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| section \<open>CCW Vector Space\<close> | |
| theory Counterclockwise_Vector | |
| imports Counterclockwise | |
| begin | |
| locale ccw_vector_space = ccw_system12 ccw S for ccw::"'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" and S + | |
| assumes translate_plus[simp]: "ccw (a + x) (b + x) (c + x) \<longleftrightarrow> ccw a b c" | |
| assumes scaleR1_eq[simp]: "0 < e \<Longrightarrow> ccw 0 (e*\<^sub>Ra) b = ccw 0 a b" | |
| assumes uminus1[simp]: "ccw 0 (-a) b = ccw 0 b a" | |
| assumes add1: "ccw 0 a b \<Longrightarrow> ccw 0 c b \<Longrightarrow> ccw 0 (a + c) b" | |
| begin | |
| lemma translate_plus'[simp]: | |
| "ccw (x + a) (x + b) (x + c) \<longleftrightarrow> ccw a b c" | |
| by (auto simp: ac_simps) | |
| lemma uminus2[simp]: "ccw 0 a (- b) = ccw 0 b a" | |
| by (metis minus_minus uminus1) | |
| lemma uminus_all[simp]: "ccw (-a) (-b) (-c) \<longleftrightarrow> ccw a b c" | |
| proof - | |
| have "ccw (-a) (-b) (-c) \<longleftrightarrow> ccw 0 (- (b - a)) (- (c - a))" | |
| using translate_plus[of "-a" a "-b" "-c"] | |
| by simp | |
| also have "\<dots> \<longleftrightarrow> ccw 0 (b - a) (c - a)" | |
| by (simp del: minus_diff_eq) | |
| also have "\<dots> \<longleftrightarrow> ccw a b c" | |
| using translate_plus[of a "-a" b c] | |
| by simp | |
| finally show ?thesis . | |
| qed | |
| lemma translate_origin: "NO_MATCH 0 p \<Longrightarrow> ccw p q r \<longleftrightarrow> ccw 0 (q - p) (r - p)" | |
| using translate_plus[of p "- p" q r] | |
| by simp | |
| lemma translate[simp]: "ccw a (a + b) (a + c) \<longleftrightarrow> ccw 0 b c" | |
| by (simp add: translate_origin) | |
| lemma translate_plus3: "ccw (a - x) (b - x) c \<longleftrightarrow> ccw a b (c + x)" | |
| using translate_plus[of a "-x" b "c + x"] by simp | |
| lemma renormalize: | |
| "ccw 0 (a - b) (c - a) \<Longrightarrow> ccw b a c" | |
| by (metis diff_add_cancel diff_self cyclic minus_diff_eq translate_plus3 uminus1) | |
| lemma cyclicI: "ccw p q r \<Longrightarrow> ccw q r p" | |
| by (metis cyclic) | |
| lemma | |
| scaleR2_eq[simp]: | |
| "0 < e \<Longrightarrow> ccw 0 xr (e *\<^sub>R P) \<longleftrightarrow> ccw 0 xr P" | |
| using scaleR1_eq[of e "-P" xr] | |
| by simp | |
| lemma scaleR1_nonzero_eq: | |
| "e \<noteq> 0 \<Longrightarrow> ccw 0 (e *\<^sub>R a) b = (if e > 0 then ccw 0 a b else ccw 0 b a)" | |
| proof cases | |
| assume "e < 0" | |
| define e' where "e' = - e" | |
| hence "e = -e'" "e' > 0" using \<open>e < 0\<close> by simp_all | |
| thus ?thesis by simp | |
| qed simp | |
| lemma neg_scaleR[simp]: "x < 0 \<Longrightarrow> ccw 0 (x *\<^sub>R b) c \<longleftrightarrow> ccw 0 c b" | |
| using scaleR1_nonzero_eq by auto | |
| lemma | |
| scaleR1: | |
| "0 < e \<Longrightarrow> ccw 0 xr P \<Longrightarrow> ccw 0 (e *\<^sub>R xr) P" | |
| by simp | |
| lemma | |
| add3: "ccw 0 a b \<and> ccw 0 a c \<Longrightarrow> ccw 0 a (b + c)" | |
| using add1[of "-b" a "-c"] uminus1[of "b + c" a] | |
| by simp | |
| lemma add3_self[simp]: "ccw 0 p (p + q) \<longleftrightarrow> ccw 0 p q" | |
| using translate[of "-p" p "p + q"] | |
| apply (simp add: cyclic) | |
| apply (metis cyclic uminus2) | |
| done | |
| lemma add2_self[simp]: "ccw 0 (p + q) p \<longleftrightarrow> ccw 0 q p" | |
| using translate[of "-p" "p + q" p] | |
| apply simp | |
| apply (metis cyclic uminus1) | |
| done | |
| lemma scale_add3[simp]: "ccw 0 a (x *\<^sub>R a + b) \<longleftrightarrow> ccw 0 a b" | |
| proof - | |
| { | |
| assume "x = 0" | |
| hence ?thesis by simp | |
| } moreover { | |
| assume "x > 0" | |
| hence ?thesis using add3_self scaleR1_eq by blast | |
| } moreover { | |
| assume "x < 0" | |
| define x' where "x' = - x" | |
| hence "x = -x'" "x' > 0" using \<open>x < 0\<close> by simp_all | |
| hence "ccw 0 a (x *\<^sub>R a + b) = ccw 0 (x' *\<^sub>R a + - b) (x' *\<^sub>R a)" | |
| by (subst uminus1[symmetric]) simp | |
| also have "\<dots> = ccw 0 (- b) a" | |
| unfolding add2_self by (simp add: \<open>x' > 0\<close>) | |
| also have "\<dots> = ccw 0 a b" | |
| by simp | |
| finally have ?thesis . | |
| } ultimately show ?thesis by arith | |
| qed | |
| lemma scale_add3'[simp]: "ccw 0 a (b + x *\<^sub>R a) \<longleftrightarrow> ccw 0 a b" | |
| and scale_minus3[simp]: "ccw 0 a (x *\<^sub>R a - b) \<longleftrightarrow> ccw 0 b a" | |
| and scale_minus3'[simp]: "ccw 0 a (b - x *\<^sub>R a) \<longleftrightarrow> ccw 0 a b" | |
| using | |
| scale_add3[of a x b] | |
| scale_add3[of a "-x" b] | |
| scale_add3[of a x "-b"] | |
| by (simp_all add: ac_simps) | |
| lemma sum: | |
| assumes fin: "finite X" | |
| assumes ne: "X\<noteq>{}" | |
| assumes ncoll: "(\<And>x. x \<in> X \<Longrightarrow> ccw 0 a (f x))" | |
| shows "ccw 0 a (sum f X)" | |
| proof - | |
| from ne obtain x where "x \<in> X" "insert x X = X" by auto | |
| have "ccw 0 a (sum f (insert x X))" | |
| using fin ncoll | |
| proof (induction X) | |
| case empty thus ?case using \<open>x \<in> X\<close> ncoll | |
| by auto | |
| next | |
| case (insert y F) | |
| hence "ccw 0 a (sum f (insert y (insert x F)))" | |
| by (cases "y = x") (auto intro!: add3) | |
| thus ?case | |
| by (simp add: insert_commute) | |
| qed | |
| thus ?thesis using \<open>insert x X = X\<close> by simp | |
| qed | |
| lemma sum2: | |
| assumes fin: "finite X" | |
| assumes ne: "X\<noteq>{}" | |
| assumes ncoll: "(\<And>x. x \<in> X \<Longrightarrow> ccw 0 (f x) a)" | |
| shows "ccw 0 (sum f X) a" | |
| using sum[OF assms(1,2), of "-a" f] ncoll | |
| by simp | |
| lemma translate_minus[simp]: | |
| "ccw (x - a) (x - b) (x - c) = ccw (-a) (-b) (-c)" | |
| using translate_plus[of "-a" x "-b" "-c"] | |
| by simp | |
| end | |
| locale ccw_convex = ccw_system ccw S for ccw and S::"'a::real_vector set" + | |
| fixes oriented | |
| assumes convex2: | |
| "u \<ge> 0 \<Longrightarrow> v \<ge> 0 \<Longrightarrow> u + v = 1 \<Longrightarrow> ccw a b c \<Longrightarrow> ccw a b d \<Longrightarrow> oriented a b \<Longrightarrow> | |
| ccw a b (u *\<^sub>R c + v *\<^sub>R d)" | |
| begin | |
| lemma convex_hull: | |
| assumes [intro, simp]: "finite C" | |
| assumes ccw: "\<And>c. c \<in> C \<Longrightarrow> ccw a b c" | |
| assumes ch: "x \<in> convex hull C" | |
| assumes oriented: "oriented a b" | |
| shows "ccw a b x" | |
| proof - | |
| define D where "D = C" | |
| have D: "C \<subseteq> D" "\<And>c. c \<in> D \<Longrightarrow> ccw a b c" by (simp_all add: D_def ccw) | |
| show "ccw a b x" | |
| using \<open>finite C\<close> D ch | |
| proof (induct arbitrary: x) | |
| case empty thus ?case by simp | |
| next | |
| case (insert c C) | |
| hence "C \<subseteq> D" by simp | |
| { | |
| assume "C = {}" | |
| hence ?case | |
| using insert | |
| by simp | |
| } moreover { | |
| assume "C \<noteq> {}" | |
| from convex_hull_insert[OF this, of c] insert(6) | |
| obtain u v d where "u \<ge> 0" "v \<ge> 0" "d \<in> convex hull C" "u + v = 1" | |
| and x: "x = u *\<^sub>R c + v *\<^sub>R d" | |
| by blast | |
| have "ccw a b d" | |
| by (auto intro: insert.hyps(3)[OF \<open>C \<subseteq> D\<close>] insert.prems \<open>d \<in> convex hull C\<close>) | |
| from insert | |
| have "ccw a b c" | |
| by simp | |
| from convex2[OF \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> \<open>u + v = 1\<close> \<open>ccw a b c\<close> \<open>ccw a b d\<close> \<open>oriented a b\<close>] | |
| have ?case by (simp add: x) | |
| } ultimately show ?case by blast | |
| qed | |
| qed | |
| end | |
| end | |