hcju/isabelleps huggingface dataset:
neville_aitken_def: fixes x :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b" shows "neville_aitken x f = neville_aitken_main x f 0"
sup_inf_distrib2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and y :: "'d"   and z :: "'d"   and x :: "'d" assumes "class.distrib_lattice inf less_eq less sup" shows "sup (inf y z) x = inf (sup y x) (sup z x)"
inext_nth_insert_Suc: fixes I :: "nat set"   and a :: "nat"   and n :: "nat" assumes "I \<noteq> {}"   and "a < iMin I" shows "insert a I \<rightarrow> Suc n = I \<rightarrow> n"
nbt_subset_carrier: fixes x :: "'a" assumes "x \<in> carrier" shows "no_better_than x carrier relation \<subseteq> carrier"
simps: fixes S :: "char list \<Rightarrow> 'av"   and X :: "char list set" shows "top_on\<^sub>s S X = (S = \<lambda>x. \<top> on X)"
a: shows "span S \<subseteq> {x. a \<bullet> x = 0}"
sum_representation_eq: fixes basis :: "'b set"   and v :: "'b"   and B :: "'b set" assumes "cindependent basis"   and "v \<in> cspan basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. crepresentation basis v b *\<^sub>C b) = v"
Qg_iTr: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "c_top G N (c_iop G N (N \<bullet> a)) (N \<bullet> a) = N"
Ide_Diagonalize_Can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Ide C (monoidal_language.Diagonalize C t)"
not_range_Inr: fixes x :: "'b + 'c" shows "(x \<notin> range Inr) = (x \<in> range Inl)"
that: fixes z :: "'a" assumes "z \<in> Q"   and "\<And>y. r y z \<Longrightarrow> y \<notin> Q" shows "thesis"
second_summand_not_universe: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "y \<noteq> u"
simps_2: fixes inc :: "nat"   and n :: "nat"   and T :: "typ"   and body :: "Core.term" shows "incr_bv inc n (Abs T body) = Abs T (incr_bv inc (n + 1) body)"
preserves_adjoint_pair: fixes f :: "'c"   and g :: "'c" assumes "C.adjoint_pair f g" shows "D.adjoint_pair (F f) (F g)"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = ((enat = 0 \<longrightarrow> P f1) \<and> (enat = eSuc (epred enat) \<longrightarrow> P (f2 (epred enat))))"
contr: shows "\<forall>x xs. inDed (x # x # xs) \<longrightarrow> inDed (x # xs)"
assms: shows "P[x::=b] \<Longrightarrow>\<^sub>\<tau> P'"
\<T>o_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b"   and b :: "'a" assumes "yoneda_lemma C S \<phi> F a" shows "yoneda_lemma.\<T>o C S \<phi> F a e b = set_category_given_img.mkArr S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C \<phi> (b, a)) (set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F b)) (\<lambda>x. set_category_given_img.Fun S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F (hom_functor.\<psi> C \<phi> (b, a) x)) e)"
suppI_contra: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "x \<notin> supp f" shows "f x = (0::'c)"
induct: fixes x1 :: "'d rexp"   and x2 :: "'d rexp"   and P :: "'d rexp \<Rightarrow> 'd rexp \<Rightarrow> bool" assumes "x1 \<approx> x2"   and "\<And>r s t. P (Alt (Alt r s) t) (Alt r (Alt s t))"   and "\<And>r s. P (Alt r s) (Alt s r)"   and "\<And>r. P (Alt r r) r"   and "\<And>r r' s s'. \<lbrakk>r \<approx> r'; P r r'; s \<approx> s'; P s s'\<rbrakk> \<Longrightarrow> P (Alt r s) (Alt r' s')"   and "\<And>r r' s s'. \<lbrakk>r \<approx> r'; P r r'; s \<approx> s'; P s s'\<rbrakk> \<Longrightarrow> P (Conc r s) (Conc r' s')"   and "\<And>r r'. \<lbrakk>r \<approx> r'; P r r'\<rbrakk> \<Longrightarrow> P (Star r) (Star r')"   and "\<And>r. P r r"   and "\<And>r s. \<lbrakk>r \<approx> s; P r s\<rbrakk> \<Longrightarrow> P s r"   and "\<And>r s t. \<lbrakk>r \<approx> s; P r s; s \<approx> t; P s t\<rbrakk> \<Longrightarrow> P r t" shows "P x1 x2"
vec_module_col_helper: fixes A :: "'b mat" shows "0\<^sub>v (dim_row A) \<in> LinearCombinations.module.span class_ring \<lparr>carrier = carrier_vec (dim_row A), mult = undefined, one = undefined, zero = 0\<^sub>v (dim_row A), add = (+), smult = (\<cdot>\<^sub>v)\<rparr> (set (cols A))"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets T" shows "emeasure (N \<Otimes>\<^sub>M T) (A \<times> B) = emeasure N A * emeasure T B"
deepR_inv: fixes n :: "nat"   and m :: "('c, 'd) FingerTreeStruc"   and pr :: "('c, 'd) Digit"   and sf :: "('c, 'd) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n pr \<and> is_measured_digit pr"   and "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4" shows "is_leveln_ftree n (deepR pr m sf) \<and> is_measured_ftree (deepR pr m sf)"
pp_bijective_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_bijective f = (\<forall>j. \<exists>i. f (i, j) \<noteq> bot \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
rel_sel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list"   and b :: "'c list" shows "list_all2 R a b = ((a = []) = (b = []) \<and> (a \<noteq> [] \<longrightarrow> b \<noteq> [] \<longrightarrow> R (hd a) (hd b) \<and> list_all2 R (tl a) (tl b)))"
comm_groupE_1: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
fps_onel: fixes f :: "('c, 'd) fps" shows "1 \<cdot> f = f"
l_r_def: fixes root_info :: "root_info" shows "root_info.l_r root_info = (case root_info of Root_Info x1 x2 \<Rightarrow> x1)"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<simeq> <\<nu>x>P \<parallel> Q"
ran_restrict_ran: fixes f :: "'c \<Rightarrow> 'd option" shows "f\<upharpoonleft>\<^bsub>ran f\<^esub> = f"
\<Sigma>_protocol: shows "\<Sigma>_OR.\<Sigma>_protocol"
reds: shows "t2 = App (Lam [x].t') s' \<and>  t1 \<longrightarrow>\<^sub>1 t' \<and>  s1 \<longrightarrow>\<^sub>1 s' \<or> t2 = t'[x::=s'] \<and>  t1 \<longrightarrow>\<^sub>1 t' \<and>  s1 \<longrightarrow>\<^sub>1 s'"
in_homI: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "J.arr f"   and "J.dom f = a"   and "J.cod f = b" shows "J.in_hom f a b"
disc_unequal: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c"   and w :: "'c" assumes "DFS_invar G param s"   and "v \<in> dom (discovered s)"   and "w \<in> dom (discovered s)"   and "v \<noteq> w" shows "the_discovered s v \<noteq> the_discovered s w"
implements_actJP_jAction: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol"   and t :: "'e Trace"   and n :: "nat"   and a :: "'b" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "IncrEnvironment.implements jkbp envInit envAction envTrans envVal jview envObs jp"   and "t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n" shows "set (pAct (jp a) (IncrEnvironment.runJP envObs jp t a)) = set (JKBP.jAction jkbp (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n)) t a)"
Merge_PostConj_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and G :: "'f set"   and P' :: "'e set"   and R :: "'e set"   and X :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/G\<^esub> P' c R,X"   and "F \<subseteq> G"   and "P \<subseteq> P'" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> X)"
proper_interval_finite_1_def: fixes x :: "Enum.finite_1 option"   and y :: "Enum.finite_1 option" shows "proper_interval x y = (x = None \<and> y = None)"
T_square: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "(p \<leadsto> x \<leadsto> q) \<cdot>  |x] (q \<leadsto> x \<leadsto> p) \<le> p \<leadsto> x \<cdot> x \<leadsto> p"
rev_singleton_conv: fixes xs :: "'d list"   and x :: "'d" shows "(rev xs = [x]) = (xs = [x])"
SeqReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e';; e\<^sub>2,s'\<rangle>"
R36_a: shows "pmf (sds R36) a = 1 / 2"
g_def: shows "g = Cn 4 r_snoc [recf.Id 4 1, Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0]]"
assms_2: fixes i :: "nat" assumes "i \<le> k" shows "finite (A i)"
init_fin_delay_bisimulation_obs: fixes t :: "'t" shows "delay_bisimulation_obs (r1.init_fin t) (r2.init_fin t) (init_fin_bisim t) (ta_bisim init_fin_bisim) r1.init_fin_\<tau>move r2.init_fin_\<tau>move"
clear_locals_sip_not_ip': fixes \<xi> :: "state"   and i :: "nat" assumes "ip \<xi> = i" shows "sip (clear_locals \<xi>) \<noteq> i"
blockNumNELemma: fixes cfile1 :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte"   and cfile2 :: "CFile"   and blockNum :: "nat"   and i :: "nat" assumes "nextFreeBlockInvariant cfile1"   and "cfile1\<lparr>data := writeCArray (data cfile1) (byteIndex div blockSize) (writeCArray (readCArray (data cfile1) (byteIndex div blockSize)) (byteIndex mod blockSize) value)\<rparr> = cfile2"   and "\<not> blockNum < nextFreeBlock cfile2"   and "blockNum < numBlocks"   and "i < blockSize"   and "byteIndex div blockSize < numBlocks"   and "byteIndex < fileSize cfile1" shows "blockNum \<noteq> byteIndex div blockSize"
simps_13: fixes I :: "\<I>"   and \<phi> :: "Formula.formula" shows "Formula.future_bounded (formula.Next I \<phi>) = Formula.future_bounded \<phi>"
symmetric_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "local.comp_inf.symmetric x"   and "local.comp_inf.symmetric y" shows "local.comp_inf.symmetric (x \<sqinter> y)"
assms_1: shows "sorted_list_of_set A = sorted_list_of_set B"
d_ex: fixes x :: "'a" assumes "x \<in> {a--b}" shows "{- d..d} \<subseteq> existence_ivl0 x"
pos_imp_ne: fixes u :: "nat" assumes "0 < u" shows "nat_to_set u \<noteq> {}"
hyps_5: shows "pred_option (is_valid_fun_call (Fubx_get F2) gd n \<Sigma>2) opt"
beta_reduction_left_unitE: fixes M' :: "'a trm" assumes "Unit \<rightarrow>\<beta> M'" shows "False"
insort_insert_key_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.insort_insert_key less_eq f x xs = (if f x \<in> f ` set xs then xs else linorder.insort_key less_eq f x xs)"
cindependent_cfinite_dim_finite: fixes S :: "'b set" assumes "cindependent S" shows "finite S"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'm" assumes "i < nr"   and "j < nc" shows "mat nr nc f $$ (i, j) = f (i, j)"
that: fixes ns' :: "'node SDG_node list" assumes "realizable n ns' n'"   and "set ns \<subseteq> set ns'" shows "thesis"
normalize_ports_generic_def: fixes normalize_pos :: "'c common_primitive match_expr \<Rightarrow> 'c common_primitive match_expr list"   and rewrite_neg :: "'c common_primitive match_expr \<Rightarrow> 'c common_primitive match_expr"   and m :: "'c common_primitive match_expr" shows "normalize_ports_generic normalize_pos rewrite_neg m = concat (map normalize_pos (normalize_match (rewrite_neg m)))"
normalized_simps_3: fixes xs :: "('c, 'd) bintree option list" shows "normalized (None # xs) = (xs \<noteq> [] \<and> normalized xs)"
base_case: shows "Jacobi (- 1) p = (- 1) ^ nat ((p - 1) div 2)"
mult_p_def: fixes p :: "int"   and x :: "int"   and y :: "int" shows "mult_p p x y = mod_nonneg_pos_int (x * y) p"
bind_SE_def: fixes f :: "'b \<Rightarrow> ('d \<times> 'b) option"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) option" shows "bind\<^sub>S\<^sub>E f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> g out \<sigma>')"
simps: fixes t :: "'d"   and ws :: "'d \<Rightarrow> 'e wait_set_status option"   and a1 :: "('d, 'e) wait_set_action"   and a2 :: "'d \<Rightarrow> 'e wait_set_status option" shows "redT_updW t ws a1 a2 = ((\<exists>t' w. a1 = Notify w \<and> a2 = ws(t' \<mapsto> PostWS WSNotified) \<and> ws t' = \<lfloor>InWS w\<rfloor>) \<or> (\<exists>w. a1 = Notify w \<and> a2 = ws \<and> (\<forall>x. ws x \<noteq> \<lfloor>InWS w\<rfloor>)) \<or> (\<exists>w. a1 = NotifyAll w \<and> a2 = (\<lambda>t. if ws t = \<lfloor>InWS w\<rfloor> then \<lfloor>PostWS WSNotified\<rfloor> else ws t)) \<or> (\<exists>w. a1 = Suspend w \<and> a2 = ws(t \<mapsto> InWS w)) \<or> (\<exists>t' w WSInterrupted. a1 = WakeUp t' \<and> a2 = ws(t' \<mapsto> PostWS WSInterrupted) \<and> ws t' = \<lfloor>InWS w\<rfloor>) \<or> (\<exists>t'. a1 = WakeUp t' \<and> a2 = ws \<and> (\<forall>x. ws t' \<noteq> \<lfloor>InWS x\<rfloor>)) \<or> a1 = Notified \<and> a2 = ws(t := None) \<or> a1 = WokenUp \<and> a2 = ws(t := None))"
subprob_measure_le_1: fixes M :: "'b pmf"   and X :: "'b set" shows "measure_pmf.prob M X \<le> 1"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y = y \<oplus> x"
simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "list_all3 P [] [] [] = True"
id_eq_prop_prop_9: fixes v :: "i"   and p :: "\<o>"   and q :: "\<o>"   and r :: "\<o>" shows "[p \<^bold>= q \<^bold>& q \<^bold>= r \<^bold>\<rightarrow> p \<^bold>= r in v]"
for_rec2'_def: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat"   and i :: "nat"   and j :: "nat" shows "for_rec2' f a n i j = (if i = 0 then RETURN a else for_rec (\<lambda>a i. for_rec (\<lambda>a. f a i) a n) a (i - 1)) \<bind> (\<lambda>a. for_rec (\<lambda>a. f a i) a j)"
hom: shows "(\<lambda>f. sum' f I) \<in> hom (sum_group I (\<lambda>i. free_Abelian_group (S i))) (free_Abelian_group (\<Union> (S ` I)))"
rref_G: shows "reduced_row_echelon_form (Gauss_Jordan A)"
RangeE: fixes b :: "'g"   and r :: "('h \<times> 'g) set" assumes "b \<in> Range r" obtains a :: "'h" where "(a, b) \<in> r"
index_mat_swapcols_3: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_col (swapcols k l A) = dim_col A"
construct_lhs_vector_clean_R: fixes p :: "real poly"   and i :: "nat"   and signs :: "rat list list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "i < length signs" shows "of_rat (w_vec_R p qs signs $ i) = of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = signs ! i})"
pochhammer_poly_Suc': fixes n :: "nat" shows "pochhammer_poly (Suc n) = pCons (0::'a) (pochhammer_poly n \<circ>\<^sub>p [:1::'a, 1::'a:])"
X_2: assumes "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map cyc_permutations xss)" shows "distincts yss"
labeled_sat_ineq_lift: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and Di :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and i :: "'a"   and D :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))\<rbrakk>\<^sub>d \<I>"   and "\<forall>(j, p)\<in>{(i, t, s)} \<union> set D \<union> set Di. \<forall>(k, q)\<in>{(i, t, s)} \<union> set D \<union> set Di. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k" shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
lemma_2_5_9_b: fixes a :: "'a"   and b :: "'a" shows "a \<le> b r\<rightarrow> a"
nsum_addTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A \<and> g j \<in> carrier A) \<longrightarrow> \<Sigma>\<^sub>e A (\<lambda>i. f i \<plusminus> g i) n = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A g n"
eff_Atm: fixes atm :: "'atom"   and s :: "'state"   and i :: "nat" shows "eff (Atm atm) s i = aval atm s"
that: fixes qs :: "'a list" assumes "length qs = length bs"   and "p = sum_list (map2 (*s) qs bs)" shows "thesis"
awalk_verts_arc2: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and e :: "'b" assumes "H.awalk u p v"   and "e \<in> set p" shows "head (G \<restriction> S) e \<in> set (H.awalk_verts u p)"
Rec_functional: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a" shows "\<exists>!y. Rec e r x y"
ordLeq_cexp2: fixes q :: "('c \<times> 'c) set"   and r :: "('d \<times> 'd) set" assumes "ctwo \<le>o q"   and "Card_order r" shows "r \<le>o q ^c r"
reachable_refl: fixes v :: "'a" assumes "v \<in> verts (del_vert u)" shows "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v"
tuple_simps_2: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.dom (tuple f g) = local.dom f"
conj_def: fixes p :: "fm"   and q :: "fm" shows "Ferrack.conj p q = (if p = F \<or> q = F then F else if p = T then q else if q = T then p else if p = q then p else And p q)"
simps: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" shows "extra_assms (v, p) = labelAtOut v |`| hyps_for (nodeOf v) p"
prems_2: shows "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b (r # sb) \<subseteq> A"
f_nonneg_base: fixes x :: "nat" assumes "x\<^sub>0 \<le> x"   and "x < x\<^sub>1" shows "0 \<le> f x"
disjointFrame_L3: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and rcv :: "nat \<Rightarrow> 'a Frame list"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and i :: "nat"   and j :: "nat" assumes "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "\<forall>i<n. FlexRayController (nReturn i) rcv (nC i) (nStore i) (nSend i) (nGet i)"   and "t mod cycleLength (nC i) mem schedule (nC i)"   and "i < n"   and "j < n"   and "i \<noteq> j" shows "nSend j t = []"
has_field_derivative_Lambert_W': fixes x :: "real"   and A :: "real set" assumes "x \<in> {- exp (- 1)<..<0}" shows "(Lambert_W' has_real_derivative inverse (x + exp (Lambert_W' x))) (at x within A)"
strict_iff_order: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term" shows "b \<rhd> a = (b \<unrhd> a \<and> a \<noteq> b)"
simps: fixes x :: "'a extB"   and y :: "'a extB" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
prefixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>p ys" obtains zs :: "'b list" where "ys = xs \<cdot> zs"
fold_insert: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and z :: "'c" assumes "comp_fun_commute f"   and "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
subgroupE_4: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Group.group G"   and "subgroup H G"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes>\<^bsub>G\<^esub> b \<in> H"
is_app_def: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and t :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_app unapp t = (unapp t \<noteq> None)"
prm_disagreement_composition: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> b"   and "b \<noteq> c"   and "a \<noteq> c" shows "ds ([a \<leftrightarrow> b] \<diamondop> [b \<leftrightarrow> c]) [a \<leftrightarrow> c] = {a, b}"
dbm_le_def: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "a \<preceq> b \<equiv> a \<prec> b \<or> a = b"
map_graph_def: fixes f :: "('g \<times> 'f) set"   and G :: "('e, 'g) labeled_graph" shows "map_graph f G = LG (on_triple f `` edges G) (f `` vertices G)"
connected_component_of_disjoint: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "disjnt (connected_component_of_set X x) (connected_component_of_set X y) = (\<not> connected_component_of X x y)"
mm_equiv_intro: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and lc\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "strong_low_bisim_mm \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "lc\<^sub>1 \<approx> lc\<^sub>2"
prv_eqv_prv_rev: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv \<phi>"   and "prv (Syntax_with_Connectives.eqv cnj imp \<chi> \<phi>)" shows "prv \<chi>"