hcju/isabelleps huggingface dataset:
eucl_of_list_prod: fixes xs :: "real list" shows "eucl_of_list xs = (eucl_of_list (take DIM('a) xs), eucl_of_list (drop DIM('a) xs))"
nth_equalityI: fixes xs :: "'a list"   and ys :: "'a list" assumes "|xs| = |ys|"   and "\<And>i. i < |xs| \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
moebius_reciprocal_def: shows "moebius_reciprocal = mk_moebius 0 1 1 0"
sup_aci_2: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
induct: fixes x1 :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and x2 :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and x3 :: "('b, 'a, 'c) ann"   and x4 :: "('b, 'a, 'c) com"   and P :: "('a \<Rightarrow> ('b, 'a, 'c) com option) \<Rightarrow> ('a \<Rightarrow> ('b, 'a, 'c) ann list option) \<Rightarrow> ('b, 'a, 'c) ann \<Rightarrow> ('b, 'a, 'c) com \<Rightarrow> bool" assumes "ann_matches x1 x2 x3 x4"   and "\<And>\<Gamma> \<Theta> a. P \<Gamma> \<Theta> (AnnExpr a) Skip"   and "\<And>\<Gamma> \<Theta> a f. P \<Gamma> \<Theta> (AnnExpr a) (Basic f)"   and "\<And>\<Gamma> \<Theta> a r. P \<Gamma> \<Theta> (AnnExpr a) (Spec r)"   and "\<And>\<Gamma> \<Theta> a. P \<Gamma> \<Theta> (AnnExpr a) Throw"   and "\<And>\<Gamma> \<Theta> a e r b. \<lbrakk>ann_matches \<Gamma> \<Theta> a e; P \<Gamma> \<Theta> a e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec r a) (Await b e)"   and "\<And>\<Gamma> \<Theta> a1 p1 a2 p2. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 p1; P \<Gamma> \<Theta> a1 p1; ann_matches \<Gamma> \<Theta> a2 p2; P \<Gamma> \<Theta> a2 p2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnComp a1 a2) (Seq p1 p2)"   and "\<And>\<Gamma> \<Theta> a1 c1 a2 c2 a b. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 c1; P \<Gamma> \<Theta> a1 c1; ann_matches \<Gamma> \<Theta> a2 c2; P \<Gamma> \<Theta> a2 c2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnBin a a1 a2) (Cond b c1 c2)"   and "\<And>\<Gamma> \<Theta> a1 c1 a2 c2. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 c1; P \<Gamma> \<Theta> a1 c1; ann_matches \<Gamma> \<Theta> a2 c2; P \<Gamma> \<Theta> a2 c2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnComp a1 a2) (Catch c1 c2)"   and "\<And>\<Gamma> \<Theta> a' e a i b. \<lbrakk>ann_matches \<Gamma> \<Theta> a' e; P \<Gamma> \<Theta> a' e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnWhile a i a') (While b e)"   and "\<And>\<Gamma> \<Theta> a' e a f b. \<lbrakk>ann_matches \<Gamma> \<Theta> a' e; P \<Gamma> \<Theta> a' e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec a a') (Guard f b e)"   and "\<And>\<Theta> f as \<Gamma> b n a. \<lbrakk>\<Theta> f = Some as; \<Gamma> f = Some b; n < length as; ann_matches \<Gamma> \<Theta> (as ! n) b; P \<Gamma> \<Theta> (as ! n) b\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnCall a n) (Call f)"   and "\<And>r \<Gamma> \<Theta> a c. \<forall>s\<in>r. ann_matches \<Gamma> \<Theta> a (c s) \<and> P \<Gamma> \<Theta> a (c s) \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec r a) (DynCom c)"   and "\<And>as cs \<Gamma> \<Theta>. \<lbrakk>length as = length cs; \<forall>i<length cs. ann_matches \<Gamma> \<Theta> (fst (as ! i)) (cs ! i) \<and> P \<Gamma> \<Theta> (fst (as ! i)) (cs ! i)\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnPar as) (Parallel cs)" shows "P x1 x2 x3 x4"
conga_distinct: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E"
k: shows "insert (Upair u v) F \<inter> F = F"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b dual"   and b :: "'c dual" assumes "rel_dual R a b" obtains xa :: "'b" and ya :: "'c" where "a = \<partial> xa" and "b = \<partial> ya" and "R xa ya"
fls_const_mult_const: fixes x :: "'a"   and y :: "'a" shows "fls_const x * fls_const y = fls_const (x * y)"
ladder_\<alpha>_is_\<gamma>: shows "ladder_\<alpha> (\<alpha> @ \<delta>) D L (Suc index) = ladder_\<gamma> (\<alpha> @ \<delta>) D L index"
1: shows "idEnv [y @ys[z1 \<and> z2]_zs \<leftarrow> Y #[z1 \<and> z2]_zs]_ys = idEnv [y \<leftarrow> Y]_ys &[z1 \<and> z2]_zs"
some_img_def: shows "S'.some_img = (SOME img. set_category_given_img (\<cdot>\<acute>) img \<AA>)"
is_aligned_mult_triv1: fixes n :: "nat"   and x :: "'b word" shows "is_aligned (2 ^ n * x) n"
agree_on_refl: fixes R :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set" shows "agree_on R f f"
declared_in_classD: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname" assumes "G\<turnstile> m declared_in C" shows "is_class G C"
dvd: fixes n :: "nat" assumes "n dvd p" shows "n = 1 \<or> n = p"
igSubstIGOpSTR_imp_igSubstIGOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igSubstIGOpSTR MOD" shows "igSubstIGOp MOD"
Diff_subset_conv: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
ConsThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>throw e' # es,s\<^sub>1\<rangle>"
path_is_rtrancl: fixes E :: "('c \<times> 'c) set"   and u :: "'c"   and l :: "'c list"   and v :: "'c" assumes "path E u l v" shows "(u, v) \<in> E\<^sup>*"
h2: shows "take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))) = take_rows_from_matrix (take_cols_from_matrix (M_mat signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (rows_to_keep (take_cols_from_matrix (M_mat signs subsets) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))))"
zero_order_2: fixes n :: "'k multiset" shows "(n \<subseteq># {#}) = (n = {#})"
finite_starsetNat_iff: fixes A :: "nat set" shows "(*s* A = hypnat_of_nat ` A) = finite A"
distinct_3: fixes x1 :: "instr"   and x31 :: "char list"   and x32 :: "char list list" shows "StepI x1 \<noteq> StepC2 x31 x32"
div_0: fixes a :: "'b" shows "(0::'b) div a = (0::'b)"
le_iff_diff_le_0: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a b = less_eq (minus a b) zero"
polynate_norm: fixes p :: "poly" shows "isnpoly (polynate p)"
order_refl: fixes less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'i" assumes "class.preorder less_eq less" shows "less_eq x x"
asymmetric_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "asymmetric A less \<equiv> \<forall>x y. less x y \<longrightarrow> less y x \<longrightarrow> x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> False"
ccval_def: fixes cc :: "('a, 'b) cconstraint" shows "Regions.ccval cc \<equiv> {v. v \<turnstile> cc}"
eucl_truncate_up_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and q :: "nat"   and b :: "'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "eucl_truncate_up q b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (truncate_up q (inner b i)) i) Basis"
cring_simprules_12: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "cring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
that: fixes p :: "(name \<times> name) list"   and T :: "('a, 'b, 'c) psi" assumes "set p \<subseteq> set xvec \<times> set yvec"   and "P = (p \<bullet> Q) \<parallel> T"   and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' R" shows "thesis"
to_tree_order_def: fixes ptr :: "(_) object_ptr" shows "to_tree_order ptr = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. Heap_Error_Monad.bind (map_M to_tree_order (map cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r children)) (\<lambda>treeorders. return (ptr # concat treeorders)))"
simps_8: fixes m :: "nat"   and p :: "fm"   and q :: "fm" shows "bound m (And p q) = (bound m p \<and> bound m q)"
open_uniformity: fixes "open" :: "'a set \<Rightarrow> bool"   and uniformity :: "('a \<times> 'a) filter"   and U :: "'a set" assumes "class.open_uniformity open uniformity" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
induct: fixes P :: "(bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "(bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and a1 :: "nat"   and a2 :: "nat" assumes "\<And>uu_ uv_. P [] uu_ uv_"   and "\<And>uw_ ux_ i' uy_ uz_ j' va_ ps i j. P ps i j \<Longrightarrow> P ((uw_, (ux_, i', uy_), uz_, j', va_) # ps) i j" shows "P a0 a1 a2"
mcalc: fixes c :: "('fsym, 'psym) lit list"   and \<sigma> :: "'tp" assumes "c \<in> \<Phi>" shows "\<sigma> \<turnstile> c"
span_add1: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and U :: "'d set"   and v :: "'d"   and w :: "'d" assumes "vectorspace K V"   and "U \<subseteq> carrier V"   and "v \<in> LinearCombinations.module.span K V U"   and "w \<in> LinearCombinations.module.span K V U" shows "v \<oplus>\<^bsub>V\<^esub> w \<in> LinearCombinations.module.span K V U"
x1: assumes "b $\<^sub>e 0 \<noteq> a $\<^sub>e 0" shows "x $\<^sub>e Suc 0 = ((b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0) * x $\<^sub>e 0 - a $\<^sub>e 0 * b $\<^sub>e Suc 0 + b $\<^sub>e 0 * a $\<^sub>e Suc 0) / (b $\<^sub>e 0 - a $\<^sub>e 0)"
fcompare_double_swap: fixes x :: "double"   and y :: "double" shows "(fcompare_double x y = ccode.Gt) = (fcompare_double y x = ccode.Lt)"
scaleR_fun_def: fixes r :: "real"   and f :: "'b \<Rightarrow> 'c" shows "r *\<^sub>R f = (\<lambda>x. r *\<^sub>R f x)"
eucl_rel_poly_dividesI: fixes y :: "'b poly"   and x :: "'b poly"   and q :: "'b poly" assumes "y \<noteq> 0"   and "x = q * y" shows "eucl_rel_poly x y (q, 0)"
invProp: assumes "normal s3" shows "G\<turnstile>invmode statM e\<rightarrow>invC\<preceq>statT"
continuous_on_Un_local_open: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "openin (top_of_set (S \<union> T)) S"   and "openin (top_of_set (S \<union> T)) T"   and "continuous_on S f"   and "continuous_on T f" shows "continuous_on (S \<union> T) f"
can_def: fixes f :: "'c local.term set" shows "can f \<equiv> arr f \<and> (\<exists>t. t \<in> f \<and> Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>)"
mbal'_gt: shows "\<forall>bk\<in>allBlocksRead s p. mbal bk \<le> mbal (dblock s' p)"
TRANS: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "trans r"
exec_step_input_StepTD: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and Cs :: "char list list"   and a :: "nat" assumes "exec_step_input P C M pc ics = StepT Cs a" shows "ics = Throwing Cs a"
lookup_pair_update_by_pair: fixes xs :: "('a \<times> 'b) list"   and k1 :: "'a"   and v :: "'b"   and k2 :: "'a" assumes "tc.oalist_inv_raw xs" shows "lookup_pair_tc (update_by_pair_tc (k1, v) xs) k2 = (if k1 = k2 then v else lookup_pair_tc xs k2)"
TTI: fixes G :: "('a, 'b) form list" shows "G \<turnstile> TT"
frontier: fixes u :: "'v"   and v :: "'v" assumes "\<pi>' Q \<pi> u u = Some v" shows "v \<in> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"
g_2: shows "\<forall>i. totalizer 0 i = \<phi> (the (g i))"
pw_leI': fixes S' :: "'a nres"   and S :: "'a nres" assumes "nofail S' \<Longrightarrow> nofail S"   and "\<And>x. \<lbrakk>nofail S'; inres S x\<rbrakk> \<Longrightarrow> inres S' x" shows "S \<le> S'"
leg1_simps_3: shows "src (tab\<^sub>1 f) = src (tab\<^sub>0 f)"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_start m rs (l, r) = (\<exists>ln rn. l = Bk # Oc \<up> Suc m \<and> tl r = Oc # Bk \<up> ln @ Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
call_return: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list"   and rrs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)"   and "matches \<gamma> m' p"   and "\<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m (Call chain) # rrs \<Rightarrow>\<^sub>z t"
simps: fixes e :: "expr"   and t :: "pdf_type"   and f :: "cexpr" shows "(e : t \<Rightarrow>\<^sub>c f) = ((\<lambda>_. UNIT) \<turnstile> e : t \<and> free_vars e = {} \<and> ([], [], \<lambda>_. UNIT, CReal 1) \<turnstile>\<^sub>c e \<Rightarrow> f)"
div_comp_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and rs1 :: "word5"   and rd :: "word5"   and operand2 :: "word32" shows "div_comp instr rs1 rd operand2 \<equiv> do curr_win \<leftarrow> get_curr_win (); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); y_val \<leftarrow> gets (cpu_reg_val Y); y_rs1 \<leftarrow> gets (\<lambda>s. word_cat y_val rs1_val); temp_64bit \<leftarrow> gets (\<lambda>s. div_comp_temp_64bit instr y_rs1 operand2); temp_high32 \<leftarrow> gets (\<lambda>s. ucast (temp_64bit >> 32)); temp_high33 \<leftarrow> gets (\<lambda>s. ucast (temp_64bit >> 31)); temp_V \<leftarrow> gets (\<lambda>s. div_comp_temp_V instr temp_high32 temp_high33); result \<leftarrow> gets (\<lambda>s. div_comp_result instr temp_V temp_64bit); rd_val \<leftarrow> gets (user_reg_val curr_win rd); new_rd_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then result else rd_val); write_reg new_rd_val curr_win rd; div_write_new_val instr result temp_V od"
satL_transE: fixes l :: "('fsym, 'psym) lit"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtL l"   and "I.wtE \<xi>"   and "\<And>x. \<lbrakk>\<not> infTp (tpOfV x); x \<in> nv2L l\<rbrakk> \<Longrightarrow> \<exists>l'. wtL l' \<and> \<not> I.satL \<xi> l' \<and> isGuard x l'"   and "Ik.satL (transE \<xi>) l" shows "I.satL \<xi> l"
rel_refl: fixes Ra :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l option" assumes "\<And>x. Ra x x" shows "rel_option Ra x x"
notin_List_update: fixes q :: "ref"   and ps :: "ref list"   and p :: "ref"   and h :: "ref \<Rightarrow> ref"   and y :: "ref" assumes "q \<notin> set ps" shows "List p (h(q := y)) ps = List p h ps"
inverse_eq_iff_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = inverse b) = (a = b)"
f_sum_S': fixes i :: "'a" shows "f abs_summable_on S' i"
ord_eq_le_trans: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a = b"   and "b \<le>s c" shows "a \<le>s c"
ex_un_null: shows "\<exists>!n. \<forall>f. VV.comp n f = n \<and> VV.comp f n = n"
finiteI_independent: fixes B :: "'b set" assumes "local.independent B" shows "finite B"
simps_1: shows "Leftist_Heap.linorder_class.heap \<langle>\<rangle> = True"
ComSecTyping_single_is_sound: fixes c :: "('exp, 'id) MWLfCom" assumes "\<turnstile>\<^bsub>\<C>\<^esub> c" shows "Strongly_Secure [c]"
correctness_OT_14: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "funct_OT_14 M C = protocol_14_OT M C"
phiNodes_of_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" shows "uninst_code.ssa.phiNodes_of p g = fold (uninst_code.ssa.phis_addN p g) (sorted_list_of_set (dom (Mapping.lookup (p g)))) Mapping.empty"
closed_imp_fip: fixes S :: "'b set"   and T :: "'b set"   and \<F> :: "'b set set" assumes "closed S"   and "T \<in> \<F>"   and "bounded T"   and "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"   and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter> \<F>' \<noteq> {}" shows "S \<inter> \<Inter> \<F> \<noteq> {}"
bot_eq_sup_iff: fixes x :: "'a"   and y :: "'a" shows "((0::'a) = x + y) = (x = (0::'a) \<and> y = (0::'a))"
aux1: fixes M :: "object atom formula set"   and \<phi> :: "object atom formula"   and \<A> :: "object atom \<Rightarrow> bool" assumes "wm_basic M"   and "is_STRIPS_fmla \<phi>"   and "valuation M \<Turnstile> \<phi>"   and "\<forall>G\<in>M. \<A> \<Turnstile> G" shows "\<A> \<Turnstile> \<phi>"
that: assumes "pnop A (2\<^sub>\<nat>) f" shows "thesis"
fr1: shows "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)"
vertical_composite_axioms: shows "vertical_composite A B F G H \<sigma> \<tau>"
mon_mono: fixes r :: "'a set \<Rightarrow> 'b set" shows "mono r = (\<forall>X Y. X \<subseteq> Y \<longrightarrow> r X \<subseteq> r Y)"
SD_efficient_same_support: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and p' :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "set_pmf p = set_pmf p'"   and "p \<in> lotteries_on alts" shows "SD_efficient R p = SD_efficient R p'"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
instances_def: fixes S :: "'a eclause set" shows "instances S = {x. \<exists>C \<sigma>. C \<in> S \<and> ground_clause (subst_cl (cl_ecl C) \<sigma>) \<and> x = (C, \<sigma>)}"
set_child_nodes_writes: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"
assms: shows "(cms, mem) \<rightarrow> (cms', mem')"
OUT_plus_current: fixes x :: "'v" shows "d_OUT (plus_current f g) x = (if x \<in> RF\<^sup>\<circ> (TER f) then d_OUT f x else d_OUT g x)"
bit_and_int_iff: fixes k :: "int"   and l :: "int"   and n :: "nat" shows "bit (k AND l) n = (bit k n \<and> bit l n)"
seq_char: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" shows "seq g f = (arr g \<and> arr f \<and> (f = Zero \<and> g \<noteq> One \<or> f \<noteq> Zero \<and> g = One))"
eval_models_aux: fixes ec :: "'s odlist"   and t :: "'s Trace"   and s :: "'s"   and \<phi> :: "('a, 'p) Kform" assumes "spr_simAbs ec = spr_jkbpCSt t"   and "s \<in> toSet ec" shows "(s \<in> toSet (eval envVal ec \<phi>)) = spr_repMC (toSet ec), s \<Turnstile> \<phi>"
simps_2: fixes i :: "char"   and "is" :: "char list" shows "internal_iface_name_match (i # is) [] = (i = CHR ''+'' \<and> is = [])"
c: shows "B \<in> \<gamma>\<^sub>T\<^sub>R K \<phi>"
lfinite_llist_of: fixes xs :: "'a list" shows "lfinite (llist_of xs)"
cluster_union: fixes C\<^sub>1 :: "'node set"   and C\<^sub>2 :: "'node set" assumes "is_cons_cluster C\<^sub>1"   and "is_cons_cluster C\<^sub>2"   and "C\<^sub>1 \<inter> C\<^sub>2 \<noteq> {}" shows "is_cons_cluster (C\<^sub>1 \<union> C\<^sub>2)"
metric_CauchyD: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "uniform_space.Cauchy uniformity X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
append_Cons: fixes x :: "'f"   and xs :: "'f list"   and ys :: "'f list" shows "(x # xs) @ ys = x # xs @ ys"
except_min_Nil: fixes xs :: "'x"   and ko :: "'o" assumes "fst (list_of_oalist xs) = []" shows "fst (list_of_oalist (except_min ko xs)) = []"
absorb_iff2: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (inf_class.inf a b = b)"
nonneg: fixes x :: "'a" assumes "x \<in> cbox a b" shows "0 \<le> f x"
alpha_abs_res_stronger1_aux: fixes as :: "atom set"   and x :: "'a"   and p' :: "perm"   and as' :: "atom set"   and x' :: "'a" assumes "(as, x) \<approx>res (=) supp p' (as', x')" shows "\<exists>p. (as, x) \<approx>res (=) supp p (as', x') \<and> supp p \<subseteq> supp x \<inter> as \<union> supp x' \<inter> as'"
full_chain_not_lnull: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "full_chain R xs" shows "\<not> lnull xs"
cring_simprules_8: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
orthonormal_basis_expand: fixes B :: "'c set"   and x :: "'c" assumes "pairwise orthogonal B"   and "\<And>i. i \<in> B \<Longrightarrow> norm i = 1"   and "x \<in> span B"   and "finite B" shows "(\<Sum>i\<in>B. (x \<bullet> i) *\<^sub>R i) = x"
po_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "po_on P A = (irreflp_on P A \<and> transp_on P A)"
syzygy_moduleI: fixes B :: "'b set"   and s :: "'b \<Rightarrow>\<^sub>0 'a" assumes "represents B s (0::'b)" shows "s \<in> syzygy_module B"
Limsup_within: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "Limsup (at x within S) f = (INF e\<in>{0<..}. Sup (f ` (S \<inter> ball x e - {x})))"
extractFrameFreshChain_1: fixes xvec :: "name list"   and P :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* P" shows "xvec \<sharp>* extractFrame P"
prefix_fininf_list: fixes w :: "'a list"   and u :: "'a list"   and v :: "'a stream" shows "(w @ u \<le>\<^sub>F\<^sub>I w @- v) = (u \<le>\<^sub>F\<^sub>I v)"