UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
lemma verts_ge_Suc0 : "\<not> [0..<length (g_V G)] = [0]"
lemma [code]: "Array.swap i x a = do { y \<leftarrow> Array.nth a i; Array.upd i x a; return y }"
lemma fps_shift_rev_shift: "m \<le> n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f" "m > n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))"
lemma ds_approx_empty[simp]: "[] \<lessapprox> []"
lemma wf_eq_check_append''[intro]: "wf\<^sub>s\<^sub>t V S \<Longrightarrow> wf\<^sub>s\<^sub>t V (S@[Equality Check t t'])"
lemma mem_read_reg_write: shows "mem_read (\<sigma> with ((r :=\<^sub>r w)#updates)) a si = mem_read (\<sigma> with updates) a si"
lemma sets_P[measurable_cong]: "x \<in> space (PiM {0..<n} M) \<Longrightarrow> sets (P n x) = sets (M n)"
lemma permute_flip_at: fixes a b c::"'a::at" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
lemma [def_pat_rules]: "hmstruct.hm_prio_of_op$prio \<equiv> PR_CONST hm_prio_of_op"
lemma \<nu>\<upsilon>_surj[meta_aux]: "surj \<nu>\<upsilon>"
lemma GF_advice_sync_lesseq: assumes "\<And>i. i \<le> n \<Longrightarrow> \<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>" assumes "\<exists>j. suffix (n + j) w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> j + n])[X]\<^sub>\<nu>" shows "\<exists>k \<ge> n. (\<forall>j \<le> n. suffix k w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> k])[X]\<^sub>\<nu>) \<and> suffix k w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> k])[X]\<^sub>\<nu>"
lemma payload_PairI: "x \<in> payload \<Longrightarrow> y \<in> payload \<Longrightarrow> Pair x y \<in> payload"
lemma Exp_Exp_Gen_inj: "Exp (Exp Gen X) X' = Z \<Longrightarrow> (\<exists> Y Y'. Z = Exp (Exp Gen Y) Y' \<and> ((X = Y \<and> X' = Y') \<or> (X = Y' \<and> X' = Y)))"
lemma line_convex_combination2: "(1 - u) \<^sub>R a + u \<^sub>R line a b i = line a b (i*u)"
lemma [simp]: "`!p \<cdot> !p = !p`"
lemma remove_cycles_cnt_id: "x \<noteq> y \<Longrightarrow> cnt y (remove_cycles xs x ys) \<le> cnt y ys + cnt y xs"
lemma iteration_star [simp]: "(x\<^sup>\<infinity>)\<^sup>\<star> = x\<^sup>\<infinity>"
lemma list_ex_simps [simp, code]: "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs" "list_ex P [] \<longleftrightarrow> False"
lemma SKIPp: "\<down>\<^sub>t(SKIP,s) = Suc 0"
lemma sup_absorb: "x \<squnion> x \<sqinter> y = x"
lemma path_image_subpath_gen: fixes g :: "_ \<Rightarrow> 'a::real_normed_vector" shows "path_image(subpath u v g) = g ` (closed_segment u v)"
lemma at_within_ball_bot_iff: fixes x y :: "'a::{real_normed_vector,perfect_space}" shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"
lemma compute_\<ff>s_inner_bounds: assumes "I_out_cb s (\<ff>s,ix,j)" assumes "j < length \<ff>s" assumes "I_in_cb s j i" shows "i-1 < length s" "j-1 < length s"
lemma (in domain) pirreducibleI: assumes "subring K R" "p \<in> carrier (K[X])" "p \<noteq> []" "p \<notin> Units (K[X])" and "\<And>q r. \<lbrakk> q \<in> carrier (K[X]); r \<in> carrier (K[X])\<rbrakk> \<Longrightarrow> p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])" shows "pirreducible K p"
lemma "real_of_dec (m, e) = int_of_integer m * 10 powr (int_of_integer e)"
lemma MSF_eq: "s.MSF E' = minimum_spanning_forest (ind E') (ind E)"
lemma well_base_bound: assumes "well_base M" and "\<forall>m \<in># M. m < n" shows "(\<Sum>m \<in># M. base ^ m) < base ^ n"
lemma inf_commutative: "-x \<sqinter> -y = -y \<sqinter> -x"
lemma finite_arg_max_eq_Max: assumes "finite (X :: 'c set)" "X \<noteq> {}" shows "(f :: 'c \<Rightarrow> real) (arg_max_on f X) = Max (f ` X)"
lemma inter_sorted_cons: "sorted (rev (x # xs)) \<Longrightarrow> inter_sorted_rev (x # xs) xs = xs"
lemma inverse_arrows_assoc: assumes "ide a" and "ide b" and "ide c" shows "inverse_arrows \<a>[a, b, c] \<a>\<^sup>-\<^sup>1[a, b, c]"
lemma ik\<^sub>s\<^sub>s\<^sub>t_trms\<^sub>s\<^sub>s\<^sub>t_subset: "ik\<^sub>s\<^sub>s\<^sub>t A \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t A"
lemma length_dupST [simp]: "length (mt_of (dupST sttp)) = 1"
lemma remove_unis_sentence: assumes inf_params: \<open>infinite (- params p)\<close> and \<open>closed 0 (put_unis m p)\<close> \<open>[] \<turnstile> put_unis m p\<close> shows \<open>[] \<turnstile> p\<close>
lemma Fix_lsl_left_slsided: "Fix \<nu> = {(x::'a::unital_quantale). lsd x}"
lemma coplanar_3: "coplanar {a,b,c}"
lemma inEIE_lessN[simp]: "e\<in>E \<or> e\<in>E\<inverse> \<Longrightarrow> case e of (u,v) \<Rightarrow> u<N \<and> v<N"
lemma Nonce_supply: "Nonce (SOME N. Nonce N \<notin> used evs) \<notin> used evs"
lemma ln_2_40_decimals: "\<bar>ln 2 - 0.6931471805599453094172321214581765680755\<bar> \<le> inverse (10^40 :: real)"
lemma prepend_domain: "a \<le> b \<Longrightarrow> x--a \<le> x--b"
lemma difibs:"(d\<inverse> \<union> f\<inverse> \<union> ov \<union> e \<union> f \<union> m \<union> b \<union> s\<inverse> \<union> s) O (b \<union> s \<union> m) \<subseteq> (b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse> \<union> d \<union> e \<union> s \<union> s\<inverse>)"
lemma is_sublist_eq: "distinct vs \<Longrightarrow> c \<noteq> y \<Longrightarrow> (nextElem vs c x = y) = is_sublist [x,y] vs"
lemma rank_senior_senior: "x \<le> n \<Longrightarrow> rank (senior x n) n = rank x n"
lemma n_star_import: assumes "n(y) * x \<le> x * n(y)" shows "n(y) * x\<^sup>\<star> = n(y) * (n(y) * x)\<^sup>\<star>"
lemma correctCompositionKS_subcomp2: assumes "correctCompositionKS C" and h1:"x \<in> subcomponents C" and "xa \<in> specSecrets C" shows "\<exists> y \<in> subcomponents C. xa \<in> specSecrets y"
lemma Gr_univalent[intro]: shows "univalent (BNF_Def.Gr A f)"
lemma inv_ik_dyn_add_chan2[elim!]: "\<lbrakk>dp2_add_chan2 s s' a1 i1 m; inv_ik_dyn s; terms_pkt m \<subseteq> synth (analz ik)\<rbrakk> \<Longrightarrow> inv_ik_dyn s'"
lemma AF_lfp: "\<^bold>A\<^bold>F p = lfp (\<lambda>s. p \<union> \<^bold>A\<^bold>X s)"
lemma conv_abscissa_1 [simp]: "conv_abscissa (1 :: 'a :: dirichlet_series fds) = -\<infinity>"
lemma dirichlet_prod_inverse': assumes "f 1 * i = 1" shows "dirichlet_prod (dirichlet_inverse f i) f = (\<lambda>n. if n = 1 then 1 else 0)"
lemma "\<^bold>E_residual_network": "\<^bold>E\<^bsub>residual_network f\<^esub> = \<^bold>E \<union> {(x, y). (y, x) \<in> \<^bold>E \<and> y \<noteq> source \<Delta>}"
lemma incrIndexList_help2[simp]: "incrIndexList ls m nmax \<Longrightarrow> hd ls = 0"
lemma "(1359::int) * -2468 = -3354012"
lemma ordD: "[\| ord p inff; \<not> inff R R'; R \<in> p; R' \<in> p \|] ==> inff R' R"
lemma Agent_Pair: "Agent X \<noteq> Pair X' Y'"
lemma equals_minim_Under: "\<lbrakk>B \<le> Field r; a \<in> B; a \<in> Under B\<rbrakk> \<Longrightarrow> a = minim B"
theorem finite_lderivs: "finite {\<guillemotleft>lderivs xs r\<guillemotright> \| xs . True}"
lemma mono_ctble_discont: fixes f :: "real \<Rightarrow> real" assumes "mono f" shows "countable {a. \<not> isCont f a}"
lemma distinct_lroft_s3: "\<lbrakk>distinct (map fst amr); distinct ifs\<rbrakk> \<Longrightarrow> distinct (lr_of_tran_s3 ifs amr)"
lemma lprodr_parametric [transfer_rule]: includes lifting_syntax shows "(rel_prod A (rel_prod B C) ===> rel_prod (rel_prod A B) C) lprodr lprodr"
lemma list_sub_implies_member: shows "\<forall> a x . set (a#x) \<subseteq> Z \<longrightarrow> a \<in> Z"
lemma rtb_rbl_ariths: "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys" "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys" "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs" "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
lemma fps_shift_times_fps_X_power: "n \<le> subdegree f \<Longrightarrow> fps_shift n f * fps_X ^ n = f"
lemma lsu_lax: "\<nu>\<^sup>\<natural> (x::'a::unital_quantale) \<cdot> \<nu>\<^sup>\<natural> y \<le> \<nu>\<^sup>\<natural> (x \<cdot> y)"
lemma eq_class_to_rel: assumes "(r, s) \<in> carrier R \<times> S" and "(r', s') \<in> carrier R \<times> S" and "(r \|\<^bsub>rel\<^esub> s) = (r' \|\<^bsub>rel\<^esub> s')" shows "(r, s) .=\<^bsub>rel\<^esub> (r', s')"
lemma lasso_run_rel_sv[relator_props]: "single_valued R \<Longrightarrow> single_valued (\<langle>R\<rangle>lasso_run_rel)"
lemma server_view1: "j \<in> J \<Longrightarrow> \<P>(\<omega> in \<PP>. visit H {j} \<omega>) = p_j"
lemma neq_Base_Object [simp]: "Base\<noteq>Object"
lemma "root 3 4 > sqrt (root 4 3) + \<lfloor>1/10 * root 3 7\<rfloor>"
lemma lesspoll_imp_lepoll: "A \<prec> B ==> A \<lesssim> B"
lemma in_snd_sym_preproc_addnewE: assumes "p \<in> set (snd (sym_preproc_addnew gs vs fs v))" assumes 1: "p \<in> set fs \<Longrightarrow> thesis" assumes 2: "\<And>g s. g \<in> set gs \<Longrightarrow> p = monom_mult 1 s g \<Longrightarrow> thesis" shows thesis
lemma measure_cases[cases type: measure]: obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
lemma tmap_id_id [id_simps]: "tmap id id = id"
lemma prime_nat_iff': "prime (p :: nat) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {2..<p}. \<not> n dvd p)"
lemma setv_col_subset_mat_elems: assumes "v \<in> set (cols M)" shows "set\<^sub>v v \<subseteq> elements_mat M"
lemma i_take_take_eq2: "n \<le> m \<Longrightarrow> (f \<Down> n) \<down> m = f \<Down> n"
lemma length_y_ty_list_vs[rule_format]: "\<forall>vs. lift_opts (map (\<lambda>(y, ty). L y) y_ty_list) = Some vs \<longrightarrow> length y_ty_list = length vs"
lemma 1: "2 < length a \<Longrightarrow> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"
lemma aux4_lm0241_prod_one: fixes f::"(nat \<Rightarrow> nat)" assumes "(\<forall>x. (1 \<ge> f x))" shows "(\<Prod>k \<le> n. (f k)) = 1 \<longrightarrow> (\<forall>k. k \<le> n \<longrightarrow> f k = 1)" (is "?P \<longrightarrow> ?Q")
lemma subst_clause'_nil[simp]: "subst_clause' [] m = {[]}"
lemma prove_nonneg_empty[simp]: "prove_nonneg prnt (Suc i) p slp []"
theorem gs1: "\<lbrakk> Pref P\<^sub>a P\<^sub>b; n = length P\<^sub>a \<rbrakk> \<Longrightarrow> \<exists>A. Gale_Shapley1 P\<^sub>a P\<^sub>b = Some A \<and> Pref.matching P\<^sub>a (list A) {<n} \<and> Pref.stable P\<^sub>a P\<^sub>b (list A) {<n} \<and> Pref.opti\<^sub>a P\<^sub>a P\<^sub>b (list A)"
lemma strict_mono_sets_subset: assumes "strict_mono_sets B f" "A \<subseteq> B" shows "strict_mono_sets A f"
lemma dtail_notelem_eq_def: assumes "e \<notin> darcs t" shows "dtail t def e = def e"
lemma vrat_identity_law_multiplication: assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x *\<^sub>\<rat> 1\<^sub>\<rat> = x"
lemma sd_exists: assumes has3A: "has 3 A" and finiteIs: "finite Is" and twoIs: "has 2 Is" and iia: "iia swf A Is" and swf: "SWF swf A Is universal_domain" and wp: "weak_pareto swf A Is universal_domain" shows "\<exists>j u v. hasw [u,v] A \<and> semidecisive swf A Is {j} u v"
lemma has_field_derivative_stirling_sum'_real [derivative_intros]: assumes "j > 0" "x > (0::real)" shows "(stirling_sum' j m has_field_derivative stirling_sum' (Suc j) m x) (at x)"
lemma lrange_sorted_split: assumes "Laligned (Node ts t) u" and "sorted_less (leaves (Node ts t))" and "split ts x = (ls, rs)" shows "lrange_filter x (leaves (Node ts t)) = lrange_filter x (leaves_list rs @ leaves t)"
lemma SBWriteNonVolatile': "\<lbrakk> sb'= sb@ [Write\<^sub>s\<^sub>b False a (D,f) (f \<theta>) A L R W]\<rbrakk> \<Longrightarrow> (Write False a (D,f) A L R W#is,\<theta>, sb, m, ghst) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb', m, ghst)"
lemma of_bool_nth: "of_bool (bit x v) = (x >> v) AND 1" for x :: \<open>'a::len word\<close>
lemma octo_mult_cnj_conv_norm: "x * cnj x = octo_of_real (norm x) ^ 2"
lemma (in discrete) carrier: "carrier = X"
lemma drop_bit_negative_int_iff [simp]: \<open>drop_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
lemma extreme_bound_mono: assumes XY: "X \<subseteq> Y" and sX: "extreme_bound X sX" and sY: "extreme_bound Y sY" shows "sX \<sqsubseteq> sY"
lemma aext_not [alpha]: "(\<not> P) \<oplus>\<^sub>p x = (\<not> (P \<oplus>\<^sub>p x))"
lemma move_matrix_le_move_matrix_iff[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= move_matrix B j i) = (A <= (B::('a::{order,zero}) matrix))"
lemma desargues_config_3D_coplanar_5 : assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, C, \<alpha>, \<beta>} = 3" and "rk {A', B', C', \<alpha>, \<beta>} = 3"
lemma has_field_derivative_Lambert_W [derivative_intros]: assumes x: "x > -exp (-1)" shows "(Lambert_W has_real_derivative inverse (x + exp (Lambert_W x))) (at x within A)"
lemma(in padic_integers) ord_Zp_p: "ord_Zp \<p> = (1::int)"
lemma path_splitting_2: "path_splitting_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_splitting_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x (p[[y]]) p0 \<and> ((p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * (p[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"

lemma verts_ge_Suc0 : "\<not> [0..<length (g_V G)] = [0]"

lemma [code]: "Array.swap i x a = do { y \<leftarrow> Array.nth a i; Array.upd i x a; return y }"

lemma fps_shift_rev_shift: "m \<le> n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f" "m > n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))"

lemma ds_approx_empty[simp]: "[] \<lessapprox> []"

lemma wf_eq_check_append''[intro]: "wf\<^sub>s\<^sub>t V S \<Longrightarrow> wf\<^sub>s\<^sub>t V (S@[Equality Check t t'])"

lemma mem_read_reg_write: shows "mem_read (\<sigma> with ((r :=\<^sub>r w)#updates)) a si = mem_read (\<sigma> with updates) a si"

lemma sets_P[measurable_cong]: "x \<in> space (PiM {0..<n} M) \<Longrightarrow> sets (P n x) = sets (M n)"

lemma permute_flip_at: fixes a b c::"'a::at" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"

lemma [def_pat_rules]: "hmstruct.hm_prio_of_op$prio \<equiv> PR_CONST hm_prio_of_op"

lemma \<nu>\<upsilon>_surj[meta_aux]: "surj \<nu>\<upsilon>"

lemma GF_advice_sync_lesseq: assumes "\<And>i. i \<le> n \<Longrightarrow> \<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>" assumes "\<exists>j. suffix (n + j) w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> j + n])[X]\<^sub>\<nu>" shows "\<exists>k \<ge> n. (\<forall>j \<le> n. suffix k w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> k])[X]\<^sub>\<nu>) \<and> suffix k w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> k])[X]\<^sub>\<nu>"

lemma payload_PairI: "x \<in> payload \<Longrightarrow> y \<in> payload \<Longrightarrow> Pair x y \<in> payload"

lemma Exp_Exp_Gen_inj: "Exp (Exp Gen X) X' = Z \<Longrightarrow> (\<exists> Y Y'. Z = Exp (Exp Gen Y) Y' \<and> ((X = Y \<and> X' = Y') \<or> (X = Y' \<and> X' = Y)))"

lemma line_convex_combination2: "(1 - u) *\<^sub>R a + u *\<^sub>R line a b i = line a b (i*u)"

lemma [simp]: "`!p \<cdot> !p = !p`"

lemma remove_cycles_cnt_id: "x \<noteq> y \<Longrightarrow> cnt y (remove_cycles xs x ys) \<le> cnt y ys + cnt y xs"

lemma iteration_star [simp]: "(x\<^sup>\<infinity>)\<^sup>\<star> = x\<^sup>\<infinity>"

lemma list_ex_simps [simp, code]: "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs" "list_ex P [] \<longleftrightarrow> False"

lemma SKIPp: "\<down>\<^sub>t(SKIP,s) = Suc 0"

lemma sup_absorb: "x \<squnion> x \<sqinter> y = x"

lemma path_image_subpath_gen: fixes g :: "_ \<Rightarrow> 'a::real_normed_vector" shows "path_image(subpath u v g) = g ` (closed_segment u v)"

lemma at_within_ball_bot_iff: fixes x y :: "'a::{real_normed_vector,perfect_space}" shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"

lemma compute_\<ff>s_inner_bounds: assumes "I_out_cb s (\<ff>s,ix,j)" assumes "j < length \<ff>s" assumes "I_in_cb s j i" shows "i-1 < length s" "j-1 < length s"

lemma (in domain) pirreducibleI: assumes "subring K R" "p \<in> carrier (K[X])" "p \<noteq> []" "p \<notin> Units (K[X])" and "\<And>q r. \<lbrakk> q \<in> carrier (K[X]); r \<in> carrier (K[X])\<rbrakk> \<Longrightarrow> p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])" shows "pirreducible K p"

lemma "real_of_dec (m, e) = int_of_integer m * 10 powr (int_of_integer e)"

lemma MSF_eq: "s.MSF E' = minimum_spanning_forest (ind E') (ind E)"

lemma well_base_bound: assumes "well_base M" and "\<forall>m \<in># M. m < n" shows "(\<Sum>m \<in># M. base ^ m) < base ^ n"

lemma inf_commutative: "-x \<sqinter> -y = -y \<sqinter> -x"

lemma finite_arg_max_eq_Max: assumes "finite (X :: 'c set)" "X \<noteq> {}" shows "(f :: 'c \<Rightarrow> real) (arg_max_on f X) = Max (f ` X)"

lemma inter_sorted_cons: "sorted (rev (x # xs)) \<Longrightarrow> inter_sorted_rev (x # xs) xs = xs"

lemma inverse_arrows_assoc: assumes "ide a" and "ide b" and "ide c" shows "inverse_arrows \<a>[a, b, c] \<a>\<^sup>-\<^sup>1[a, b, c]"

lemma ik\<^sub>s\<^sub>s\<^sub>t_trms\<^sub>s\<^sub>s\<^sub>t_subset: "ik\<^sub>s\<^sub>s\<^sub>t A \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t A"

lemma length_dupST [simp]: "length (mt_of (dupST sttp)) = 1"

lemma remove_unis_sentence: assumes inf_params: \<open>infinite (- params p)\<close> and \<open>closed 0 (put_unis m p)\<close> \<open>[] \<turnstile> put_unis m p\<close> shows \<open>[] \<turnstile> p\<close>

lemma Fix_lsl_left_slsided: "Fix \<nu> = {(x::'a::unital_quantale). lsd x}"

lemma coplanar_3: "coplanar {a,b,c}"

lemma inEIE_lessN[simp]: "e\<in>E \<or> e\<in>E\<inverse> \<Longrightarrow> case e of (u,v) \<Rightarrow> u<N \<and> v<N"

lemma Nonce_supply: "Nonce (SOME N. Nonce N \<notin> used evs) \<notin> used evs"

lemma ln_2_40_decimals: "\<bar>ln 2 - 0.6931471805599453094172321214581765680755\<bar> \<le> inverse (10^40 :: real)"

lemma prepend_domain: "a \<le> b \<Longrightarrow> x--a \<le> x--b"

lemma difibs:"(d\<inverse> \<union> f\<inverse> \<union> ov \<union> e \<union> f \<union> m \<union> b \<union> s\<inverse> \<union> s) O (b \<union> s \<union> m) \<subseteq> (b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse> \<union> d \<union> e \<union> s \<union> s\<inverse>)"

lemma is_sublist_eq: "distinct vs \<Longrightarrow> c \<noteq> y \<Longrightarrow> (nextElem vs c x = y) = is_sublist [x,y] vs"

lemma rank_senior_senior: "x \<le> n \<Longrightarrow> rank (senior x n) n = rank x n"

lemma n_star_import: assumes "n(y) * x \<le> x * n(y)" shows "n(y) * x\<^sup>\<star> = n(y) * (n(y) * x)\<^sup>\<star>"

lemma correctCompositionKS_subcomp2: assumes "correctCompositionKS C" and h1:"x \<in> subcomponents C" and "xa \<in> specSecrets C" shows "\<exists> y \<in> subcomponents C. xa \<in> specSecrets y"

lemma Gr_univalent[intro]: shows "univalent (BNF_Def.Gr A f)"

lemma inv_ik_dyn_add_chan2[elim!]: "\<lbrakk>dp2_add_chan2 s s' a1 i1 m; inv_ik_dyn s; terms_pkt m \<subseteq> synth (analz ik)\<rbrakk> \<Longrightarrow> inv_ik_dyn s'"

lemma AF_lfp: "\<^bold>A\<^bold>F p = lfp (\<lambda>s. p \<union> \<^bold>A\<^bold>X s)"

lemma conv_abscissa_1 [simp]: "conv_abscissa (1 :: 'a :: dirichlet_series fds) = -\<infinity>"

lemma dirichlet_prod_inverse': assumes "f 1 * i = 1" shows "dirichlet_prod (dirichlet_inverse f i) f = (\<lambda>n. if n = 1 then 1 else 0)"

lemma "\<^bold>E_residual_network": "\<^bold>E\<^bsub>residual_network f\<^esub> = \<^bold>E \<union> {(x, y). (y, x) \<in> \<^bold>E \<and> y \<noteq> source \<Delta>}"

lemma incrIndexList_help2[simp]: "incrIndexList ls m nmax \<Longrightarrow> hd ls = 0"

lemma "(1359::int) * -2468 = -3354012"

lemma ordD: "[| ord p inff; \<not> inff R R'; R \<in> p; R' \<in> p |] ==> inff R' R"

lemma Agent_Pair: "Agent X \<noteq> Pair X' Y'"

lemma equals_minim_Under: "\<lbrakk>B \<le> Field r; a \<in> B; a \<in> Under B\<rbrakk> \<Longrightarrow> a = minim B"

theorem finite_lderivs: "finite {\<guillemotleft>lderivs xs r\<guillemotright> | xs . True}"

lemma mono_ctble_discont: fixes f :: "real \<Rightarrow> real" assumes "mono f" shows "countable {a. \<not> isCont f a}"

lemma distinct_lroft_s3: "\<lbrakk>distinct (map fst amr); distinct ifs\<rbrakk> \<Longrightarrow> distinct (lr_of_tran_s3 ifs amr)"

lemma lprodr_parametric [transfer_rule]: includes lifting_syntax shows "(rel_prod A (rel_prod B C) ===> rel_prod (rel_prod A B) C) lprodr lprodr"

lemma list_sub_implies_member: shows "\<forall> a x . set (a#x) \<subseteq> Z \<longrightarrow> a \<in> Z"

lemma rtb_rbl_ariths: "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys" "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys" "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs" "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"

lemma fps_shift_times_fps_X_power: "n \<le> subdegree f \<Longrightarrow> fps_shift n f * fps_X ^ n = f"

lemma lsu_lax: "\<nu>\<^sup>\<natural> (x::'a::unital_quantale) \<cdot> \<nu>\<^sup>\<natural> y \<le> \<nu>\<^sup>\<natural> (x \<cdot> y)"

lemma eq_class_to_rel: assumes "(r, s) \<in> carrier R \<times> S" and "(r', s') \<in> carrier R \<times> S" and "(r |\<^bsub>rel\<^esub> s) = (r' |\<^bsub>rel\<^esub> s')" shows "(r, s) .=\<^bsub>rel\<^esub> (r', s')"

lemma lasso_run_rel_sv[relator_props]: "single_valued R \<Longrightarrow> single_valued (\<langle>R\<rangle>lasso_run_rel)"

lemma server_view1: "j \<in> J \<Longrightarrow> \<P>(\<omega> in \<PP>. visit H {j} \<omega>) = p_j"

lemma neq_Base_Object [simp]: "Base\<noteq>Object"

lemma "root 3 4 > sqrt (root 4 3) + \<lfloor>1/10 * root 3 7\<rfloor>"

lemma lesspoll_imp_lepoll: "A \<prec> B ==> A \<lesssim> B"

lemma in_snd_sym_preproc_addnewE: assumes "p \<in> set (snd (sym_preproc_addnew gs vs fs v))" assumes 1: "p \<in> set fs \<Longrightarrow> thesis" assumes 2: "\<And>g s. g \<in> set gs \<Longrightarrow> p = monom_mult 1 s g \<Longrightarrow> thesis" shows thesis

lemma measure_cases[cases type: measure]: obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"

lemma tmap_id_id [id_simps]: "tmap id id = id"

lemma prime_nat_iff': "prime (p :: nat) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {2..<p}. \<not> n dvd p)"

lemma setv_col_subset_mat_elems: assumes "v \<in> set (cols M)" shows "set\<^sub>v v \<subseteq> elements_mat M"

lemma i_take_take_eq2: "n \<le> m \<Longrightarrow> (f \<Down> n) \<down> m = f \<Down> n"

lemma length_y_ty_list_vs[rule_format]: "\<forall>vs. lift_opts (map (\<lambda>(y, ty). L y) y_ty_list) = Some vs \<longrightarrow> length y_ty_list = length vs"

lemma 1: "2 < length a \<Longrightarrow> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"

lemma aux4_lm0241_prod_one: fixes f::"(nat \<Rightarrow> nat)" assumes "(\<forall>x. (1 \<ge> f x))" shows "(\<Prod>k \<le> n. (f k)) = 1 \<longrightarrow> (\<forall>k. k \<le> n \<longrightarrow> f k = 1)" (is "?P \<longrightarrow> ?Q")

lemma subst_clause'_nil[simp]: "subst_clause' [] m = {[]}"

lemma prove_nonneg_empty[simp]: "prove_nonneg prnt (Suc i) p slp []"

theorem gs1: "\<lbrakk> Pref P\<^sub>a P\<^sub>b; n = length P\<^sub>a \<rbrakk> \<Longrightarrow> \<exists>A. Gale_Shapley1 P\<^sub>a P\<^sub>b = Some A \<and> Pref.matching P\<^sub>a (list A) {<n} \<and> Pref.stable P\<^sub>a P\<^sub>b (list A) {<n} \<and> Pref.opti\<^sub>a P\<^sub>a P\<^sub>b (list A)"

lemma strict_mono_sets_subset: assumes "strict_mono_sets B f" "A \<subseteq> B" shows "strict_mono_sets A f"

lemma dtail_notelem_eq_def: assumes "e \<notin> darcs t" shows "dtail t def e = def e"

lemma vrat_identity_law_multiplication: assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x *\<^sub>\<rat> 1\<^sub>\<rat> = x"

lemma sd_exists: assumes has3A: "has 3 A" and finiteIs: "finite Is" and twoIs: "has 2 Is" and iia: "iia swf A Is" and swf: "SWF swf A Is universal_domain" and wp: "weak_pareto swf A Is universal_domain" shows "\<exists>j u v. hasw [u,v] A \<and> semidecisive swf A Is {j} u v"

lemma has_field_derivative_stirling_sum'_real [derivative_intros]: assumes "j > 0" "x > (0::real)" shows "(stirling_sum' j m has_field_derivative stirling_sum' (Suc j) m x) (at x)"

lemma lrange_sorted_split: assumes "Laligned (Node ts t) u" and "sorted_less (leaves (Node ts t))" and "split ts x = (ls, rs)" shows "lrange_filter x (leaves (Node ts t)) = lrange_filter x (leaves_list rs @ leaves t)"

lemma SBWriteNonVolatile': "\<lbrakk> sb'= sb@ [Write\<^sub>s\<^sub>b False a (D,f) (f \<theta>) A L R W]\<rbrakk> \<Longrightarrow> (Write False a (D,f) A L R W#is,\<theta>, sb, m, ghst) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb', m, ghst)"

lemma of_bool_nth: "of_bool (bit x v) = (x >> v) AND 1" for x :: \<open>'a::len word\<close>

lemma octo_mult_cnj_conv_norm: "x * cnj x = octo_of_real (norm x) ^ 2"

lemma (in discrete) carrier: "carrier = X"

lemma drop_bit_negative_int_iff [simp]: \<open>drop_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int

lemma extreme_bound_mono: assumes XY: "X \<subseteq> Y" and sX: "extreme_bound X sX" and sY: "extreme_bound Y sY" shows "sX \<sqsubseteq> sY"

lemma aext_not [alpha]: "(\<not> P) \<oplus>\<^sub>p x = (\<not> (P \<oplus>\<^sub>p x))"

lemma move_matrix_le_move_matrix_iff[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= move_matrix B j i) = (A <= (B::('a::{order,zero}) matrix))"

lemma desargues_config_3D_coplanar_5 : assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, C, \<alpha>, \<beta>} = 3" and "rk {A', B', C', \<alpha>, \<beta>} = 3"

lemma has_field_derivative_Lambert_W [derivative_intros]: assumes x: "x > -exp (-1)" shows "(Lambert_W has_real_derivative inverse (x + exp (Lambert_W x))) (at x within A)"

lemma(in padic_integers) ord_Zp_p: "ord_Zp \<p> = (1::int)"

lemma path_splitting_2: "path_splitting_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_splitting_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x (p[[y]]) p0 \<and> ((p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * (p[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"