hcju/isabelleps huggingface dataset:
asian_borel: fixes T :: "nat" shows "asian_option T \<in> borel_measurable (G T)"
P'_FreeS: shows "order.greater_eq (FreeGroup S) (Abs_freeword ` map (charpair S) ` R)"
XXX_2: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (override_on Pd Pds' ds') Ch"
AAcc1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and a' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and i :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>a,s\<rangle> -ta\<rightarrow> \<langle>a',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>a\<lfloor>i\<rceil>,s\<rangle> -ta\<rightarrow> \<langle>a'\<lfloor>i\<rceil>,s'\<rangle>"
r_into_rtranclp: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
mem_cball: fixes y :: "'b"   and x :: "'b"   and e :: "real" shows "(y \<in> cball x e) = (dist x y \<le> e)"
assms_4: shows "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>r result"
bool_cast: shows "G\<turnstile>S\<preceq>? PrimT Boolean"
simps_2: fixes vb :: "('a, 'b) graph_node"   and vc :: "'a"   and vd :: "nat"   and ve :: "'c"   and va :: "('a, 'b, 'c, 'd) itnode rose_tree list" shows "isHNode (INode vb vc vd ve va) = False"
has_integral_periodic_offset_weak: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and i :: "real"   and c :: "real" assumes "has_bochner_integral (lebesgue_on {a..b}) f i"   and "\<And>x. f (x + (b - a)) = f x"   and "\<bar>c\<bar> \<le> b - a" shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f (x + c)) i"
p_def: fixes k :: "nat" shows "p k = (\<Sum>i<n. x i ^ k)"
p: shows "p = (\<Sum>x\<in>keys (mapping_of p). MPoly_Type.monom x (MPoly_Type.coeff p x))"
Suc_rtos_mod_r_1: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "Suc (rtos r s i) < r * s"   and "i mod s = s - Suc 0"   and "i div s < r - Suc 0" shows "Suc (rtos r s i) mod r = Suc (i div s)"
Lcm_rat_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s'" shows "\<Gamma>\<turnstile>Catch c\<^sub>1 c\<^sub>2 \<down> Normal s"
disc_iff_2: fixes R :: "('f, 'g, 'h) Resumption.resumption set" shows "(\<not> Resumption.resumption.is_Done (resumption_lub R)) = (\<not> (\<forall>r\<in>R. Resumption.resumption.is_Done r))"
simps: fixes f :: "'c \<Rightarrow> 'a"   and q :: "'c"   and l :: "'b"   and qs :: "'c list" shows "remap_rule f q \<rightarrow> l qs = f q \<rightarrow> l map f qs"
intro: fixes U :: "'a set"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a" assumes "lattice_ow U inf le ls sup"   and "order_top_ow U le ls top" shows "bounded_lattice_top_ow U inf le ls sup top"
FAssRedsStatic: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
protocol_inverse: fixes m0 :: "'a"   and m1 :: "'a"   and a :: "nat"   and b :: "nat"   and s1 :: "nat"   and r1 :: "nat"   and v :: "bool" assumes "m0 \<in> carrier \<G>"   and "m1 \<in> carrier \<G>" shows "(\<^bold>g [^] (a * b mod order \<G>)) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1 \<otimes> (if v then m0 else m1) \<otimes> inv (((\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1) [^] b) = (if v then m0 else m1)"
idiff_0_right: fixes n :: "eint" shows "n - 0 = n"
is_monoid: fixes R :: "('b, 'c) ring_scheme" assumes "ring R" shows "Group.monoid R"
nop_onto_vrange: shows "\<R>\<^sub>\<circ> vint_plus = \<int>\<^sub>\<circ>"
arith_simps_14: fixes m :: "num"   and n :: "num" shows "num.Bit1 m * num.Bit0 n = num.Bit0 (num.Bit1 m * n)"
inj_apfst: fixes f :: "'b \<Rightarrow> 'd" shows "inj (apfst f) = inj f"
rel_envT_def: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "local.rel_envT R M = BNF_Def.vimage2p run_env run_env (R ===> M)"
Bv_carrier: shows "set Bv \<subseteq> carrier_vec d"
inheritedIncLevel: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and A :: "(nat \<times> 'b) set" assumes "inherited subs P"   and "P A" shows "P (incLevel ` A)"
hyps: shows "snd (snd (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<k])) = bezout"
set_insort_wrt: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "set (insort_wrt R xs) = set xs"
mkPoint_img_2: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and x :: "'b"   and a :: "'b" assumes "set_category S \<AA>"   and "partial_magma.in_hom S x (set_category_data.unity S) a" shows "set_category.mkPoint S \<AA> a (set_category.img S \<AA> x) = x"
vars0_def: shows "d.vars0 = vars1 \<union> (vars2 \<union> vars3)"
avl_delete: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "avl (delete x t) \<and> height t = height (delete x t) + (if decr t (delete x t) then 1 else 0)"
simps_7: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and F' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' - set X"
remove_vertex_walk: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and x :: "'c" assumes "Digraph G"   and "Digraph.walk G xs"   and "x \<notin> set xs" shows "Digraph.walk (Digraph.remove_vertex G x) xs"
prefix_of_node_history_def: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" shows "xs prefix of i \<equiv> \<exists>ys. xs @ ys = history i"
isNSCont_isCont_iff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "isNSCont f a = isCont f a"
matrix_to_iarray_pair_def: fixes X :: "(('c, 'e) vec, 'f) vec \<times> (('d, 'g) vec, 'h) vec" shows "matrix_to_iarray_pair X = (matrix_to_iarray (fst X), matrix_to_iarray (snd X))"
to_Gromov_completion_def: fixes x :: "'a" shows "to_Gromov_completion x = abs_Gromov_completion (\<lambda>n. x)"
singleton_dotP: fixes x :: "'a vec"   and v :: "'a" assumes "dim_vec x = 1" shows "[v]\<^sub>v \<bullet> x = v * x $ 0"
pupdate_DefaultPUpdate: shows "(\<lambda>D. DefaultPData (Data.DataSpace D)) \<in> pupdate"
d_weak_loc: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = (0::'a)) = (x \<cdot> d y = (0::'a))"
setdist_eq_0_closed_compact: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "compact T" shows "(setdist S T = 0) = (S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {})"
echelon_foldl_condition4: fixes k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec"   and mb :: "'d" assumes "\<forall>m>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'d). A $ m $ mod_type_class.from_nat k = (0::'e)"   and "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"   and "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'d) \<le> mb"   and "A $ mb $ mod_type_class.from_nat k \<noteq> (0::'e)" shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) A)"
case_eq_if: fixes f :: "'c word \<Rightarrow> nat \<Rightarrow> 'b"   and prefix_match :: "'c prefix_match" shows "(case prefix_match of PrefixMatch x xa \<Rightarrow> f x xa) = f (pfxm_prefix prefix_match) (pfxm_length prefix_match)"
comp_right_isotone: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "less_eq x y" shows "less_eq (times z x) (times z y)"
ln_less_zero: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "ln x < 0"
ipoly_min_int_poly: fixes x :: "'b" assumes "algebraic x" shows "ipoly (min_int_poly x) x = (0::'b)"
singletonI: fixes a :: "'b" shows "a \<in> \<eta> a"
K_def: shows "K = U \<inter> \<omega> -` {min_prio}"
\<tau>exec_methE: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>exec_meth ci P ins xt t h s s'" obtains stk :: "'addr val list" and loc :: "'addr val list" and pc :: "nat" and xcp :: "'addr option" where "s = (stk, loc, pc, xcp)" and "exec_meth ci P ins xt t h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'" and "xcp = None \<Longrightarrow> \<tau>instr P h stk (ins ! pc)"
ide_trg: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.ide (S.trg \<mu>)"
aezfun_common_supp_spanning_set: fixes A :: "'a set" assumes "finite A" shows "\<exists>as. distinct as \<and> set as = A \<and> (\<forall>x\<in>aezfun_setspan A. \<exists>bs. length bs = length as \<and> x = sum_list (map2 (\<delta>\<delta>) bs as))"
elims_1: fixes x :: "protocol"   and xa :: "8 word"   and y :: "bool" assumes "match_proto x xa = y" obtains uu_ :: "8 word" where "x = ProtoAny" and "xa = uu_" and "y"   | p :: "8 word" and p_p :: "8 word" where "x = Proto p" and "xa = p_p" and "y = (p_p = p)"
simps_7: shows "cospan_shape.comp cospan_shape.TT cospan_shape.TT = cospan_shape.TT"
intro: fixes R :: "('e, 'g) ring_scheme"   and M :: "('e, 'f, 'h) module_scheme" assumes "\<And>a x y. \<lbrakk>a \<in> carrier R; x \<in> carrier M; y \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<otimes>\<^bsub>M\<^esub> y = a \<odot>\<^bsub>M\<^esub> (x \<otimes>\<^bsub>M\<^esub> y)" shows "algebra_axioms R M"
sign_simps_5: fixes ah :: "'b"   and bh :: "'b" shows "ah * bh = bh * ah"
MP_thin: fixes HA :: "fm set"   and A :: "fm"   and B :: "fm"   and HB :: "fm set"   and H :: "fm set" assumes "HA \<turnstile> A IMP B"   and "HB \<turnstile> A"   and "HA \<union> HB \<subseteq> H" shows "H \<turnstile> B"
binomial_addition_formula: fixes n :: "nat"   and k :: "nat" assumes "0 < n" shows "n choose Suc k = n - 1 choose Suc k + (n - 1 choose k)"
ord_cong: fixes k1 :: "'a"   and k2 :: "'a"   and n :: "'a" assumes "[k1 = k2] (mod n)" shows "Pocklington.ord n k1 = Pocklington.ord n k2"
rep_eq: fixes x :: "('g, 'e) generator"   and xa :: "('g, 'f) generator"   and xb :: "'f" shows "generator (append_trans x xa xb) = append_raw (generator x) (generator xa) xb"
convex_def: fixes n :: "nat"   and S :: "'b vec set" shows "gram_schmidt.convex n S = (gram_schmidt.convex_hull n S = S)"
HAVOC_def: shows "HAVOC = [: x \<leadsto> y . True :]"
cfg_approx: fixes c :: "cmd"   and prog :: "'a program_scheme"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "c \<in> cfg_approx prog"   and "cfg c a c'" shows "c' \<in> cfg_approx prog"
m1a_trans_def: shows "m1a_trans \<equiv> \<Union>A B Ra Rb Rs Na Kab nls nla nlb. (m1a_step1 Ra A B Na \<union> m1a_step2 Rb A B \<union> m1a_step3 Rs A B Kab Na nls \<union> m1a_step4 Ra A B Na Kab nla \<union> m1a_step5 Rb A B Kab nlb \<union> m1a_leak Rs)\<^sup>="
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "ergodic_qmpt M T"   and "conservative M T" shows "ergodic_conservative M T"
quality_increases_refl: fixes \<xi> :: "state" shows "quality_increases \<xi> \<xi>"
trans_f: shows "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt (f z) (f y) \<longrightarrow> gt (f y) (f x) \<longrightarrow> gt (f z) (f x)"
\<F>\<G>_implies_FG: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<psi>\<in>\<F>\<G> \<phi> w. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[\<G>\<F> \<phi> w]\<^sub>\<nu>)"
1: shows "length is1 = card {i \<in> A. i < length is1 + length is2}"
pt_fresh_bij: fixes pi :: "('b \<times> 'b) list"   and a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
le_cSup_finite: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq x (Sup X)"
CEM_def: shows "CEM \<equiv> CM \<and> EM"
Un_upper1: fixes A :: "'e set"   and B :: "'e set" shows "A \<subseteq> A \<union> B"
prems_1: shows "b x = Some y \<or> (\<exists>m\<in>set (a # ms). m x = Some y)"
timpl_closure_Fun_inv'_2: fixes g :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and f :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set"   and i :: "nat" assumes "Fun g S \<in> timpl_closure (Fun f T) TI"   and "i < length S" shows "S ! i \<in> timpl_closure (T ! i) TI"
viewL_correct_2: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t \<noteq> Empty" shows "\<exists>a s. FingerTreeStruc.viewL t = Some (a, s) \<and> FingerTreeStruc.ft_invar s \<and> FingerTreeStruc.toList t = a # FingerTreeStruc.toList s"
assms: fixes t :: "real"   and e :: "real" assumes "0 \<le> t"   and "0 < e" shows "\<exists>tt\<ge>t. dist (f tt) p < e"
len_gt_false: assumes "length (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) < dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))" shows "False"
is_aligned_no_wrap: fixes ptr :: "'b word"   and sz :: "nat"   and off :: "'b word" assumes "is_aligned ptr sz"   and "off < 2 ^ sz" shows "unat ptr + unat off < 2 ^ LENGTH('b)"
Id_on_Gr: fixes A :: "'a set" shows "Id_on A = BNF_Def.Gr A id"
simps_23: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a" shows "(case a\<^sub>3 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3 | a\<^sub>4 \<Rightarrow> f4 | a\<^sub>5 \<Rightarrow> f5) = f3"
root_main': fixes x :: "int"   and n :: "int" assumes "0 \<le> x"   and "0 \<le> n" shows "root_main' x n = root_newton_int_main x n"
ne_SUP_mono: fixes U\<^sub>2 :: "'c set"   and U\<^sub>3 :: "'d set"   and A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'al"   and g :: "'d \<Rightarrow> 'al" assumes "U\<^sub>2 \<noteq> {}"   and "U\<^sub>3 \<noteq> {}"   and "A \<subseteq> U\<^sub>2"   and "B \<subseteq> U\<^sub>3"   and "\<forall>x\<in>U\<^sub>2. f x \<in> U"   and "\<forall>x\<in>U\<^sub>3. g x \<in> U"   and "\<And>n. \<lbrakk>n \<in> U\<^sub>2; n \<in> A\<rbrakk> \<Longrightarrow> \<exists>x\<in>B. f n \<le>\<^sub>o\<^sub>w g x" shows "\<Squnion>\<^sub>o\<^sub>w (f ` A) \<le>\<^sub>o\<^sub>w \<Squnion>\<^sub>o\<^sub>w (g ` B)"
singular_simplex_singular_face: fixes p :: "nat"   and X :: "'c topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c"   and k :: "nat" assumes "singular_simplex p X f"   and "1 \<le> p"   and "k \<le> p" shows "singular_simplex (p - Suc 0) X (singular_face p k f)"
Order: fixes G :: "('a, 'b) form list"   and G' :: "('a, 'b) form list" assumes "\<stileturn> G"   and "set G = set G'" shows "\<stileturn> G'"
intro: fixes bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "\<And>m m'. bind m (\<lambda>_. m') = m'" shows "monad_discard_axioms bind"
assms_3: fixes x :: "'a" assumes "x \<in> S" shows "f (f x) = f x"
opset_subset: fixes Y :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set"   and X :: "('c \<times> 'd) set" assumes "opset Y deps"   and "X \<subseteq> Y" shows "opset X deps"
diagonal_to_Smith_aux_PQ_append: fixes xs :: "nat list"   and ys :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_aux_PQ (xs @ ys) bezout (P, A, Q) = diagonal_to_Smith_aux_PQ ys bezout (diagonal_to_Smith_aux_PQ xs bezout (P, A, Q))"
int_pow_neg: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
borel_measurable_inner: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M"
seqI': fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>" shows "D.seq g f"
inject_11: fixes x2 :: "name"   and x3 :: "coname"   and x4 :: "trm"   and x1 :: "coname"   and y2 :: "name"   and y3 :: "coname"   and y4 :: "trm"   and y1 :: "coname" shows "(ImpR x2.<x3>.x4 x1 = ImpR y2.<y3>.y4 y1) = ([x2].[x3].x4 = [y2].[y3].y4 \<and> x1 = y1)"
construction_uniqueness: fixes Q :: "'p"   and A :: "'p"   and X :: "'p"   and B :: "'p"   and C :: "'p"   and Y :: "'p" assumes "Q \<noteq> A"   and "Bet Q A X"   and "Cong A X B C"   and "Bet Q A Y"   and "Cong A Y B C" shows "X = Y"
id: shows "\<parallel>f\<parallel>\<^sub>\<infinity> = \<bar>coeff f i\<bar>"
le_TC_trans: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
ring_hom_imp_img_ring: fixes h :: "'a \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring (S\<lparr>carrier := h ` carrier R, zero := h \<zero>\<rparr>)"
abs_eq: shows "0 = Abs_euclidean_space (\<lambda>_. 0)"
valid_c: shows "\<Gamma>|\<^bsub>M\<^esub>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
prems_2: shows "\<forall>a\<in>set (y : ys). \<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>"
simps_7: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and vc :: "('b, 'c) Combinators"   and vd :: "('b, 'c) Combinators list" shows "removeShadowRules1_alternative_rev (AllowPortFromTo v va vb # vc # vd) = AllowPortFromTo v va vb # removeShadowRules1_alternative_rev (vc # vd)"
in_closed_iff_infdist_zero: fixes A :: "'a set"   and x :: "'a" assumes "closed A"   and "A \<noteq> {}" shows "(x \<in> A) = (infdist x A = 0)"
gfp_exp_upperbound: fixes P :: "'b \<Rightarrow> real"   and t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "P \<tturnstile> t P"   and "unitary P" shows "P \<tturnstile> gfp_exp t"
Un_insert_right: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A \<union> insert a B = insert a (A \<union> B)"
g_def: shows "g = Cn 3 r_ifz [Cn 3 r_nth [recf.Id 3 2, recf.Id 3 0], Cn 3 r_ifeq [recf.Id 3 0, recf.Id 3 1, Cn 3 S [recf.Id 3 0], recf.Id 3 1], recf.Id 3 0]"
fract_collapse_3: fixes k :: "'a" shows "Fraction_Field.Fract k (0::'a) = 0"
enat_of_option_inj: fixes x :: "nat option"   and y :: "nat option" shows "(enat_of_option x = enat_of_option y) = (x = y)"
IH1: fixes P :: "'a mat"   and A :: "'a mat"   and Q :: "'a mat" assumes "A' $h from_nat i $h from_nat i dvd A' $h from_nat j $h from_nat j"   and "HMA_M3 (P, A, Q) (P', A', Q')"   and "i < min CARD('nr) CARD('nc)"   and "\<forall>j\<in>set xs. j < min CARD('nr) CARD('nc)" shows "HMA_M3 (diagonal_to_Smith_i_PQ_JNF xs i bezout (P, A, Q)) (diagonal_to_Smith_i_PQ xs i bezout (P', A', Q'))"
strict_order_iff: fixes a :: "'b"   and b :: "'b" shows "order.greater b a = (a = inf a b \<and> a \<noteq> b)"
prems_2: shows "vars (c' # F') \<subseteq> vars valuation"
mult2_ns_one_step: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "ns O s \<subseteq> s"   and "refl ns"   and "trans s" shows "((X, Y) \<in> mult2_ns ns s) = (\<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)))"
is_inf_prefix: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "length zs = length (xs \<sqinter> ys)"   and "\<And>i. i < length (xs \<sqinter> ys) \<Longrightarrow> zs ! i = xs ! i \<and> zs ! i = ys ! i" shows "zs = xs \<sqinter> ys"
discr0_MtransT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>*t s'" shows "s \<approx> s'"
non1: fixes z :: "complex" assumes "z \<in> S" shows "f z \<noteq> 1 \<and> f z \<noteq> - 1"
af_letter_build: fixes x :: "'b set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "(x ## w) \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n af_letter \<phi> x"
hull_same: fixes S :: "'d set \<Rightarrow> bool"   and s :: "'d set" assumes "S s" shows "S hull s = s"
encPf: fixes pf :: "'proof" assumes "pf \<in> proof" shows "encPf pf \<in> num"
o_assoc: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'e" shows "(f \<circ>\<circ>\<circ> (\<circ>)) g h = f \<circ> g \<circ> h"
ucast_up_ucast: fixes w :: "'b word" assumes "is_up UCAST('b \<rightarrow> 'c)" shows "UCAST('c \<rightarrow> 'd) (UCAST('b \<rightarrow> 'c) w) = UCAST('b \<rightarrow> 'd) w"
pbij_alt_intro: fixes X :: "'d \<Longrightarrow> 'e" assumes "weak_lens X"   and "\<And>s. s \<in> \<S>\<^bsub>X\<^esub> \<Longrightarrow> create\<^bsub>X\<^esub> (get\<^bsub>X\<^esub> s) = s" shows "pbij_lens X"
Diff_eq_empty_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A - B = {}) = order.greater_eq B A"
weaken_append: fixes R :: "(formula list \<times> formula list set) set"   and A :: "formula list"   and Gamma :: "formula list" assumes "Contrs \<subseteq> R"   and "Perms \<subseteq> R"   and "A @ Gamma @ Gamma \<in> deductions R" shows "A @ Gamma \<in> deductions R"
singleton_insert_inj_eq: fixes b :: "'c"   and a :: "'c"   and A :: "'c set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"