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

Statement: stringlengths 7 24.3k
lemma conf_hext: "\<lbrakk> h \<unlhd> h'; P,h \<turnstile> v :\<le> T \<rbrakk> \<Longrightarrow> P,h' \<turnstile> v :\<le> T"
lemma resid_Arr_Src [simp]: assumes "Arr t" shows "t \\ Src t = t"
lemma bin_fst_nemp: "u\<^sub>0 \<noteq> \<epsilon>" and bin_snd_nemp: "u\<^sub>1 \<noteq> \<epsilon>"
lemma (in Group) nsg1:"\<lbrakk>G \<guillemotright> H; b \<in> carrier G; h \<in> H; \<forall>a\<in> carrier G. \<forall>h\<in>H. (a \<cdot> h)\<cdot> (\<rho> a) \<in> H\<rbrakk> \<Longrightarrow> b \<cdot> h \<cdot> (\<rho> b) \<in> H"
lemma ln_exp_cf6_lower_bound_pos: assumes "0\<le>x" shows "ln (exp_cf6 x) \<le> x"
lemma next_performopD: assumes "next_performop' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self" shows "\<Delta>mint_msg \<noteq> {#} \<or> zmset_of \<Delta>mint_self - zmset_of \<Delta>neg \<noteq> {#}\<^sub>z" "\<forall>t. int (count \<Delta>neg t) \<le> zcount (c_caps c0 p) t" "minting_self (c_caps c0 p) \<Delta>mint_self" "minting_msg (c_caps c0 p) \<Delta>mint_msg" "c_temp c1 = (c_temp c0)(p := c_temp c0 p + (timestamps (zmset_of \<Delta>mint_msg) + zmset_of \<Delta>mint_self - zmset_of \<Delta>neg))" "c_msg c1 = c_msg c0" "c_glob c1 = c_glob c0" "c_data_msg c1 = c_data_msg c0 + \<Delta>mint_msg" "c_caps c1 = (c_caps c0)(p := c_caps c0 p + (zmset_of \<Delta>mint_self - zmset_of \<Delta>neg))"
lemma some_trg_composable: assumes "arr \<sigma>" shows "\<mu> \<star> \<sigma> \<noteq> null \<longleftrightarrow> \<mu> \<star> some_trg \<sigma> \<noteq> null"
lemma tail_correct: assumes "ft_invar t" "t \<noteq> Empty" shows "toList (tail t) = tl (toList t)" and "ft_invar (tail t)"
lemma "\<turnstile>\<^sub>1 { %l s. True } strip ( (a ::= (N 1)) ;; b ::= (V a) ) {time ( (a ::= (N 1)) ;; b ::= (V a) ) \<Down> %l s. s b = 1}"
lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
lemma AxiomP_sf [iff]: "Sigma_fm (AxiomP t)"
lemma ltrm_of_X_minus: assumes "a \<in> carrier R" assumes "\<one> \<noteq>\<zero>" shows "ltrm (X_minus a) = X"
theorem Extend_subset: \<open>S \<subseteq> Extend S C f\<close>
lemma quantale_hom_closed_map: fixes f :: "'a::quantale_with_dual \<Rightarrow> 'b::quantale_with_dual" shows "(f \<in> quantale_homset) \<Longrightarrow> (radj f \<in> quantale_closed_maps)"
lemma ls_trans_r: assumes "locally_sym R (A + B)" shows "locally_sym R B"
lemma \<nu>\<kappa>_proper[meta_aux]: "proper (x\<^sup>P)"
lemma listset_empty_iff: "listset xs = {} \<longleftrightarrow> {} \<in> set xs"
lemma no_path_12[simp]: "\<not> task.path 1 2 pth"
lemma ineM_Un1: assumes "ineM P A E" shows "ineM P (A Un B) E"
lemma assumes "\<And>x y. P (y + x)" shows "\<And>x y. P (x + y :: nat)"
lemma decide_origin: assumes send: "\<forall>p. \<mu> p \<in> get_msgs (send1 r) cfg (HOs r) (HOs r) p" and step: "\<forall>p. next1 r p (cfg p) (\<mu> p) (cfg' p)" and step_r: "two_step r = Suc 0" shows "D cfg cfg' \<subseteq> {p. \<exists>v. decide (cfg' p) = Some v \<and> (\<forall>q \<in> HOs r p. vote (cfg q) = Some v)}"
lemma [code]: "toy_class_raw.truncate = toy_class_raw_rec (co4 K toy_class_raw.make)"
lemma pointbased_mk_regular2: "pointbased (mk_regular2 P F)"
lemma path_components_eq_connected_components_of: "locally_path_connected_space X \<Longrightarrow> (path_components_of X = connected_components_of X)"
lemma test_bremdup1_refine_aux: "(test_bremdup1, my_bremdup_impl_loc.test_remdup) \<in> [my_bremdup_impl_loc]\<^sub>a nat_assn\<^sup>k \<rightarrow> nat_assn"
lemma has_integral_vec1_D: fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector" assumes "((f \<circ> vec) has_integral y) ((\<lambda>x. x $ 1) ` S)" shows "(f has_integral y) S"
lemma recurrent_iff_U_eq_1: "recurrent s \<longleftrightarrow> U s s = 1"
lemma goes_wrong_denot_empty: assumes gw: "goes_wrong e" and fv_e: "FV e = {}" shows "E e [] = {}"
lemma "Thm (TYPE(bool)) 1 (Imp (All (FIn 0 1)) (FExists (FIn 0 1)))"
lemma Strong_Soundness_aux: "\<lbrakk> (Some c, s) -*\<rightarrow> (co, t); s \<in> pre c; \<turnstile> c q \<rbrakk> \<Longrightarrow> if co = None then t \<in> q else t \<in> pre (the co) \<and> \<turnstile> (the co) q"
lemma (in CRR_market) assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" shows bernoulli_gen_filtration: "filtration N G" and bernoulli_sigma_finite: "\<forall>n. sigma_finite_subalgebra N (G n)"
lemma chine_in_hom [intro]: shows "\<guillemotleft>chine : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright>" and "\<guillemotleft>chine: chine \<Rightarrow> chine\<guillemotright>"
lemma CCfix_restr: assumes "domA \<Gamma> \<subseteq> S" shows "cc_restr S (CCfix \<Gamma>\<cdot>(ae, G)) = cc_restr S (CCfix \<Gamma>\<cdot>(ae f\|` S, cc_restr S G))"
lemma gp: assumes a0: "a$0 = (0::'a::field)" shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)" (is "?one oo a = _")
lemma generate_maskedDB_elim: "\<lbrakk>roundup emBits 8 * 8 - emBits \<le> length x; ( roundup emBits 8) * 8 - (length (sha1 M)) - 8 = length (maskedDB_zero x emBits)\<rbrakk> \<Longrightarrow> generate_maskedDB (maskedDB_zero x emBits @ y @ z) emBits (length(sha1 M)) = maskedDB_zero x emBits"
lemma sameDom_swapBinp_gSwapBinp[simp]: assumes "wlsBinp delta binp'" and "gWlsBinp MOD delta binp" shows "sameDom (swapBinp zs z1 z2 binp') (gSwapBinp MOD zs z1 z2 binp' binp)"
lemma list_of_oalist_OAlist: "list_of_oalist (OAlist xs) = sort_oalist_ko xs"
lemma ldistinct_LNil_code [code]: "ldistinct LNil = True"
lemma "\<Gamma>\<turnstile> \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace> WHILE \<acute>M \<noteq> a INV \<lbrace>\<acute>S = \<acute>M * b\<rbrace> DO \<acute>S :== \<acute>S + b;; \<acute>M :== \<acute>M + 1 OD \<lbrace>\<acute>S = a * b\<rbrace>"
lemma surj_int_encode: "surj int_encode"
lemma mexecd_heap_read_typed: "JVM_heap_base.mexecd addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate (\<lambda>_ :: 'heap. typeof_addr) (heap_base.heap_read_typed (\<lambda>_ :: 'heap. typeof_addr) heap_read P) heap_write P t xcpfrsh ta xcpfrsh' \<longleftrightarrow> JVM_heap_base.mexecd addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate (\<lambda>_ :: 'heap. typeof_addr) heap_read heap_write P t xcpfrsh ta xcpfrsh' \<and> (\<forall>ad al v T. ReadMem ad al v \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<longrightarrow> heap_base'.addr_loc_type TYPE('heap) typeof_addr P ad al T \<longrightarrow> heap_base'.conf TYPE('heap) typeof_addr P v T)"
lemma run_gsubt_cl: assumes "run \<A> s t" and "p \<in> gposs t" shows "run \<A> (gsubt_at s p) (gsubt_at t p)"
lemma t_seq2_below_t_seq: assumes "p \<in> Test_expression" and "q \<in> Pre_expression" shows "tseq2 (-pq) p x (p\<star>x\<guillemotleft>q) (-pq\<squnion>p(x\<guillemotleft>(p\<star>x\<guillemotleft>q)aL)) m \<le> tseq (-p) x (p\<star>x\<guillemotleft>q) (-p\<squnion>(x\<guillemotleft>(p\<star>x\<guillemotleft>q)*aL)) m"
lemma "bit (0b11000 :: 10 word) n = (n = 4 \<or> n = 3)"
lemma sum_np_eq: assumes hC: "C \<subseteq> PR" shows "(\<Sum>l\<in>{..<np}. f l) = (\<Sum>l\<in>C. f l) + (\<Sum>l\<in>({..<np}-C). f l)"
lemma (in complete_lattice) "UNIV-complete UNIV (\<le>)"
lemma pendulum_dyn: "\<^bold>{\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<^bold>} EVOL \<phi> G T \<^bold>{\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<^bold>}"
lemma sc_SupI_directed: assumes A: "\<And>a. a \<in> A \<Longrightarrow> sc a" and directed: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> \<exists>c\<in>A. a \<le> c \<and> b \<le> c" shows "sc (Sup A)"
lemma (in Ring) subring_Ring:"Subring R S \<Longrightarrow> Ring S"
lemma i_shrink_last_const: "0 < k \<Longrightarrow> (\<lambda>x. m) \<div>\<^bsub>il\<^esub> k = (\<lambda>x. m)"
lemma Exp_Number: "Exp X Y \<noteq> Number X'"
lemma add_epsilon: assumes A: "\<And>x y. le x y \<Longrightarrow> le y x \<Longrightarrow> f x = f y" shows "\<exists>\<epsilon>>0. vnm_utility carrier le (\<lambda>x. u x + \<epsilon> * f x)"
lemma conj_eqvt: shows "pi\<bullet>(A\<and>B) = ((pi\<bullet>A)\<and>(pi\<bullet>B))"
lemma Iset_star_n: "(star_n X \<in> Iset (star_n A)) = (eventually (\<lambda>n. X n \<in> A n) \<U>)"
lemma point_to_polys_length: "length (point_to_polys as) = length as"
lemma list_of_aux_Lazy_llist [code]: "list_of_aux xs (Lazy_llist ys) = (case ys () of None \<Rightarrow> rev xs \| Some (y, ys) \<Rightarrow> list_of_aux (y # xs) ys)"
lemma last_messageI2: " \<lbrakk> i < length xs; xs ! i \<noteq> \<NoMsg>; \<And>j. \<lbrakk> i < j; j < length xs \<rbrakk> \<Longrightarrow> xs ! j = \<NoMsg> \<rbrakk> \<Longrightarrow> last_message xs = xs ! i"
lemma wf_constr_bvars_disj: "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta> \<Longrightarrow> (subst_domain \<theta> \<union> range_vars \<theta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
lemma indets_monom_mult: assumes "c \<noteq> 0" and "p \<noteq> (0::('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b::semiring_no_zero_divisors)" shows "indets (punit.monom_mult c t p) = keys t \<union> indets p"
lemma card_eq_const_sum: fixes k:: real assumes "finite A" shows "k*card A = sum (\<lambda>x. k) A"
lemma irreflp_transp_imp_antisymP: "irreflp p \<Longrightarrow> transp p \<Longrightarrow> antisymp p"
lemma Fejer_kernel_0 [simp]: "Fejer_kernel 0 x = 0" "Fejer_kernel n 0 = n/2"
lemma wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_pairs: "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F) \<Longrightarrow> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (pair ` set F)"
lemma update_id[simp]: "update (\<lambda>x. x) = return ()"
lemma fgl_invar_collapse_ce_aux: assumes INV: "invar v0 D0 (p, D, pE)" assumes NE[simp]: "p\<noteq>[]" assumes NONTRIV: "vE p D pE \<inter> (last p \<times> last p) \<noteq> {}" assumes ACC: "\<forall>i<num_acc. \<exists>q\<in>last p. i\<in>acc q" shows "fgl_invar_part (Some (\<Union>(set (butlast p)), last p), p, D, pE)"
lemma red1_preserves_sync_ok: "\<lbrakk> uf,P,t \<turnstile>1 \<langle>e, s\<rangle> -ta\<rightarrow> \<langle>e', s'\<rangle>; sync_ok e \<rbrakk> \<Longrightarrow> sync_ok e'" and reds1_preserves_sync_oks: "\<lbrakk> uf,P,t \<turnstile>1 \<langle>es, s\<rangle> [-ta\<rightarrow>] \<langle>es', s'\<rangle>; sync_oks es \<rbrakk> \<Longrightarrow> sync_oks es'"
lemma transfer_FOREACHoi_plain[refine_transfer]: assumes A: "set_iterator_genord iterate s ordR" assumes R: "\<And>x \<sigma>. RETURN (fi x \<sigma>) \<le> f x \<sigma>" shows "RETURN (iterate (\<lambda>_. True) fi \<sigma>) \<le> FOREACHoi ordR I s f \<sigma>"
lemma constant_fun_closed: assumes "c \<in> carrier Q\<^sub>p" shows "constant_function (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) c \<in> carrier (SA m)"
lemma stutter_equiv_within_interval: assumes f: "stutter_sampler f \<sigma>" and lo: "f k \<le> n" and hi: "n < f (Suc k)" shows "\<sigma>[n ..] \<approx> \<sigma>[f k ..]"
lemma mult_mat_vec_smult_vec_assoc: fixes A :: "'a::comm_ring_1 mat" assumes A: "A \<in> carrier_mat n m" and w: "w \<in> carrier_vec m" shows "A \<^sub>v (a \<cdot>\<^sub>v w) = a \<cdot>\<^sub>v (A \<^sub>v w)"
lemma algebraic_csqrt [intro]: "algebraic x \<Longrightarrow> algebraic (csqrt x)"
lemma integral_less_real: fixes f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f" "continuous_on {a..b} g" and "{a<..<b} \<noteq> {}" and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> f x < g x" shows "integral {a..b} f < integral {a..b} g"
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = add_mset x A"
lemma INF_mult_left_ennreal: assumes "I = {} \<Longrightarrow> c \<noteq> 0" and "\<lbrakk> c = \<top>; \<exists>i\<in>I. f i > 0 \<rbrakk> \<Longrightarrow> \<exists>p>0. \<forall>i\<in>I. f i \<ge> p" shows "c * (INF i\<in>I. f i) = (INF i\<in>I. c * f i ::ennreal)"
lemma similar_mat_wit_pow_id: "similar_mat_wit A B P Q \<Longrightarrow> A ^\<^sub>m k = P * B ^\<^sub>m k * Q"
lemma restrict_nonempty_product: fixes f g :: "('a::finite,'b::idempotent_semiring) square" assumes "\<not> List.member ls l" shows "(k#ks)\<langle>f\<rangle>(l#ls) \<odot> (l#ls)\<langle>g\<rangle>(m#ms) = ([k]\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>[m] \<oplus> [k]\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>[m]) \<oplus> ([k]\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>ms \<oplus> [k]\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>ms) \<oplus> (ks\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>[m] \<oplus> ks\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>[m]) \<oplus> (ks\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>ms \<oplus> ks\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>ms)"
lemma WTrt_binop_widen_mono: "\<lbrakk> P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T; P \<turnstile> T1' \<le> T1; P \<turnstile> T2' \<le> T2 \<rbrakk> \<Longrightarrow> \<exists>T'. P \<turnstile> T1'\<guillemotleft>bop\<guillemotright>T2' : T' \<and> P \<turnstile> T' \<le> T"
theorem wls_rawInduct[case_names Var Op Abs]: assumes Var: "\<And> xs x. phi (asSort xs) (Var xs x)" and Op: "\<And> delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)" and Abs: "\<And> s xs x X. \<lbrakk>isInBar (xs,s); wls s X; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs,s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs,s') A \<longrightarrow> phiAbs (xs,s') A)"
lemma length_entries_rbtreeify_f: "n \<le> length kvs \<Longrightarrow> length (entries (fst (rbtreeify_f n kvs))) = n" and length_entries_rbtreeify_g: "n \<le> Suc (length kvs) \<Longrightarrow> length (entries (fst (rbtreeify_g n kvs))) = n - 1"
lemma inj_on_set: "inj_on set (Collect small)"
lemma left_unique_rel_converter: "\<lbrakk> left_total A; left_unique B; left_unique C; left_total R \<rbrakk> \<Longrightarrow> left_unique (rel_converter A B C R)"
lemma new_psubst_image: \<open>new c p \<Longrightarrow> d \<notin> image f (params p) \<Longrightarrow> new d (psubst (f(c := d)) p)\<close>
lemma resl_comp2 [simp]: "(x \<leftarrow> y) \<cdot> y \<leftarrow> y = x \<leftarrow> y"
lemma ring_finite_field_ops32: "ring_ops (finite_field_ops32 pp) mod_ring_rel32"
lemma (in composition_series) composition_series_length_one: shows "(length \<GG> = 1) = (\<GG> = [{\<one>}])"
lemma undeff_equiv: "(\<phi>\<noteq>undeff) = (\<exists>f. \<phi>=Afml f)"
lemma Posix_simp: assumes "s \<in> (fst (simp r)) \<rightarrow> v" shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"
lemma match_abss_ticl_abs_Inter_subset: assumes TI: "set TI = {(a,b). (a,b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}" and \<delta>: "match_abss OCC TI s t = Some \<delta>" and x: "x \<in> fv s" shows "\<Inter>(ticl_abs TI ` \<delta> x) \<subseteq> \<delta> x"
lemma map_to_set_map_of : "distinct (map fst l) \<Longrightarrow> map_to_set (map_of l) = set l"
lemma filterlim_at_leftI: assumes "filterlim (\<lambda>x. f x - c) (at_left 0) F" shows "filterlim f (at_left (c::real)) F"
theorem knights_circuit_exists: assumes "min n m \<ge> 5" "even (n*m)" shows "\<exists>ps. knights_circuit (board n m) ps"
lemma "(\<exists>z w. \<forall>x y. P x y = (x=z \<and> y=w)) \<longrightarrow> (\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P x y = (y=w)) = (x=z))"
lemma list_all_map: "list_all (h o i) l \<longleftrightarrow> list_all h (map i l)"
lemma [simp]: shows rFail_neq_React: "rFail \<noteq> React f" and React_neq_rFail: "React f \<noteq> rFail"
lemma fresh_literal_if: "fresh xs y = (if y \<in> xs \<and> finite xs then fresh (xs - {y}) (upChar_literal y) else y)"
lemma IsProperInXY_intro[IsProper_intros]: "IsProperInXY (\<lambda> x y . \<chi> \<comment> \<open>only \<open>x\<close>\<close> \<comment> \<open>one place:\<close> (\<lambda> F . \<lparr>F,x\<rparr>) \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,x,x\<rparr>) (\<lambda> F a . \<lparr>F,x,a\<rparr>) (\<lambda> F a . \<lparr>F,a,x\<rparr>) \<comment> \<open>three place three \<open>x\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,x\<rparr>) \<comment> \<open>three place two \<open>x\<close>:\<close> (\<lambda> F a . \<lparr>F,x,x,a\<rparr>) (\<lambda> F a . \<lparr>F,x,a,x\<rparr>) (\<lambda> F a . \<lparr>F,a,x,x\<rparr>) \<comment> \<open>three place one \<open>x\<close>:\<close> (\<lambda> F a b. \<lparr>F,x,a,b\<rparr>) (\<lambda> F a b. \<lparr>F,a,x,b\<rparr>) (\<lambda> F a b . \<lparr>F,a,b,x\<rparr>) \<comment> \<open>only \<open>y\<close>\<close> \<comment> \<open>one place:\<close> (\<lambda> F . \<lparr>F,y\<rparr>) \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,y,y\<rparr>) (\<lambda> F a . \<lparr>F,y,a\<rparr>) (\<lambda> F a . \<lparr>F,a,y\<rparr>) \<comment> \<open>three place three \<open>y\<close>:\<close> (\<lambda> F . \<lparr>F,y,y,y\<rparr>) \<comment> \<open>three place two \<open>y\<close>:\<close> (\<lambda> F a . \<lparr>F,y,y,a\<rparr>) (\<lambda> F a . \<lparr>F,y,a,y\<rparr>) (\<lambda> F a . \<lparr>F,a,y,y\<rparr>) \<comment> \<open>three place one \<open>y\<close>:\<close> (\<lambda> F a b. \<lparr>F,y,a,b\<rparr>) (\<lambda> F a b. \<lparr>F,a,y,b\<rparr>) (\<lambda> F a b . \<lparr>F,a,b,y\<rparr>) \<comment> \<open>\<open>x\<close> and \<open>y\<close>\<close> \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,x,y\<rparr>) (\<lambda> F . \<lparr>F,y,x\<rparr>) \<comment> \<open>three place \<open>(x,y)\<close>:\<close> (\<lambda> F a . \<lparr>F,x,y,a\<rparr>) (\<lambda> F a . \<lparr>F,x,a,y\<rparr>) (\<lambda> F a . \<lparr>F,a,x,y\<rparr>) \<comment> \<open>three place \<open>(y,x)\<close>:\<close> (\<lambda> F a . \<lparr>F,y,x,a\<rparr>) (\<lambda> F a . \<lparr>F,y,a,x\<rparr>) (\<lambda> F a . \<lparr>F,a,y,x\<rparr>) \<comment> \<open>three place \<open>(x,x,y)\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,y\<rparr>) (\<lambda> F . \<lparr>F,x,y,x\<rparr>) (\<lambda> F . \<lparr>F,y,x,x\<rparr>) \<comment> \<open>three place \<open>(x,y,y)\<close>:\<close> (\<lambda> F . \<lparr>F,x,y,y\<rparr>) (\<lambda> F . \<lparr>F,y,x,y\<rparr>) (\<lambda> F . \<lparr>F,y,y,x\<rparr>) \<comment> \<open>three place \<open>(x,x,x)\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,x\<rparr>) \<comment> \<open>three place \<open>(y,y,y)\<close>:\<close> (\<lambda> F . \<lparr>F,y,y,y\<rparr>))"
lemma (in is_tdghm) tdghm_NTMap_app_in_Arr[dg_cs_intros]: assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
lemma Always_Constrains_weaken: "[\| F \<in> Always C; F \<in> A Co A'; C \<inter> B \<subseteq> A; C \<inter> A' \<subseteq> B' \|] ==> F \<in> B Co B'"
lemma arr_Par_comp_Par[dg_Par_cs_intros]: assumes "arr_Par \<alpha> S" and "arr_Par \<alpha> T" shows "arr_Par \<alpha> (S \<circ>\<^sub>P\<^sub>a\<^sub>r T)"
lemma infinite_sumset_iff: shows "infinite (sumset A B) \<longleftrightarrow> infinite (A \<inter> G) \<and> B \<inter> G \<noteq> {} \<or> A \<inter> G \<noteq> {} \<and> infinite (B \<inter> G)"

lemma conf_hext: "\<lbrakk> h \<unlhd> h'; P,h \<turnstile> v :\<le> T \<rbrakk> \<Longrightarrow> P,h' \<turnstile> v :\<le> T"

lemma resid_Arr_Src [simp]: assumes "Arr t" shows "t \\ Src t = t"

lemma bin_fst_nemp: "u\<^sub>0 \<noteq> \<epsilon>" and bin_snd_nemp: "u\<^sub>1 \<noteq> \<epsilon>"

lemma (in Group) nsg1:"\<lbrakk>G \<guillemotright> H; b \<in> carrier G; h \<in> H; \<forall>a\<in> carrier G. \<forall>h\<in>H. (a \<cdot> h)\<cdot> (\<rho> a) \<in> H\<rbrakk> \<Longrightarrow> b \<cdot> h \<cdot> (\<rho> b) \<in> H"

lemma ln_exp_cf6_lower_bound_pos: assumes "0\<le>x" shows "ln (exp_cf6 x) \<le> x"

lemma next_performopD: assumes "next_performop' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self" shows "\<Delta>mint_msg \<noteq> {#} \<or> zmset_of \<Delta>mint_self - zmset_of \<Delta>neg \<noteq> {#}\<^sub>z" "\<forall>t. int (count \<Delta>neg t) \<le> zcount (c_caps c0 p) t" "minting_self (c_caps c0 p) \<Delta>mint_self" "minting_msg (c_caps c0 p) \<Delta>mint_msg" "c_temp c1 = (c_temp c0)(p := c_temp c0 p + (timestamps (zmset_of \<Delta>mint_msg) + zmset_of \<Delta>mint_self - zmset_of \<Delta>neg))" "c_msg c1 = c_msg c0" "c_glob c1 = c_glob c0" "c_data_msg c1 = c_data_msg c0 + \<Delta>mint_msg" "c_caps c1 = (c_caps c0)(p := c_caps c0 p + (zmset_of \<Delta>mint_self - zmset_of \<Delta>neg))"

lemma some_trg_composable: assumes "arr \<sigma>" shows "\<mu> \<star> \<sigma> \<noteq> null \<longleftrightarrow> \<mu> \<star> some_trg \<sigma> \<noteq> null"

lemma tail_correct: assumes "ft_invar t" "t \<noteq> Empty" shows "toList (tail t) = tl (toList t)" and "ft_invar (tail t)"

lemma "\<turnstile>\<^sub>1 { %l s. True } strip ( (a ::= (N 1)) ;; b ::= (V a) ) {time ( (a ::= (N 1)) ;; b ::= (V a) ) \<Down> %l s. s b = 1}"

lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"

lemma AxiomP_sf [iff]: "Sigma_fm (AxiomP t)"

lemma ltrm_of_X_minus: assumes "a \<in> carrier R" assumes "\<one> \<noteq>\<zero>" shows "ltrm (X_minus a) = X"

theorem Extend_subset: \<open>S \<subseteq> Extend S C f\<close>

lemma quantale_hom_closed_map: fixes f :: "'a::quantale_with_dual \<Rightarrow> 'b::quantale_with_dual" shows "(f \<in> quantale_homset) \<Longrightarrow> (radj f \<in> quantale_closed_maps)"

lemma ls_trans_r: assumes "locally_sym R (A + B)" shows "locally_sym R B"

lemma \<nu>\<kappa>_proper[meta_aux]: "proper (x\<^sup>P)"

lemma listset_empty_iff: "listset xs = {} \<longleftrightarrow> {} \<in> set xs"

lemma no_path_12[simp]: "\<not> task.path 1 2 pth"

lemma ineM_Un1: assumes "ineM P A E" shows "ineM P (A Un B) E"

lemma assumes "\<And>x y. P (y + x)" shows "\<And>x y. P (x + y :: nat)"

lemma decide_origin: assumes send: "\<forall>p. \<mu> p \<in> get_msgs (send1 r) cfg (HOs r) (HOs r) p" and step: "\<forall>p. next1 r p (cfg p) (\<mu> p) (cfg' p)" and step_r: "two_step r = Suc 0" shows "D cfg cfg' \<subseteq> {p. \<exists>v. decide (cfg' p) = Some v \<and> (\<forall>q \<in> HOs r p. vote (cfg q) = Some v)}"

lemma [code]: "toy_class_raw.truncate = toy_class_raw_rec (co4 K toy_class_raw.make)"

lemma pointbased_mk_regular2: "pointbased (mk_regular2 P F)"

lemma path_components_eq_connected_components_of: "locally_path_connected_space X \<Longrightarrow> (path_components_of X = connected_components_of X)"

lemma test_bremdup1_refine_aux: "(test_bremdup1, my_bremdup_impl_loc.test_remdup) \<in> [my_bremdup_impl_loc]\<^sub>a nat_assn\<^sup>k \<rightarrow> nat_assn"

lemma has_integral_vec1_D: fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector" assumes "((f \<circ> vec) has_integral y) ((\<lambda>x. x $ 1) ` S)" shows "(f has_integral y) S"

lemma recurrent_iff_U_eq_1: "recurrent s \<longleftrightarrow> U s s = 1"

lemma goes_wrong_denot_empty: assumes gw: "goes_wrong e" and fv_e: "FV e = {}" shows "E e [] = {}"

lemma "Thm (TYPE(bool)) 1 (Imp (All (FIn 0 1)) (FExists (FIn 0 1)))"

lemma Strong_Soundness_aux: "\<lbrakk> (Some c, s) -*\<rightarrow> (co, t); s \<in> pre c; \<turnstile> c q \<rbrakk> \<Longrightarrow> if co = None then t \<in> q else t \<in> pre (the co) \<and> \<turnstile> (the co) q"

lemma (in CRR_market) assumes "N = bernoulli_stream q" and "0 < q" and "q < 1" shows bernoulli_gen_filtration: "filtration N G" and bernoulli_sigma_finite: "\<forall>n. sigma_finite_subalgebra N (G n)"

lemma chine_in_hom [intro]: shows "\<guillemotleft>chine : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright>" and "\<guillemotleft>chine: chine \<Rightarrow> chine\<guillemotright>"

lemma CCfix_restr: assumes "domA \<Gamma> \<subseteq> S" shows "cc_restr S (CCfix \<Gamma>\<cdot>(ae, G)) = cc_restr S (CCfix \<Gamma>\<cdot>(ae f|` S, cc_restr S G))"

lemma gp: assumes a0: "a$0 = (0::'a::field)" shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)" (is "?one oo a = _")

lemma generate_maskedDB_elim: "\<lbrakk>roundup emBits 8 * 8 - emBits \<le> length x; ( roundup emBits 8) * 8 - (length (sha1 M)) - 8 = length (maskedDB_zero x emBits)\<rbrakk> \<Longrightarrow> generate_maskedDB (maskedDB_zero x emBits @ y @ z) emBits (length(sha1 M)) = maskedDB_zero x emBits"

lemma sameDom_swapBinp_gSwapBinp[simp]: assumes "wlsBinp delta binp'" and "gWlsBinp MOD delta binp" shows "sameDom (swapBinp zs z1 z2 binp') (gSwapBinp MOD zs z1 z2 binp' binp)"

lemma list_of_oalist_OAlist: "list_of_oalist (OAlist xs) = sort_oalist_ko xs"

lemma ldistinct_LNil_code [code]: "ldistinct LNil = True"

lemma "\<Gamma>\<turnstile> \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace> WHILE \<acute>M \<noteq> a INV \<lbrace>\<acute>S = \<acute>M * b\<rbrace> DO \<acute>S :== \<acute>S + b;; \<acute>M :== \<acute>M + 1 OD \<lbrace>\<acute>S = a * b\<rbrace>"

lemma surj_int_encode: "surj int_encode"

lemma mexecd_heap_read_typed: "JVM_heap_base.mexecd addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate (\<lambda>_ :: 'heap. typeof_addr) (heap_base.heap_read_typed (\<lambda>_ :: 'heap. typeof_addr) heap_read P) heap_write P t xcpfrsh ta xcpfrsh' \<longleftrightarrow> JVM_heap_base.mexecd addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate (\<lambda>_ :: 'heap. typeof_addr) heap_read heap_write P t xcpfrsh ta xcpfrsh' \<and> (\<forall>ad al v T. ReadMem ad al v \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<longrightarrow> heap_base'.addr_loc_type TYPE('heap) typeof_addr P ad al T \<longrightarrow> heap_base'.conf TYPE('heap) typeof_addr P v T)"

lemma run_gsubt_cl: assumes "run \<A> s t" and "p \<in> gposs t" shows "run \<A> (gsubt_at s p) (gsubt_at t p)"

lemma t_seq2_below_t_seq: assumes "p \<in> Test_expression" and "q \<in> Pre_expression" shows "tseq2 (-p*q) p x (p\<star>x\<guillemotleft>q) (-p*q\<squnion>p*(x\<guillemotleft>(p\<star>x\<guillemotleft>q)*aL)) m \<le> tseq (-p) x (p\<star>x\<guillemotleft>q) (-p\<squnion>(x\<guillemotleft>(p\<star>x\<guillemotleft>q)*aL)) m"

lemma "bit (0b11000 :: 10 word) n = (n = 4 \<or> n = 3)"

lemma sum_np_eq: assumes hC: "C \<subseteq> PR" shows "(\<Sum>l\<in>{..<np}. f l) = (\<Sum>l\<in>C. f l) + (\<Sum>l\<in>({..<np}-C). f l)"

lemma (in complete_lattice) "UNIV-complete UNIV (\<le>)"

lemma pendulum_dyn: "\<^bold>{\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<^bold>} EVOL \<phi> G T \<^bold>{\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<^bold>}"

lemma sc_SupI_directed: assumes A: "\<And>a. a \<in> A \<Longrightarrow> sc a" and directed: "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> \<exists>c\<in>A. a \<le> c \<and> b \<le> c" shows "sc (Sup A)"

lemma (in Ring) subring_Ring:"Subring R S \<Longrightarrow> Ring S"

lemma i_shrink_last_const: "0 < k \<Longrightarrow> (\<lambda>x. m) \<div>\<^bsub>il\<^esub> k = (\<lambda>x. m)"

lemma Exp_Number: "Exp X Y \<noteq> Number X'"

lemma add_epsilon: assumes A: "\<And>x y. le x y \<Longrightarrow> le y x \<Longrightarrow> f x = f y" shows "\<exists>\<epsilon>>0. vnm_utility carrier le (\<lambda>x. u x + \<epsilon> * f x)"

lemma conj_eqvt: shows "pi\<bullet>(A\<and>B) = ((pi\<bullet>A)\<and>(pi\<bullet>B))"

lemma Iset_star_n: "(star_n X \<in> Iset (star_n A)) = (eventually (\<lambda>n. X n \<in> A n) \<U>)"

lemma point_to_polys_length: "length (point_to_polys as) = length as"

lemma list_of_aux_Lazy_llist [code]: "list_of_aux xs (Lazy_llist ys) = (case ys () of None \<Rightarrow> rev xs | Some (y, ys) \<Rightarrow> list_of_aux (y # xs) ys)"

lemma last_messageI2: " \<lbrakk> i < length xs; xs ! i \<noteq> \<NoMsg>; \<And>j. \<lbrakk> i < j; j < length xs \<rbrakk> \<Longrightarrow> xs ! j = \<NoMsg> \<rbrakk> \<Longrightarrow> last_message xs = xs ! i"

lemma wf_constr_bvars_disj: "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta> \<Longrightarrow> (subst_domain \<theta> \<union> range_vars \<theta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"

lemma indets_monom_mult: assumes "c \<noteq> 0" and "p \<noteq> (0::('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b::semiring_no_zero_divisors)" shows "indets (punit.monom_mult c t p) = keys t \<union> indets p"

lemma card_eq_const_sum: fixes k:: real assumes "finite A" shows "k*card A = sum (\<lambda>x. k) A"

lemma irreflp_transp_imp_antisymP: "irreflp p \<Longrightarrow> transp p \<Longrightarrow> antisymp p"

lemma Fejer_kernel_0 [simp]: "Fejer_kernel 0 x = 0" "Fejer_kernel n 0 = n/2"

lemma wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_pairs: "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F) \<Longrightarrow> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (pair ` set F)"

lemma update_id[simp]: "update (\<lambda>x. x) = return ()"

lemma fgl_invar_collapse_ce_aux: assumes INV: "invar v0 D0 (p, D, pE)" assumes NE[simp]: "p\<noteq>[]" assumes NONTRIV: "vE p D pE \<inter> (last p \<times> last p) \<noteq> {}" assumes ACC: "\<forall>i<num_acc. \<exists>q\<in>last p. i\<in>acc q" shows "fgl_invar_part (Some (\<Union>(set (butlast p)), last p), p, D, pE)"

lemma red1_preserves_sync_ok: "\<lbrakk> uf,P,t \<turnstile>1 \<langle>e, s\<rangle> -ta\<rightarrow> \<langle>e', s'\<rangle>; sync_ok e \<rbrakk> \<Longrightarrow> sync_ok e'" and reds1_preserves_sync_oks: "\<lbrakk> uf,P,t \<turnstile>1 \<langle>es, s\<rangle> [-ta\<rightarrow>] \<langle>es', s'\<rangle>; sync_oks es \<rbrakk> \<Longrightarrow> sync_oks es'"

lemma transfer_FOREACHoi_plain[refine_transfer]: assumes A: "set_iterator_genord iterate s ordR" assumes R: "\<And>x \<sigma>. RETURN (fi x \<sigma>) \<le> f x \<sigma>" shows "RETURN (iterate (\<lambda>_. True) fi \<sigma>) \<le> FOREACHoi ordR I s f \<sigma>"

lemma constant_fun_closed: assumes "c \<in> carrier Q\<^sub>p" shows "constant_function (carrier (Q\<^sub>p\<^bsup>m\<^esup>)) c \<in> carrier (SA m)"

lemma stutter_equiv_within_interval: assumes f: "stutter_sampler f \<sigma>" and lo: "f k \<le> n" and hi: "n < f (Suc k)" shows "\<sigma>[n ..] \<approx> \<sigma>[f k ..]"

lemma mult_mat_vec_smult_vec_assoc: fixes A :: "'a::comm_ring_1 mat" assumes A: "A \<in> carrier_mat n m" and w: "w \<in> carrier_vec m" shows "A *\<^sub>v (a \<cdot>\<^sub>v w) = a \<cdot>\<^sub>v (A *\<^sub>v w)"

lemma algebraic_csqrt [intro]: "algebraic x \<Longrightarrow> algebraic (csqrt x)"

lemma integral_less_real: fixes f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f" "continuous_on {a..b} g" and "{a<..<b} \<noteq> {}" and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> f x < g x" shows "integral {a..b} f < integral {a..b} g"

lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = add_mset x A"

lemma INF_mult_left_ennreal: assumes "I = {} \<Longrightarrow> c \<noteq> 0" and "\<lbrakk> c = \<top>; \<exists>i\<in>I. f i > 0 \<rbrakk> \<Longrightarrow> \<exists>p>0. \<forall>i\<in>I. f i \<ge> p" shows "c * (INF i\<in>I. f i) = (INF i\<in>I. c * f i ::ennreal)"

lemma similar_mat_wit_pow_id: "similar_mat_wit A B P Q \<Longrightarrow> A ^\<^sub>m k = P * B ^\<^sub>m k * Q"

lemma restrict_nonempty_product: fixes f g :: "('a::finite,'b::idempotent_semiring) square" assumes "\<not> List.member ls l" shows "(k#ks)\<langle>f\<rangle>(l#ls) \<odot> (l#ls)\<langle>g\<rangle>(m#ms) = ([k]\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>[m] \<oplus> [k]\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>[m]) \<oplus> ([k]\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>ms \<oplus> [k]\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>ms) \<oplus> (ks\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>[m] \<oplus> ks\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>[m]) \<oplus> (ks\<langle>f\<rangle>[l] \<odot> [l]\<langle>g\<rangle>ms \<oplus> ks\<langle>f\<rangle>ls \<odot> ls\<langle>g\<rangle>ms)"

lemma WTrt_binop_widen_mono: "\<lbrakk> P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T; P \<turnstile> T1' \<le> T1; P \<turnstile> T2' \<le> T2 \<rbrakk> \<Longrightarrow> \<exists>T'. P \<turnstile> T1'\<guillemotleft>bop\<guillemotright>T2' : T' \<and> P \<turnstile> T' \<le> T"

theorem wls_rawInduct[case_names Var Op Abs]: assumes Var: "\<And> xs x. phi (asSort xs) (Var xs x)" and Op: "\<And> delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)" and Abs: "\<And> s xs x X. \<lbrakk>isInBar (xs,s); wls s X; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs,s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs,s') A \<longrightarrow> phiAbs (xs,s') A)"

lemma length_entries_rbtreeify_f: "n \<le> length kvs \<Longrightarrow> length (entries (fst (rbtreeify_f n kvs))) = n" and length_entries_rbtreeify_g: "n \<le> Suc (length kvs) \<Longrightarrow> length (entries (fst (rbtreeify_g n kvs))) = n - 1"

lemma inj_on_set: "inj_on set (Collect small)"

lemma left_unique_rel_converter: "\<lbrakk> left_total A; left_unique B; left_unique C; left_total R \<rbrakk> \<Longrightarrow> left_unique (rel_converter A B C R)"

lemma new_psubst_image: \<open>new c p \<Longrightarrow> d \<notin> image f (params p) \<Longrightarrow> new d (psubst (f(c := d)) p)\<close>

lemma resl_comp2 [simp]: "(x \<leftarrow> y) \<cdot> y \<leftarrow> y = x \<leftarrow> y"

lemma ring_finite_field_ops32: "ring_ops (finite_field_ops32 pp) mod_ring_rel32"

lemma (in composition_series) composition_series_length_one: shows "(length \<GG> = 1) = (\<GG> = [{\<one>}])"

lemma undeff_equiv: "(\<phi>\<noteq>undeff) = (\<exists>f. \<phi>=Afml f)"

lemma Posix_simp: assumes "s \<in> (fst (simp r)) \<rightarrow> v" shows "s \<in> r \<rightarrow> ((snd (simp r)) v)"

lemma match_abss_ticl_abs_Inter_subset: assumes TI: "set TI = {(a,b). (a,b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}" and \<delta>: "match_abss OCC TI s t = Some \<delta>" and x: "x \<in> fv s" shows "\<Inter>(ticl_abs TI ` \<delta> x) \<subseteq> \<delta> x"

lemma map_to_set_map_of : "distinct (map fst l) \<Longrightarrow> map_to_set (map_of l) = set l"

lemma filterlim_at_leftI: assumes "filterlim (\<lambda>x. f x - c) (at_left 0) F" shows "filterlim f (at_left (c::real)) F"

theorem knights_circuit_exists: assumes "min n m \<ge> 5" "even (n*m)" shows "\<exists>ps. knights_circuit (board n m) ps"

lemma "(\<exists>z w. \<forall>x y. P x y = (x=z \<and> y=w)) \<longrightarrow> (\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P x y = (y=w)) = (x=z))"

lemma list_all_map: "list_all (h o i) l \<longleftrightarrow> list_all h (map i l)"

lemma [simp]: shows rFail_neq_React: "rFail \<noteq> React f" and React_neq_rFail: "React f \<noteq> rFail"

lemma fresh_literal_if: "fresh xs y = (if y \<in> xs \<and> finite xs then fresh (xs - {y}) (upChar_literal y) else y)"

lemma IsProperInXY_intro[IsProper_intros]: "IsProperInXY (\<lambda> x y . \<chi> \<comment> \<open>only \<open>x\<close>\<close> \<comment> \<open>one place:\<close> (\<lambda> F . \<lparr>F,x\<rparr>) \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,x,x\<rparr>) (\<lambda> F a . \<lparr>F,x,a\<rparr>) (\<lambda> F a . \<lparr>F,a,x\<rparr>) \<comment> \<open>three place three \<open>x\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,x\<rparr>) \<comment> \<open>three place two \<open>x\<close>:\<close> (\<lambda> F a . \<lparr>F,x,x,a\<rparr>) (\<lambda> F a . \<lparr>F,x,a,x\<rparr>) (\<lambda> F a . \<lparr>F,a,x,x\<rparr>) \<comment> \<open>three place one \<open>x\<close>:\<close> (\<lambda> F a b. \<lparr>F,x,a,b\<rparr>) (\<lambda> F a b. \<lparr>F,a,x,b\<rparr>) (\<lambda> F a b . \<lparr>F,a,b,x\<rparr>) \<comment> \<open>only \<open>y\<close>\<close> \<comment> \<open>one place:\<close> (\<lambda> F . \<lparr>F,y\<rparr>) \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,y,y\<rparr>) (\<lambda> F a . \<lparr>F,y,a\<rparr>) (\<lambda> F a . \<lparr>F,a,y\<rparr>) \<comment> \<open>three place three \<open>y\<close>:\<close> (\<lambda> F . \<lparr>F,y,y,y\<rparr>) \<comment> \<open>three place two \<open>y\<close>:\<close> (\<lambda> F a . \<lparr>F,y,y,a\<rparr>) (\<lambda> F a . \<lparr>F,y,a,y\<rparr>) (\<lambda> F a . \<lparr>F,a,y,y\<rparr>) \<comment> \<open>three place one \<open>y\<close>:\<close> (\<lambda> F a b. \<lparr>F,y,a,b\<rparr>) (\<lambda> F a b. \<lparr>F,a,y,b\<rparr>) (\<lambda> F a b . \<lparr>F,a,b,y\<rparr>) \<comment> \<open>\<open>x\<close> and \<open>y\<close>\<close> \<comment> \<open>two place:\<close> (\<lambda> F . \<lparr>F,x,y\<rparr>) (\<lambda> F . \<lparr>F,y,x\<rparr>) \<comment> \<open>three place \<open>(x,y)\<close>:\<close> (\<lambda> F a . \<lparr>F,x,y,a\<rparr>) (\<lambda> F a . \<lparr>F,x,a,y\<rparr>) (\<lambda> F a . \<lparr>F,a,x,y\<rparr>) \<comment> \<open>three place \<open>(y,x)\<close>:\<close> (\<lambda> F a . \<lparr>F,y,x,a\<rparr>) (\<lambda> F a . \<lparr>F,y,a,x\<rparr>) (\<lambda> F a . \<lparr>F,a,y,x\<rparr>) \<comment> \<open>three place \<open>(x,x,y)\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,y\<rparr>) (\<lambda> F . \<lparr>F,x,y,x\<rparr>) (\<lambda> F . \<lparr>F,y,x,x\<rparr>) \<comment> \<open>three place \<open>(x,y,y)\<close>:\<close> (\<lambda> F . \<lparr>F,x,y,y\<rparr>) (\<lambda> F . \<lparr>F,y,x,y\<rparr>) (\<lambda> F . \<lparr>F,y,y,x\<rparr>) \<comment> \<open>three place \<open>(x,x,x)\<close>:\<close> (\<lambda> F . \<lparr>F,x,x,x\<rparr>) \<comment> \<open>three place \<open>(y,y,y)\<close>:\<close> (\<lambda> F . \<lparr>F,y,y,y\<rparr>))"

lemma (in is_tdghm) tdghm_NTMap_app_in_Arr[dg_cs_intros]: assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"

lemma Always_Constrains_weaken: "[| F \<in> Always C; F \<in> A Co A'; C \<inter> B \<subseteq> A; C \<inter> A' \<subseteq> B' |] ==> F \<in> B Co B'"

lemma arr_Par_comp_Par[dg_Par_cs_intros]: assumes "arr_Par \<alpha> S" and "arr_Par \<alpha> T" shows "arr_Par \<alpha> (S \<circ>\<^sub>P\<^sub>a\<^sub>r T)"

lemma infinite_sumset_iff: shows "infinite (sumset A B) \<longleftrightarrow> infinite (A \<inter> G) \<and> B \<inter> G \<noteq> {} \<or> A \<inter> G \<noteq> {} \<and> infinite (B \<inter> G)"