hcju/isabelleps huggingface dataset:
converse_tranclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r y b \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>+\<^sup>+ z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
obj_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.obj (trg\<^sub>C \<mu>)"
well_analyzed_decomp_rm\<^sub>e\<^sub>s\<^sub>t_fv: fixes A :: "('fun, 'var) extstrand_step list" assumes "well_analyzed A" shows "fv\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) = fv\<^sub>e\<^sub>s\<^sub>t A"
init_nsc: fixes nsc :: "('i \<times> 'a ns_constraint) list" shows "checked_nsc nsc {} (init_nsc nsc)"
image_gen: fixes S :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "finite S" shows "sum h S = (\<Sum>y\<in>g ` S. sum h {x \<in> S. g x = y})"
const_imply3: fixes P :: "'d \<Rightarrow> 'e"   and Q :: "'d \<Rightarrow> 'f"   and R :: "'d \<Rightarrow> 'g" assumes "\<And>\<tau> \<tau>'. \<lbrakk>P \<tau> = P \<tau>'; Q \<tau> = Q \<tau>'\<rbrakk> \<Longrightarrow> R \<tau> = R \<tau>'"   and "const P"   and "const Q" shows "const R"
assms_2: shows "\<forall>(c::'a::refinement_lattice) d::'a::refinement_lattice. (c \<parallel> (i::'a::refinement_lattice)) ; (d \<parallel> i) \<sqsubseteq> c ; d \<parallel> i"
idx_image_self: fixes xs :: "'a list" assumes "distinct xs" shows "(the \<circ>\<circ> idx) xs ` set xs = {..<length xs}"
product_def: shows "List.product \<equiv> rec_list (\<lambda>_. []) (\<lambda>x xs xsa ys. map (Pair x) ys @ xsa ys)"
wt_u_ge_t: shows "wt t \<le> wt u"
normalized_fracts_def: shows "normalized_fracts = {(a, b). coprime a b \<and> unit_factor b = (1::'a)}"
has_fields_Object: fixes P :: "'a prog"   and D :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs" shows "P \<turnstile> Object has_fields FDTs"
split_option_ex: fixes P :: "'d option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
rep_eq: fixes x :: "'b metric_completion"   and xa :: "'b metric_completion" shows "dist x xa = lim (\<lambda>n. dist (rep_metric_completion x n) (rep_metric_completion xa n))"
rel_envirI: fixes p :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>x. p (f x) (g x)" shows "rel_envir p f g"
has_locks_lock_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "may_lock l t" shows "has_locks (lock_lock l t) t = Suc (has_locks l t)"
semIfSC: fixes e :: "'exp"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "IfSideCondition e c1 c2" shows "\<forall>d. e \<equiv>\<^bsub>d\<^esub> e"
accI: fixes x :: "'k"   and r :: "('k \<times> 'k) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
P: shows "\<forall>x\<in>set (x # xs). P x"
cartesian_closed_category_axioms: shows "cartesian_closed_category (\<cdot>)"
polynomial_function_const: fixes c :: "'d" shows "polynomial_function (\<lambda>x. c)"
path_is_rtrancl: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and l :: "'b list"   and v :: "'b" assumes "path E u l v" shows "(u, v) \<in> E\<^sup>*"
WTThrow: fixes 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 :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list" assumes "P,E \<turnstile> e :: Class C" shows "P,E \<turnstile> throw e :: Void"
cinner_zero_left: fixes x :: "'a" shows "cinner (0::'a) x = 0"
C: shows "\<forall>u\<in>U. (xs, xs') \<in> R u"
mult_scalar_rec_left: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "p \<odot> q = monom_mult (lookup p t) t q + except p {t} \<odot> q"
intro_elim_6_e: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<psi> \<^bold>\<equiv> \<chi> in v]" shows "[\<phi> \<^bold>\<equiv> \<chi> in v]"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and S :: "real set" shows " D f = f' on S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
cases: fixes x :: "('b, 'c) result" obtains v2 :: "'b" where "x = Rval v2"   | e :: "'c error_result" where "x = Rerr e"
preserves_comp: fixes g :: "'j"   and f :: "'j" assumes "J.seq g f" shows "\<chi>'.A.map (g \<cdot>\<^sub>J f) = \<chi>'.A.map g \<cdot> \<chi>'.A.map f"
ev_blocks_partI: fixes m :: "nat"   and A :: "'a mat" assumes "\<And>i j k. \<lbrakk>i < j; j < k; k < m; A $$ (k, k) = A $$ (i, i)\<rbrakk> \<Longrightarrow> A $$ (j, j) = A $$ (i, i)" shows "local.ev_blocks_part m A"
Right_Coset_memI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H" shows "h \<cdot> x \<in> H |\<cdot> x"
sq_norm_g2_im1: shows "\<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 = \<parallel>fs.gs.gso i\<parallel>\<^sup>2 + fs.gs.\<mu> i (i - 1) * fs.gs.\<mu> i (i - 1) * \<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2"
conditionalize: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A" shows "sum g A = (\<Sum>a. if a \<in> A then g a else (0::'d))"
pkt_def: shows "pkt \<equiv> \<lambda>(no, sid). Pkt no sid"
quasinorm_on_def: fixes F :: "'c set"   and C :: "real"   and N :: "'c \<Rightarrow> ennreal" shows "quasinorm_on F C N = ((\<forall>x y. x \<in> F \<and> y \<in> F \<longrightarrow> x + y \<in> F \<and> N (x + y) \<le> ennreal C * N x + ennreal C * N y) \<and> (\<forall>c x. x \<in> F \<longrightarrow> c *\<^sub>R x \<in> F \<and> N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * N x) \<and> 1 \<le> C \<and> (0::'c) \<in> F)"
len_gt: assumes "1 < length l" shows "\<exists>n<length (a # l) - 1. (a # l) ! n < x \<and> x < (a # l) ! (n + 1)"
fin_sat_def: fixes S :: "'b formula set" shows "fin_sat S \<equiv> \<forall>s\<subseteq>S. finite s \<longrightarrow> sat s"
prems: shows "subseq (l1 @ l2 @ l3) (a # l)"
Body: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and n :: "nat"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>Init D\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2"   and "s3 = (if \<exists>l. fst s2 = Some (Jump (Break l)) \<or> fst s2 = Some (Jump (Cont l)) then abupd (\<lambda>x. Some (Error CrossMethodJump)) s2 else s2)" shows "G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>the (locals (snd s2) Result)\<midarrow>n\<rightarrow> abupd (absorb Ret) s3"
e: shows "p \<equiv> lagrange_interpolation_poly zs_ws"
assms: shows "(x, y) \<in> steps rel (Suc n)"
rel_nabla_coinduct_var: fixes P :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set" assumes "P \<subseteq> relfdia X P" shows "P \<subseteq> rel_nabla X"
mprefix_Un_distr: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "Mprefix (A \<union> B) P = (Mprefix A P [+] Mprefix B P)"
blocks_subset: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_set_system \<U> \<A> \<V> \<B>" shows "\<A> \<subseteq># \<B>"
emeasure_buffon_set_long: shows "emeasure lborel (buffon_set l d) = 4 * ennreal (l * (1 - sqrt (1 - (d / l)\<^sup>2)) + arccos (d / l) * d)"
good_rec: fixes K :: "'a set" assumes "K \<in> sets M"   and "K \<subseteq> A"   and "0 < emeasure M K" shows "0 < good_time K \<and> next_good_set K \<subseteq> A \<and> next_good_set K \<in> sets M \<and> 0 < emeasure M (next_good_set K)"
nth_strip_while: fixes n :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "n < length (strip_while P xs)" shows "strip_while P xs ! n = xs ! n"
upd_3: fixes m :: "nat"   and N :: "nat"   and n :: "nat"   and j :: "nat"   and i :: "nat"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "m \<le> N"   and "n \<le> N"   and "j \<le> n"   and "i \<le> m" shows "DBM_upd M1 f m n N i j = DBM_upd M1 f i j N i j"
dom_map_upd: fixes tab :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" shows "dom (tab(x \<mapsto> y)) = dom tab \<union> {x}"
good_item_simps_11: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "good X" shows "good (X #[xa \<and> y]_xsa)"
const_type: fixes c :: "'fun" assumes "arity c = 0" shows "\<exists>a. \<forall>T. \<Gamma> (Fun c T) = Var a"
inf_bot_right: fixes x :: "'c" shows "top_homomorphism ((\<squnion>) x)"
less_add_iff1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus" shows "less (plus (times a e) c) (plus (times b e) d) = less (plus (times (minus a b) e) c) d"
IH: assumes "distinct xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> x < oid" shows "distinct (insert_spec xs (oid, ref))"
leg0_in_hom_2: shows "\<guillemotleft>F f : F f \<Rightarrow>\<^sub>D F f\<guillemotright>"
inverse_mod_coprime_exp: fixes p :: "int"   and n :: "nat"   and x :: "int" assumes "p ^ n = p ^ n"   and "prime p"   and "n \<noteq> 0"   and "comm_monoid_mult_class.coprime x p" shows "q.M (inverse_mod x (p ^ n) * x) = 1"
norm_triangle_ineq3: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a - norm b\<bar> \<le> norm (minus a b)"
Lxy_append: fixes xs :: "'a list"   and ys :: "'a list"   and S :: "'a set" shows "Lxy (xs @ ys) S = Lxy xs S @ Lxy ys S"
in_sccsI: fixes c :: "('a, 'b) pre_digraph" assumes "induced_subgraph c (G \<restriction> S)"   and "strongly_connected c"   and "\<nexists>c'. induced_subgraph c' (G \<restriction> S) \<and> strongly_connected c' \<and> verts c \<subset> verts c'" shows "c \<in> H.sccs"
simps: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b" shows "Domainp r a = (\<exists>aa b. a = aa \<and> r aa b)"
map_inv1: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "inv1 (RBT_Impl.map f t) = inv1 t"
remove: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
fls_mult_assoc_base_factor: fixes a :: "'a fls"   and b :: "'a fls"   and c :: "'a fls" shows "fls_base_factor a * fls_base_factor b * fls_base_factor c = fls_base_factor a * (fls_base_factor b * fls_base_factor c)"
elims: fixes x :: "val option"   and xa :: "proc \<times> val"   and y :: "val option" assumes "x \<bullet> xa = y" obtains r :: "proc \<times> val" where "x = \<bottom>" and "xa = r__" and "y = Some (snd r__)"   | v :: "val" and r :: "proc \<times> val" where "x = Some v" and "xa = r" and "y = Some v"
mono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<sqsubseteq> f y)"
common_part_fusion: fixes x :: "i"   and y :: "i" assumes "O x y" shows "\<forall>w. P w (\<sigma>v. P v x \<and> P v y) = (P w x \<and> P w y)"
to_subsetS: fixes D :: "'c Order"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "Worder D"   and "T \<subseteq> carrier D"   and "ord_isom D (SIod D T) f" shows "\<forall>a. a \<in> carrier D \<longrightarrow> a \<preceq>\<^bsub>D\<^esub> f a"
make_affine_def: fixes p :: "int \<times> int \<times> int" shows "make_affine p = (let (x, y, z) = p in if z = \<zero> then Infinity else Point (x \<oslash> z) (y \<oslash> z))"
endofunctor_axioms: shows "endofunctor (\<cdot>\<^sub>B) src\<^sub>B"
assert_bound_atoms_equiv_bounds: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "(snd ` ats) \<doteq> \<B> s" shows "(snd ` (ats \<union> {a})) \<doteq> \<B> (assert_bound a s)"
to_nat_0: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep zero = 0"
Turn_into_distance_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real" shows "Turn_into_distance f \<equiv> ((\<forall>x y. 0 \<le> f x y) \<and> (\<forall>x y. f x y = f y x)) \<and> (\<forall>x. f x x = 0) \<and> (\<forall>x z y. f x z \<le> sqrt 2 * max (f x y) (f y z))"
Un_Int_assoc_eq: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "(A \<inter> B \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
src_ipPart_motivation: fixes rs :: "'a simple_rule list"   and B :: "'a word set"   and s1 :: "'a word"   and s2 :: "'a word"   and p :: "('a, 'b) simple_packet_scheme" assumes "\<forall>A\<in>(\<lambda>(base, len). ipset_from_cidr base len) ` src ` match_sel ` set rs. B \<subseteq> A \<or> B \<inter> A = {}"   and "s1 \<in> B"   and "s2 \<in> B" shows "simple_fw rs (p\<lparr>p_src := s1\<rparr>) = simple_fw rs (p\<lparr>p_src := s2\<rparr>)"
subset_\<P>\<Q>k: fixes k :: "nat"   and u :: "nat" shows "\<P> k u \<subseteq> \<Q> k"
nearly_healthyI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P Q. \<lbrakk>unitary P; unitary Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "nearly_healthy t"
path_component_eq_eq: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "(path_component S x = path_component S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y)"
Ord_trans: fixes i :: "V"   and j :: "V"   and k :: "V" assumes "i \<in> elts j"   and "j \<in> elts k"   and "Ord k" shows "i \<in> elts k"
height_bal_tree: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node ts t)" shows "height (Node ts t) = Suc (height t)"
simps_7: fixes p :: "('a, 'b) form" shows "params (Neg p) = params p"
that: fixes w\<^sub>1 :: "'item list" assumes "u\<^sub>1 \<preceq>\<^sub>F w\<^sub>1"   and "v\<^sub>1 \<preceq>\<^sub>F w\<^sub>1" shows "thesis"
lconjby_Abs_freelist_relator_freeword: fixes s :: "'w"   and t :: "'w"   and xs :: "'w list" assumes "s \<in> S"   and "t \<in> S"   and "xs \<in> lists S" shows "lconjby (Abs_freelist xs) (pair_relator_freeword s t) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
weak_sup_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "weak_sup_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall>A\<subseteq>carrier X. A \<noteq> {} \<longrightarrow> f (\<Squnion>\<^bsub>X\<^esub>A) = \<Squnion>\<^bsub>Y\<^esub>f ` A)"
ord_isom_surj_to: fixes E :: "('g, 'h) Order_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "Order E"   and "ord_isom D E f" shows "surj_to f (carrier D) (carrier E)"
implies_elim: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma>\<^sub>1 :: "Core.term set"   and A :: "Core.term"   and B :: "Core.term"   and \<Gamma>\<^sub>2 :: "Core.term set" assumes "\<Theta>,\<Gamma>\<^sub>1 \<turnstile> A \<longmapsto> B"   and "\<Theta>,\<Gamma>\<^sub>2 \<turnstile> A" shows "\<Theta>,\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"
ltakeWhile_lappend: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist" shows "ltakeWhile P (lappend xs ys) = (if \<exists>x\<in>lset xs. \<not> P x then ltakeWhile P xs else lappend xs (ltakeWhile P ys))"
dim_le1I: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and v :: "'d" assumes "vectorspace K V"   and "LinearCombinations.module.span K V {v} = carrier V"   and "v \<in> carrier V" shows "vectorspace.dim K V \<le> 1"
equiv_env_def: fixes \<rho> :: "(nat \<times> exp) list"   and \<rho>' :: "(nat \<times> exp) list" shows "equiv_env \<rho> \<rho>' \<equiv> sdom \<rho> = sdom \<rho>' \<and> (\<forall>x. x \<in> sdom \<rho> \<longrightarrow> lookup \<rho> x = lookup \<rho>' x)"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(\<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>y1\<^bold>\<rangle>\<^sub>0) = (x1 = y1)"
base: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "S p q" shows "trace_eq_simcl S p q"
uniform_limit_on_empty: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and F :: "'b filter" shows "uniform_limit {} f g F"
less_sets_grab: fixes N :: "nat set"   and n :: "nat" assumes "infinite N" shows "fst (grab N n) \<lless> snd (grab N n)"
hoare3_valid_def: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>3 {P} c { Q}) = (\<forall>ps n. P (ps, n) \<longrightarrow> (\<exists>ps' m. (c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' \<and> m \<le> n \<and> Q (ps', n - m)))"
ring_hom_cring_axioms_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "ring_hom_cring_axioms R S h \<equiv> h \<in> ring_hom R S"
map_subst_lemma: fixes x :: "'b"   and \<sigma> :: "('b \<times> 'b trm) list"   and f :: "'b trm \<Rightarrow> 'b trm" shows "Var x \<lhd> \<sigma> \<noteq> Var x \<or> Var x \<lhd> \<sigma> \<noteq> Var x \<lhd> map_subst f \<sigma> \<longrightarrow> Var x \<lhd> map_subst f \<sigma> = f (Var x \<lhd> \<sigma>)"
mult_n_ncols: fixes fmul :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and A :: "'d matrix"   and B :: "'b matrix" assumes "\<forall>a. fmul (0::'d) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)" shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
inject_7: fixes x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and y71 :: "'d list"   and y72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and y73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "(\<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle> = \<forall>y71\<langle>\<or>\<noteq>: y72 \<or>\<notin>: y73\<rangle>) = (x71 = y71 \<and> x72 = y72 \<and> x73 = y73)"
matrix_match_def: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" shows "matrix_match A1 A2 B1 B2 \<equiv> mat (row_length A1) (length A1) A1 \<and> mat (row_length A2) (length A2) A2 \<and> mat (row_length B1) (length B1) B1 \<and> mat (row_length B2) (length B2) B2 \<and> length A1 = row_length A2 \<and> length B1 = row_length B2 \<and> A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"
path_defs_2: fixes g :: "real \<Rightarrow> 'c" shows "pathstart g = g 0"
add_mono_right: fixes c :: "'b"   and d :: "'b"   and a :: "'b" assumes "c \<le> d" shows "a + c \<le> a + d"
SUP_L_P_least: fixes i :: "'b"   and u :: "'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and Q :: "'e" assumes "\<And>v. pair v i \<sqsubset> u \<Longrightarrow> P v i \<sqsubseteq> Q" shows "SUP_L_P P u i \<sqsubseteq> Q"
eq_fin_concat_eq_fin_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* (u @ v\<^sub>1) (u @ v\<^sub>2)" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* v\<^sub>1 v\<^sub>2"
prems_1: 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) (dbproj i (dl # D)))\<rbrakk>\<^sub>d \<I>"
e2ls_def: fixes e :: "nat" shows "e2ls e \<equiv> pdec22 (e2frame e)"
carrier_dim_vec: fixes v :: "'c vec"   and n :: "nat" shows "(v \<in> carrier_vec n) = (dim_vec v = n)"
m_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + (y + z) = y + (x + z)"
length_wall_crossings: fixes Cs :: "'a set list" shows "length (wall_crossings Cs) = length Cs - 1"
c_cons_def: shows "c_cons = (\<lambda>x u. list_to_nat (x # nat_to_list u))"
g_arith_def: fixes a :: "real"   and b :: "real" shows "g_arith a b = (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b - p1d (a, b) (gx a b, gy a b)"
ltake_ltl: fixes n :: "enat"   and xs :: "'c llist" shows "ltake n (ltl xs) = ltl (ltake (eSuc n) xs)"
interval_measure_UNIV: fixes F :: "real \<Rightarrow> real"   and m :: "real" assumes "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F"   and "(F \<longlongrightarrow> 0) at_bot"   and "(F \<longlongrightarrow> m) at_top"   and "0 \<le> m" shows "emeasure (interval_measure F) UNIV = ennreal m"
ide_in_hom: fixes a :: "'s" shows "S.ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
inf_sup_distrib1: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
sum_content_null: fixes a :: "'a"   and b :: "'a"   and p :: "('a \<times> 'a set) set"   and f :: "'a \<Rightarrow> 'b" assumes "content (cbox a b) = 0"   and "p tagged_division_of cbox a b" shows "(\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) = (0::'b)"
arrE: fixes f :: "'a" assumes "S.arr f" shows "left \<omega> f"
pr_in_hom_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.in_hom C (pr0 a b) (elementary_category_with_binary_products.prod C pr0 pr1 a b) b"
complex_split: fixes a :: "real"   and b :: "real"   and c :: "complex" shows "(Complex a b = c) = (a = Re c \<and> b = Im c)"
DisjCont: fixes A :: "fm" shows "boolean_axiomsp (A OR A IMP A)"
simps_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "R\<^sup>k = (R, R)"