemma singular_simplex_simplex_map:
"⟦singular_simplex p X f; continuous_map X X' g⟧
⟹ 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:
"⟦singular_simplex p X c;
⋀x. x ∈ topspace X ⟹ f x = g x⟧
⟹ simplex_map p f c = simplex_map p g c"
by (auto simp: singular_simplex_def simplex_map_def continuous_map_def Pi_iff)

lemma simplex_map_id_gen:
"⟦singular_simplex p X c;
⋀x. x ∈ topspace X ⟹ f x = x⟧
⟹ 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 = (λc. restrict c (standard_simplex p))"
by (auto simp: simplex_map_def)

lemma simplex_map_compose:
"simplex_map p (h ∘ g) = simplex_map p h ∘ simplex_map p g"
unfolding simplex_map_def by force

lemma singular_face_simplex_map:
"⟦1 ≤ p; k ≤ p⟧
⟹ singular_face p k (simplex_map p f c) = simplex_map (p - Suc 0) f (c ∘ 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 ≤ 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 ⇒ ('b ⇒ 'a) ⇒ (((nat ⇒ real) ⇒ 'b) ⇒⇩0 int) ⇒ 'a chain"
where "chain_map p g c ≡ frag_extend (frag_of ∘ simplex_map p g) c"

lemma singular_chain_chain_map:
"⟦singular_chain p X c; continuous_map X X' g⟧ ⟹ singular_chain p X' (chain_map p g c)"
unfolding chain_map_def
by (force simp add: singular_chain_def subset_iff
intro!: singular_chain_extend 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)"

lemma chain_map_cmul [simp]:
"chain_map p g (frag_cmul a c) = frag_cmul a (chain_map p g c)"

lemma chain_map_minus: "chain_map p g (-c) = - (chain_map p g c)"

"chain_map p g (a+b) = chain_map p g a + chain_map p g b"

lemma chain_map_diff:
"chain_map p g (a-b) = chain_map p g a - chain_map p g b"

lemma chain_map_sum:
"finite I ⟹ chain_map p g (sum f I) = sum (chain_map p g ∘ f) I"

lemma chain_map_eq:
"⟦singular_chain p X c; ⋀x. x ∈ topspace X ⟹ f x = g x⟧
⟹ chain_map p f c = chain_map p g c"
unfolding singular_chain_def
proof (induction rule: frag_induction)
case (one x)
then show ?case
by (metis (no_types, lifting) chain_map_of mem_Collect_eq simplex_map_eq)
qed (auto simp: chain_map_diff)

lemma chain_map_id_gen:
"⟦singular_chain p X c; ⋀x. x ∈ topspace X ⟹ f x = x⟧
⟹  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 ⟹ chain_map p id c = c"

lemma chain_map_id:
"chain_map p id = frag_extend (frag_of ∘ (λf. restrict f (standard_simplex p)))"
by (auto simp: chain_map_def)

lemma chain_map_compose:
"chain_map p (h ∘ g) = chain_map p h ∘ chain_map p g"
proof
show "chain_map p (h ∘ g) c = (chain_map p h ∘ 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
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 ∘ 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
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 ≤ i" "i ≤ p" "p ≠ 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
qed auto

lemma singular_relcycle_chain_map:
assumes "singular_relcycle p X S c" "continuous_map X X' g" "g ` S ⊆ 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 ⊆ 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 "∀t. g ` topspace (subtopology t S) ⊆ 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‹Homology of one-point spaces degenerates except for \$p = 0\$.›

lemma singular_simplex_singleton:
assumes "topspace X = {a}"
shows "singular_simplex p X f ⟷ f = restrict (λx. a) (standard_simplex p)" (is "?lhs = ?rhs")
proof
assume L: ?lhs
then show ?rhs
proof -
have "continuous_map (subtopology (product_topology (λn. euclideanreal) UNIV) (standard_simplex p)) X f"
using ‹singular_simplex p X f› singular_simplex_def by blast
then have "⋀c. c ∉ standard_simplex p ∨ f c = a"
by (simp add: assms continuous_map_def Pi_iff)
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)
qed

lemma singular_chain_singleton:
assumes "topspace X = {a}"
shows "singular_chain p X c ⟷
(∃b. c = frag_cmul b (frag_of(restrict (λx. a) (standard_simplex p))))"
(is "?lhs = ?rhs")
proof
let ?f = "restrict (λx. a) (standard_simplex p)"
assume L: ?lhs
with assms have "Poly_Mapping.keys c ⊆ {?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 "∃b. frag_extend frag_of c = frag_cmul b (frag_of (λx∈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 ∨ odd p then 0
else frag_extend (λf. frag_of(restrict (λx. a) (standard_simplex (p -1)))) c)"
(is "?lhs = ?rhs")
proof (cases "p = 0")
case False
have "?lhs = frag_extend (λf. if odd p then 0 else frag_of(restrict (λx. a) (standard_simplex (p -1)))) c"
proof (simp only: chain_boundary_def False if_False, rule frag_extend_eq)
fix f
assume "f ∈ Poly_Mapping.keys c"
with assms have "singular_simplex p X f"
by (auto simp: singular_chain_def)
then have *: "⋀k. k ≤ p ⟹ singular_face p k f = (λx∈standard_simplex (p -1). a)"
using False singular_simplex_singular_face
by (fastforce simp flip: singular_simplex_singleton [OF tX])
define c where "c ≡ frag_of (λx∈standard_simplex (p -1). a)"
have "(∑k≤p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k f)))
= (∑k≤p. frag_cmul ((-1) ^ k) c)"
by (auto simp: c_def * intro: sum.cong)
also have "… = (if odd p then 0 else c)"
by (induction p) (auto simp: c_def restrict_def)
finally show "(∑k≤p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k f)))
= (if odd p then 0 else frag_of (λx∈standard_simplex (p -1). a))"
unfolding c_def .
qed
also have "… = ?rhs"
by (auto simp: False frag_extend_eq_0)
finally show ?thesis .

lemma singular_cycle_singleton:
assumes "topspace X = {a}"
shows "singular_relcycle p X {} c ⟷ singular_chain p X c ∧ (p = 0 ∨ odd p ∨ c = 0)"
proof -
have "c = 0" if "singular_chain p X c" and "chain_boundary p c = 0" and "even p" and "p ≠ 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"
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 ⟷ singular_chain p X c ∧ (odd p ∨ c = 0)"
proof (cases "singular_chain p X c")
case True
have "∃d. singular_chain (Suc p) X d ∧ chain_boundary (Suc p) d = c"
if "singular_chain p X c" and "odd p"
proof -
obtain b where b: "c = frag_cmul b (frag_of(restrict (λx. a) (standard_simplex p)))"
by (metis True assms singular_chain_singleton)
let ?d = "frag_cmul b (frag_of (λx∈standard_simplex (Suc p). a))"
have scd: "singular_chain (Suc p) X ?d"
by (metis assms singular_chain_singleton)
moreover have "chain_boundary (Suc p) ?d = c"
by (simp add: assms scd chain_boundary_of_singleton [of X a "Suc p"] b frag_extend_cmul ‹odd p›)
ultimately show ?thesis
by metis
qed
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 ≤ p"
shows "singular_relboundary p X {} c ⟷ singular_relcycle p X {} c" (is "?lhs = ?rhs")
proof
show "?lhs ⟹ ?rhs"
show "?rhs ⟹ ?lhs"
by (metis assms not_one_le_zero singular_boundary_singleton singular_cycle_singleton)
qed

lemma singular_boundary_set_eq_cycle_singleton:
assumes "topspace X = {a}" "1 ≤ p"
shows "singular_relboundary_set p X {} = singular_relcycle_set p X {}"
using singular_boundary_eq_cycle_singleton [OF assms]
by blast

subsection‹Simplicial chains›

text‹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.›

definition oriented_simplex
where "oriented_simplex p l ≡ (λx∈standard_simplex p. λi. (∑j≤p. l j i * x j))"

definition simplicial_simplex
where
"simplicial_simplex p S f ≡
singular_simplex p (subtopology (powertop_real UNIV) S) f ∧
(∃l. f = oriented_simplex p l)"

lemma simplicial_simplex:
"simplicial_simplex p S f ⟷ f ` (standard_simplex p) ⊆ S ∧ (∃l. f = oriented_simplex p l)"
(is "?lhs = ?rhs")
proof
assume R: ?rhs
have "continuous_map (subtopology (powertop_real UNIV) (standard_simplex p))
(powertop_real UNIV) (λx i. ∑j≤p. l j i * x j)" for l :: " nat ⇒ 'a ⇒ real"
unfolding continuous_map_componentwise
by (force intro: continuous_intros continuous_map_from_subtopology continuous_map_product_projection)
with R show ?lhs
unfolding simplicial_simplex_def singular_simplex_subtopology
by (auto simp add: singular_simplex_def oriented_simplex_def)

lemma simplicial_simplex_empty [simp]: "¬ simplicial_simplex p {} f"

definition simplicial_chain
where "simplicial_chain p S c ≡ Poly_Mapping.keys c ⊆ Collect (simplicial_simplex p S)"

lemma simplicial_chain_0 [simp]: "simplicial_chain p S 0"

lemma simplicial_chain_of [simp]:
"simplicial_chain p S (frag_of c) ⟷ simplicial_simplex p S c"

lemma simplicial_chain_cmul:
"simplicial_chain p S c ⟹ simplicial_chain p S (frag_cmul a c)"
by (auto simp: simplicial_chain_def)

lemma simplicial_chain_diff:
"⟦simplicial_chain p S c1; simplicial_chain p S c2⟧ ⟹ simplicial_chain p S (c1 - c2)"
unfolding simplicial_chain_def  by (meson UnE keys_diff subset_iff)

lemma simplicial_chain_sum:
"(⋀i. i ∈ I ⟹ simplicial_chain p S (f i)) ⟹ simplicial_chain p S (sum f I)"
unfolding simplicial_chain_def
using order_trans [OF keys_sum [of f I]]

lemma simplicial_simplex_oriented_simplex:
"simplicial_simplex p S (oriented_simplex p l)
⟷ ((λx i. ∑j≤p. l j i * x j) ` standard_simplex p ⊆ S)"
by (auto simp: simplicial_simplex oriented_simplex_def)

lemma simplicial_imp_singular_simplex:
"simplicial_simplex p S f
⟹ singular_simplex p (subtopology (powertop_real UNIV) S) f"

lemma simplicial_imp_singular_chain:
"simplicial_chain p S c
⟹ 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' ⟷ (∀i. i ≤ p ⟶ l i = l' i)"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
show ?rhs
proof clarify
fix i
assume "i ≤ p"
let ?fi = "(λj. if j = i then 1 else 0)"
have "(∑j≤p. l j k * ?fi j) = (∑j≤p. l' j k * ?fi j)" for k
using L ‹i ≤ p›
by (simp add: fun_eq_iff oriented_simplex_def split: if_split_asm)
with ‹i ≤ p› show "l i = l' i"
by (simp add: if_distrib ext cong: if_cong)
qed
qed (auto simp: oriented_simplex_def)

lemma singular_face_oriented_simplex:
assumes "1 ≤ p" "k ≤ p"
shows "singular_face p k (oriented_simplex p l) =
oriented_simplex (p -1) (λj. if j < k then l j else l (Suc j))"
proof -
have "(∑j≤p. l j i * simplical_face k x j)
= (∑j≤p - Suc 0. (if j < k then l j else l (Suc j)) i * x j)"
if "x ∈ 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]
by (simp add: if_distrib [of "λx. _ * x"] if_distrib [of "λf. f i * _"] atLeast0AtMost cong: if_cong)
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 ⇒ real) ⇒ nat ⇒ real"
assumes ss: "simplicial_simplex p S f" and p: "1 ≤ p" "k ≤ 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 "singular_face p k f = oriented_simplex (p - Suc 0) (λi. if i < k then m i else m (Suc i))"
using feq p singular_face_oriented_simplex by auto
ultimately
show ?thesis
using p simplicial_simplex_def singular_simplex_singular_face by blast
qed

lemma simplicial_chain_boundary:
"simplicial_chain p S c ⟹ 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 ≤ p" for i
using that one
by (force simp: simplicial_simplex_def singular_simplex_singular_face singular_face_oriented_simplex)
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
next
case (diff a b)
then show ?case
by (metis chain_boundary_diff simplicial_chain_def simplicial_chain_diff)
qed auto

subsection‹The cone construction on simplicial simplices.›

consts simplex_cone :: "[nat, nat ⇒ real, [nat ⇒ real, nat] ⇒ real, nat ⇒ real, nat] ⇒ real"
specification (simplex_cone)
simplex_cone:
"⋀p v l. simplex_cone p v (oriented_simplex p l) =
oriented_simplex (Suc p) (λi. if i = 0 then v else l(i -1))"
proof -
have *: "⋀x. ∀xv. ∃y. (λl. oriented_simplex (Suc x)
(λi. if i = 0 then xv else l (i - 1))) =
y ∘ oriented_simplex x"
by (simp add: oriented_simplex_eq flip: choice_iff function_factors_left)
then show ?thesis
unfolding o_def by (metis(no_types))
qed

lemma simplicial_simplex_simplex_cone:
assumes f: "simplicial_simplex p S f"
and T: "⋀x u. ⟦0 ≤ u; u ≤ 1; x ∈ S⟧ ⟹ (λi. (1 - u) * v i + u * x i) ∈ T"
shows "simplicial_simplex (Suc p) T (simplex_cone p v f)"
proof -
obtain l where l: "⋀x. x ∈ standard_simplex p ⟹ oriented_simplex p l x ∈ S"
and feq: "f = oriented_simplex p l"
using f by (auto simp: simplicial_simplex)
have "oriented_simplex p l x ∈ S" if "x ∈ standard_simplex p" for x
using f that by (auto simp: simplicial_simplex feq)
then have S: "⋀x. ⟦⋀i. 0 ≤ x i ∧ x i ≤ 1; ⋀i. i>p ⟹ x i = 0; sum x {..p} = 1⟧
⟹ (λi. ∑j≤p. l j i * x j) ∈ S"
have "oriented_simplex (Suc p) (λi. if i = 0 then v else l (i -1)) x ∈ T"
if "x ∈ standard_simplex (Suc p)" for x
proof (simp add: that oriented_simplex_def sum.atMost_Suc_shift del: sum.atMost_Suc)
have x01: "⋀i. 0 ≤ x i ∧ x i ≤ 1" and x0: "⋀i. i > Suc p ⟹ 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 ∈ S"
using f by force
show "(λi. v i * x 0 + (∑j≤p. l j i * x (Suc j))) ∈ 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.atLeast_Suc_atMost)
then have [simp]: "x (Suc j) = 0" if "j≤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] ‹a ∈ S› by (auto simp: True)
next
case False
then have "(λi. v i * x 0 + (∑j≤p. l j i * x (Suc j))) = (λi. (1 - (1 - x 0)) * v i + (1 - x 0) * (inverse (1 - x 0) * (∑j≤p. l j i * x (Suc j))))"
by (force simp: field_simps)
also have "… ∈ T"
proof (rule T)
have "x 0 < 1"
by (simp add: False less_le x01)
have xle: "x (Suc i) ≤ (1 - x 0)" for i
proof (cases "i ≤ p")
case True
have "sum x {0, Suc i} ≤ 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)
have "(λi. (∑j≤p. l j i * (x (Suc j) * inverse (1 - x 0)))) ∈ S"
proof (rule S)
have "x 0 + (∑j≤p. x (Suc j)) = sum x {..Suc p}"
by (metis sum.atMost_Suc_shift)
with x1 have "(∑j≤p. x (Suc j)) = 1 - x 0"
by simp
with False show "(∑j≤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 ‹x 0 < 1› in ‹auto simp: field_split_simps›)
then show "(λi. inverse (1 - x 0) * (∑j≤p. l j i * x (Suc j))) ∈ S"
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 ≡ frag_extend (frag_of ∘ simplex_cone p v)"

lemma simplicial_chain_simplicial_cone:
assumes c: "simplicial_chain p S c"
and T: "⋀x u. ⟦0 ≤ u; u ≤ 1; x ∈ S⟧ ⟹ (λi. (1 - u) * v i + u * x i) ∈ 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
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 (λu∈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 (λj. if j = 0 then v else l j)) = (λu∈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 (λu∈standard_simplex 0. v)"
by (simp add: assms simplicial_cone_def chain_boundary_of ‹p = 0› 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
(λj. 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 ≤ p" for x
using ‹0 < p› that
by (auto simp: Suc_leI singular_face_oriented_simplex simplex_cone oriented_simplex_eq)
have 1: "frag_extend (frag_of ∘ simplex_cone (p - Suc 0) v)
(∑k = 0..p. frag_cmul ((-1) ^ k) (frag_of (singular_face p k (oriented_simplex p l))))
= - (∑k = Suc 0..Suc p. frag_cmul ((-1) ^ k)
(frag_of (singular_face (Suc p) k (simplex_cone p v (oriented_simplex p l)))))"
unfolding sum.atLeast_Suc_atMost_Suc_shift
by (auto simp: 0 simplex_cone singular_face_oriented_simplex frag_extend_sum frag_extend_cmul simp flip: sum_negf)
moreover have 2: "singular_face (Suc p) 0 (simplex_cone p v (oriented_simplex p l))
= oriented_simplex p l"
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 ‹p > 0›
apply (simp add: assms simplicial_cone_def chain_boundary_of atMost_atLeast0 del: sum.atMost_Suc)
apply (subst sum.atLeast_Suc_atMost [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 (λu∈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
⟹ chain_boundary (Suc p) (simplicial_cone p v c) =
c - (if p = 0 then frag_extend (λf. frag_of (λu∈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 ≤ r"
shows "simplex_map p g (oriented_simplex p l) = oriented_simplex p (g ∘ l)"
proof -
obtain m where geq: "g = oriented_simplex r m"
using g by (auto simp: simplicial_simplex_def)
have "g (λi. ∑j≤p. l j i * x j) i = (∑j≤p. g (l j) i * x j)"
if "x ∈ standard_simplex p" for x i
proof -
have ssr: "(λi. ∑j≤p. l j i * x j) ∈ standard_simplex r"
using l that standard_simplex_mono [OF ‹q ≤ r›]
unfolding simplicial_simplex_oriented_simplex by auto
have lss: "l j ∈ standard_simplex r" if "j≤p" for j
proof -
have q: "(λx i. ∑j≤p. l j i * x j) ` standard_simplex p ⊆ standard_simplex q"
using l by (simp add: simplicial_simplex_oriented_simplex)
let ?x = "(λi. if i = j then 1 else 0)"
have p: "l j ∈ (λx i. ∑j≤p. l j i * x j) ` standard_simplex p"
proof
show "l j = (λi. ∑j≤p. l j i * ?x j)"
using ‹j≤p› by (force simp: if_distrib cong: if_cong)
show "?x ∈ standard_simplex p"
qed
show ?thesis
using standard_simplex_mono [OF ‹q ≤ r›] q p
by blast
qed
have "g (λi. ∑j≤p. l j i * x j) i = (∑j≤r. ∑n≤p. m j i * (l n j * x n))"
by (simp add: geq oriented_simplex_def sum_distrib_left ssr)
also have "... =  (∑j≤p. ∑n≤r. m n i * (l j n * x j))"
by (rule sum.swap)
also have "... = (∑j≤p. g (l j) i * x j)"
by (simp add: geq oriented_simplex_def sum_distrib_right mult.assoc lss)
finally show ?thesis .
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 ∈ standard_simplex q" and "q ≤ 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 ∈ 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 ‹simplicial_simplex p (standard_simplex q) f› feq by blast
then have 1: "simplicial_simplex (Suc p) (standard_simplex q)
(oriented_simplex (Suc p) (λi. 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 ‹q ≤ r›]
simplex_map_oriented_simplex [of p q m r S g, OF 0 g ‹q ≤ r›]
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‹Barycentric subdivision of a linear ("simplicial") simplex's image›

definition simplicial_vertex
where "simplicial_vertex i f = f(λj. if j = i then 1 else 0)"

lemma simplicial_vertex_oriented_simplex:
"simplicial_vertex i (oriented_simplex p l) = (if i ≤ 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
(λf. simplicial_cone p
(λi. (∑j≤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)
(λi. (∑j≤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
⟹ 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: "⋀x. x ∈ standard_simplex (Suc p) ⟹ (λi. (∑j≤Suc p. l j i * x j)) ∈ 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 "(λi. (1 - u) * ((∑j≤Suc p. simplicial_vertex j f i) / (real p + 2)) + u * y i) ∈ S"
if "0 ≤ u" "u ≤ 1" and y: "y ∈ f ` standard_simplex (Suc p)" for y u
proof -
obtain x where x: "x ∈ standard_simplex (Suc p)" and yeq: "y = oriented_simplex (Suc p) l x"
using y feq by blast
have "(λi. ∑j≤Suc p. l j i * ((if j ≤ Suc p then (1 - u) * inverse (p + 2) + u * x j else 0))) ∈ S"
proof (rule l)
have "inverse (2 + real p) ≤ 1" "(2 + real p) * ((1 - u) * inverse (2 + real p)) + u = 1"
then show "(λj. if j ≤ Suc p then (1 - u) * inverse (real (p + 2)) + u * x j else 0) ∈ standard_simplex (Suc p)"
using x ‹0 ≤ u› ‹u ≤ 1›
by (simp add: sum.distrib standard_simplex_def linepath_le_1 flip: sum_distrib_left del: sum.atMost_Suc)
qed
moreover have "(λi. ∑j≤Suc p. l j i * ((1 - u) * inverse (2 + real p) + u * x j))
= (λi. (1 - u) * (∑j≤Suc p. l j i) / (real p + 2) + u * (∑j≤Suc p. l j i * x j))"
proof
fix i
have "(∑j≤Suc p. l j i * ((1 - u) * inverse (2 + real p) + u * x j))
= (∑j≤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 "… = (1 - u) * (∑j≤Suc p. l j i) / (real p + 2) + u * (∑j≤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
qed
ultimately show ?case
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
⟹ 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
using chain_boundary_simplicial_cone Suc
by (auto simp: chain_boundary_of frag_extend_diff simplicial_cone_def)
next
case (diff a b)
then show ?case
by (simp add: simplicial_subdivision_diff chain_boundary_diff frag_extend_diff)
qed auto
qed auto

text ‹A MESS AND USED ONLY ONCE›
lemma simplicial_subdivision_shrinks:
"⟦simplicial_chain p S c;
⋀f x y. ⟦f ∈ Poly_Mapping.keys c; x ∈ standard_simplex p; y ∈ standard_simplex p⟧ ⟹ ¦f x k - f y k¦ ≤ d;
f ∈ Poly_Mapping.keys(simplicial_subdivision p c);
x ∈ standard_simplex p; y ∈ standard_simplex p⟧
⟹ ¦f x k - f y k¦ ≤ (p / (Suc p)) * d"
proof (induction p arbitrary: d c f x y)
case (Suc p)
define Sigp where "Sigp ≡ λf::```