hcju/isabelleps huggingface dataset:
scaleR_conv_of_real: fixes r :: "real"   and x :: "'c" shows "r *\<^sub>R x = of_real r * x"
IH_1: shows "insertion f (replace_coeff (insertion f) p1) = insertion f (reduce_nested_mpoly p1)"
to_nat_le: 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"   and y :: "'a"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less y (mod_type.from_nat Abs k)" shows "mod_type.to_nat Rep y < k"
maps_axioms: shows "maps (Trie None []) update_trie lookup_tries invar_trie"
subst_cls_lists_empty_iff: fixes Cs :: "'a literal multiset list"   and \<eta>s :: "'s list" shows "(Cs \<cdot>\<cdot>cl \<eta>s = []) = (Cs = [] \<or> \<eta>s = [])"
vwalk_to_vpath_nonempty: fixes p :: "'a list" assumes "p \<noteq> []" shows "vwalk_to_vpath p \<noteq> []"
valid_regions_distinct_spec: fixes R :: "('c \<Rightarrow> real) set"   and R' :: "('c \<Rightarrow> real) set"   and v :: "'c \<Rightarrow> real" assumes "R \<in> \<R>"   and "R' \<in> \<R>"   and "v \<in> R"   and "v \<in> R'" shows "R = R'"
simple_refinement_safe_def: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and abs_steps :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val) \<Rightarrow> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val) \<Rightarrow> 'a" shows "simple_refinement_safe \<R>\<^sub>A \<R> abs_steps \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A mem\<^sub>2\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<and> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> stops\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C = stops\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<and> abs_steps \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C = abs_steps \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<and> (\<forall>mds\<^sub>1\<^sub>C' mds\<^sub>2\<^sub>C' mem\<^sub>1\<^sub>C' mem\<^sub>2\<^sub>C' c\<^sub>1\<^sub>C' c\<^sub>2\<^sub>C'. \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>1\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<and> \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>2\<^sub>C', mds\<^sub>2\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> c\<^sub>1\<^sub>C' = c\<^sub>2\<^sub>C' \<and> mds\<^sub>1\<^sub>C' = mds\<^sub>2\<^sub>C')"
simps_2: fixes uu :: "('c \<times> nat) tree"   and uv :: "'c"   and lv :: "nat"   and uw :: "('c \<times> nat) tree" shows "lvl \<langle>uu, (uv, lv), uw\<rangle> = lv"
wP: fixes s :: "'a" assumes "P s" shows "Q s \<le> t R s"
\<gamma>_ivl_nice: fixes l :: "int extended"   and h :: "int extended" shows "\<gamma>_ivl [l, h] = {i. l \<le> Fin i \<and> Fin i \<le> h}"
less_eq_set_def: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<le> (\<lambda>x. x \<in> B))"
arr_MkArr: fixes F :: "'a2 \<Rightarrow> 'b"   and G :: "'a2 \<Rightarrow> 'b"   and \<tau> :: "'a2 \<Rightarrow> 'b" shows "A2_B.arr (A2_B.MkArr F G \<tau>) = natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>"
squarefree_def: fixes n :: "'a" shows "squarefree n = (\<forall>x. x\<^sup>2 dvd n \<longrightarrow> x dvd (1::'a))"
infsetsum_reindex_bij_betw: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "bij_betw g A B" shows "(\<Sum>\<^sub>ax\<in>A. f (g x)) = infsetsum f B"
observable_transition_def: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" shows "P \<Rightarrow>{\<alpha>} P' \<equiv> \<exists>Q Q'. P \<Rightarrow> Q \<and> Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle> \<and> Q' \<Rightarrow> P'"
lappend_assoc: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" shows "xs @\<^sub>l ys @\<^sub>l zs = xs @\<^sub>l (ys @\<^sub>l zs)"
vsv_valeq_v11I: assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r; r\<lparr>x\<rparr> = r\<lparr>y\<rparr>\<rbrakk> \<Longrightarrow> x = y" shows "v11 r"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) RBT.rbt" shows "impl_of (RBT.insert x xa xb) = rbt_insert x xa (impl_of xb)"
close_supp: fixes T :: "ty"   and \<Gamma> :: "(W.var \<times> tyS) list" shows "supp (close \<Gamma> T) = set (ftv T) \<inter> set (ftv \<Gamma>)"
assms_1: shows "coprime e ((P - 1) * (Q - 1))"
Inter_greatest: fixes A :: "'b set set"   and C :: "'b set" assumes "\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X" shows "C \<subseteq> \<Inter> A"
proj_state_succ: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fst a \<subseteq>\<^sub>f s" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s a)"
simps_2: fixes f :: "'fsym"   and Tl :: "'fsym trm list" shows "tNN (Fn f Tl) = (if unprot (resOf f) \<or> protFw (resOf f) then Fn (Oldf f) (map tNN Tl) else Fn (Tag (resOf f)) [Fn (Oldf f) (map tNN Tl)])"
map_graph_selectors_1: fixes f :: "('g \<times> 'e) set"   and G :: "('f, 'g) labeled_graph" shows "vertices (map_graph f G) = f `` vertices G"
diff: shows "(Var x' - p)\<^sup>2 - (Var x' - p) \<in> More_Modules.ideal polynomial_bool"
float_distinct_6: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_infinity a \<and> is_denormal a)"
Lev2_Suc: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and x :: "'e" shows "Lev2 s1 s2 (Suc n) x = {Inl (tobd_F22 s2 x b1) # kl |b1 kl. b1 \<in> F2set2 (s2 x) \<and> kl \<in> Lev1 s1 s2 n b1} \<union> {Inr (tobd_F23 s2 x b2) # kl |b2 kl. b2 \<in> F2set3 (s2 x) \<and> kl \<in> Lev2 s1 s2 n b2}"
assms_2: shows "{x \<in> space M. P x} \<in> events"
intFI_def: shows "intFI \<equiv> MonotProblem.intFI GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf GE_\<Phi>"
enum_distinct: fixes enum :: "'e list"   and enum_all :: "('e \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('e \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "distinct enum"
total_action_order: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "total_onP (actions E) (action_order E)"
hcomp_simps_2: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.hseq \<nu> \<mu>" shows "trg\<^sub>C (\<nu> \<star>\<^sub>C \<mu>) = trg\<^sub>C \<nu>"
add_unique_scalars: fixes vs :: "'v list"   and v :: "'v"   and v' :: "'v" assumes "set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs"   and "v \<in> V"   and "v' \<in> V" shows "(THE ds. length ds = length vs \<and> v + v' = ds \<bullet>\<cdot> vs) = map2 (+) (THE ds. length ds = length vs \<and> v = ds \<bullet>\<cdot> vs) (THE ds. length ds = length vs \<and> v' = ds \<bullet>\<cdot> vs)"
type_weaken': fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile> t : T"   and "\<turnstile> (\<Delta> @ \<Gamma>) ok" shows "\<Delta> @ \<Gamma> \<turnstile> t : T"
step2_a: shows "DataRefinement ({. Loop' .} \<circ> Q4'_a) R2'_a R2'_a Q4''_a"
igAbsIPresIGWls_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
error_def: fixes uu :: "('a, 'b) IEEE.float itself"   and x :: "real" shows "error uu x = valof (IEEE.round To_nearest x) - x"
degen_skip_arbitrary_condition: fixes l :: "nat"   and cs :: "('b \<Rightarrow> bool) list"   and w :: "'b stream"   and k :: "nat" assumes "l < length cs"   and "infs (degen cs) (w ||| k ## sscan (count cs) w k)" obtains u :: "'b list" and a :: "'b" and v :: "'b stream" where "w = u @- a ## v" and "fold (count cs) u k = l" and "(cs ! l) a"
simps_2: fixes P :: "(('b, 'c) vec, 'c) vec"   and A :: "(('b, 'd) vec, 'c) vec"   and n :: "nat"   and i :: "'c"   and j :: "'d" shows "row_add_iterate_PA (P, A) (Suc n) i j = (if Suc n = mod_type_class.to_nat i then row_add_iterate_PA (P, A) n i j else row_add_iterate_PA (row_add P (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j), row_add A (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j)) n i j)"
simps_20: fixes x1 :: "'b"   and x7 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<r>\<^bold>[x7\<^bold>]"
CH_domain: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and A :: "'d set"   and h :: "'f" assumes "ContractsWithIRC Xd Xh Pd Ch" shows "Contracts.CH Ch A \<inter> {x. Xh x = h} = Ch h (A \<inter> {x. Xh x = h})"
primekeytrick: fixes M :: "nat"   and A :: "nat"   and P :: "nat"   and B :: "nat" assumes "M * A mod P = M * B mod P"   and "M mod P \<noteq> 0"   and "prime P" shows "A mod P = B mod P"
hom_add_eq_zero: fixes x :: "'a"   and y :: "'a"   and c :: "'a" assumes "x + y = (0::'a)" shows "c * x + c * y = (0::'a)"
order_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.order less_eq less \<equiv> class.preorder less_eq less \<and> class.order_axioms less_eq"
xfer_run: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts" obtains p' :: "pr_operation' list" where "((f, l), p', f', l') \<in> trcl pr_algo_lts'" and "p = map project_operation p'"
emeasure_neq_0_sets: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A \<noteq> 0" shows "A \<in> sets M"
sum_incoming_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "sum g (cf.incoming u) = (\<Sum>v\<in>cf.E\<inverse> `` {u}. g (v, u))"
simps: fixes G :: "prog"   and a1 :: "ty"   and a2 :: "ty" shows "G\<turnstile>a1\<preceq>a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>I J. a1 = Iface I \<and> a2 = Iface J \<and> G\<turnstile>I\<preceq>I J) \<or> (\<exists>I. a1 = Iface I \<and> a2 = Class Object) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> G\<turnstile>C\<preceq>\<^sub>C D) \<or> (\<exists>C I. a1 = Class C \<and> a2 = Iface I \<and> G\<turnstile>C\<leadsto>I) \<or> (\<exists>R. a1 = NT \<and> a2 = RefT R) \<or> (\<exists>T. a1 = T.[] \<and> a2 = Class Object) \<or> (\<exists>S T. a1 = RefT S.[] \<and> a2 = RefT T.[] \<and> G\<turnstile>RefT S\<preceq>RefT T))"
monomial_0_iff: fixes t :: "'c"   and c :: "'d" shows "(monomial c t = 0) = (c = (0::'d))"
iffalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom" assumes "BMap (E b m) = False" shows "SRabbr (if b then c1 else c2 fi) m [c2] m"
id: shows "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m n"
fps_X_fps_compose_startby0: fixes a :: "'b fps" assumes "a $ 0 = (0::'b)" shows "fps_X oo a = a"
list_update_same_conv: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < |xs|" shows "(xs[i := x] = xs) = (xs ! i = x)"
vD_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> vD rt" shows "nsqn rt ip = sqn rt ip"
assms_1: shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
simps_4: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula"   and G :: "'b formula" shows "\<A> \<Turnstile> F \<^bold>\<and> G = (\<A> \<Turnstile> F \<and> \<A> \<Turnstile> G)"
fold_empty: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "Finite_Set.fold f z {} = z"
assms_1: fixes x :: "'a" assumes "x \<in> vertices G" shows "(f \<circ> g) x = x"
irreducible\<^sub>d_smultI: fixes c :: "'b"   and p :: "'b poly" assumes "c \<noteq> (0::'b)"   and "irreducible\<^sub>d (Polynomial.smult c p)" shows "irreducible\<^sub>d p"
empty_is_zero: fixes u :: "nat" assumes "nat_to_set u = {}" shows "u = 0"
sys_reg_mod_def: fixes data_w32 :: "word32"   and sys :: "sys_reg"   and state :: "'b sparc_state" shows "sys_reg_mod data_w32 sys state \<equiv> state\<lparr>sys_reg := (sys_reg state)(sys := data_w32)\<rparr>"
intro: fixes S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0" shows "set_iterator_genord iti S0 R"
induct: fixes x :: "'c \<Rightarrow> real"   and P :: "('c \<Rightarrow> real) \<Rightarrow> bool" assumes "real_polynomial_function x"   and "\<And>f. bounded_linear f \<Longrightarrow> P f"   and "\<And>c. P (\<lambda>x. c)"   and "\<And>f g. \<lbrakk>real_polynomial_function f; P f; real_polynomial_function g; P g\<rbrakk> \<Longrightarrow> P (\<lambda>x. f x + g x)"   and "\<And>f g. \<lbrakk>real_polynomial_function f; P f; real_polynomial_function g; P g\<rbrakk> \<Longrightarrow> P (\<lambda>x. f x * g x)" shows "P x"
List_conv_islist_list: fixes h :: "'c \<Rightarrow> 'c"   and p :: "'c"   and as :: "'c list" shows "List h p as = (islist h p \<and> as = list h p)"
lang_flatten_PLUS: fixes X :: "'b rexp set"   and n :: "nat" assumes "finite X" shows "lang n (PLUS (sorted_list_of_set X)) = \<Union> (lang n ` X)"
newman_ingham_aux2: fixes m :: "nat"   and w :: "complex"   and z :: "complex"   and f :: "nat \<Rightarrow> complex"   and C :: "real" assumes "1 \<le> m"   and "1 \<le> Re w"   and "0 < Re z"   and "\<And>n. 1 \<le> n \<Longrightarrow> cmod (f n) \<le> C" shows "cmod (\<Sum>n = 1..m. f n / of_nat n powr (w - z)) \<le> C * real m powr Re z * (1 / real m + 1 / Re z)"
assigns_r_comp: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and P :: "('a, 'b) urel" shows "\<langle>\<sigma>\<rangle>\<^sub>a ;; P = \<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> P"
Least_Hermite_of_row_i: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass" shows "(LEAST n. Hermite_of_row_i ass res A i $ i $ n \<noteq> (0::'b)) = (LEAST n. A $ i $ n \<noteq> (0::'b))"
fst_sym_preproc_aux_complete: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and vs :: "'t list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "Keys (set fs) = set ks \<union> set vs" shows "set (fst (sym_preproc_aux gs ks (vs, fs))) = Keys (set (snd (sym_preproc_aux gs ks (vs, fs))))"
zero_less_two: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (plus one one)"
trace_pdo_eq_imp_eq: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat n n; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> trace (A * \<rho>) = trace (B * \<rho>)" shows "A = B"
"2_3": shows "decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)"
fold_is_None: fixes x :: "'a option" shows "(x = None) = is_None x"
is_all_iff_strip_all_vars_not_empty: fixes t :: "Core.term" shows "is_all t = (strip_all_vars t \<noteq> [])"
med_a0im1_ir_def: fixes o1 :: "m1_state" shows "med_a0im1_ir o1 \<equiv> \<lparr>signals = ir_runs2sigs (runs o1), corrupted = Domain (m1_state.leak o1) \<times> UNIV\<rparr>"
SeqWRP_Zero: fixes s :: "name" shows "{} \<turnstile> SyntaxN.Ex s (SeqWRP (Var s) Zero Zero)"
prems_4: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
l_inv_ex: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "\<exists>y\<in>carrier (relative_homology_group (int n) (nsphere n) (equator n)). y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>"
inf_sup_aci_7: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "sup_prod x (sup_prod y z) = sup_prod y (sup_prod x z)"
typing_alt_simps_23: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and its :: "String.literal list"   and body :: "'a OCL_Types.type option \<times> 'a expr"   and ys :: "'a OCL_Types.type \<times> 'a OCL_Types.type \<times> 'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>I (src, its, body) : ys) = (\<exists>\<tau> \<sigma> its_ty bodya. body = (Some its_ty, bodya) \<and> (\<exists>\<rho>. ys = (\<tau>, \<sigma>, \<rho>) \<and> \<Gamma> \<turnstile>\<^sub>E src : \<tau> \<and> element_type \<tau> \<sigma> \<and> \<sigma> \<le> its_ty \<and> \<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, its_ty)) its) \<turnstile>\<^sub>E bodya : \<rho>))"
lens_plus_eq_right: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<bowtie> Y"   and "Y \<approx>\<^sub>L Z" shows "X +\<^sub>L Y \<approx>\<^sub>L X +\<^sub>L Z"
linear_orderable_7_implies_1: assumes "top \<sqinter> id top = top \<and> top \<squnion> id top = top" shows "((top \<le> top \<and> top \<sqinter> top \<le> top) \<and> top \<sqinter> id top \<le> top) \<and> top \<squnion> id top = top"
Mf: shows "set_borel_measurable borel {g a..g b} f"
float_of_fp64_def: shows "float_of_fp64 \<equiv> map_fun id Abs_float (\<lambda>x. apsnd word_split (word_split x))"
Inf_dual_var: fixes X :: "'a set" shows "- \<Sqinter> X = \<Squnion> (uminus ` X)"
I: fixes r :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set"   and seq :: "'b sequent" assumes "r \<in> R" shows "extendRule seq r \<in> R*"
vwb_lens_def: fixes x :: "'e \<Longrightarrow> 'f" shows "vwb_lens x \<equiv> wb_lens x \<and> mwb_lens x"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_inv: fixes shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme" shows "cast (cast shadow_root) = Some shadow_root"
p_imp_p: fixes P :: "'d upred" shows "(P \<Rightarrow> P) = true"
simps_2: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "floor_2 (Irrational n rai) = floor_1 rai"
open_on_good: fixes Q :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "open_on (LEXEQ strict_suffix) (good Q) (SEQ (lists A))"
B: fixes g :: "'a \<Rightarrow> real" assumes "g \<in> {g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}" shows "LINT x|M. f x * g x \<le> Norm (\<LL> p M) f"
sf_4: assumes "p = 0" shows "c = (0::'a) \<and> bs = []"
prod_exp: fixes x :: "real"   and y :: "real" shows "4 * (x * y) = (x + y)\<^sup>2 - (x - y)\<^sup>2"
"13_2": fixes x :: "'a state"   and y :: "(v list, v) result"   and x1 :: "v list" assumes "(x, y) = fix_clock st (fun_evaluate st env [e1])"   and "y = Rval x1" shows "clock (fst (fun_evaluate x (update_v (\<lambda>_. nsOptBind xo (hd x1) (sem_env.v env)) env) [e2])) \<le> clock x"
radj_Inf_pres: fixes g :: "'d \<Rightarrow> 'c" assumes "\<exists>f. f \<stileturn> g" shows "Inf_pres g"
iterated_serial_on_in_A: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b"   and n :: "nat" assumes "serial_on A r"   and "a \<in> A" shows "iterated_serial_on A r a n \<in> A"
vfsequence_at_not_last: fixes i :: "V"   and x :: "V" assumes "i \<in>\<^sub>\<circ> vcard xs" shows "(xs #\<^sub>\<circ> x)\<lparr>i\<rparr> = xs\<lparr>i\<rparr>"
reachable_def: fixes as :: "'a list"   and s :: "'a rexp" shows "reachable as s = snd (the (rtrancl_while (\<lambda>_. True) (\<lambda>s. map (\<lambda>a. delta a s) as) (init s)))"
atC_not_empty: fixes c :: "('h, 'g, 'i, 'j) com" shows "atC c \<noteq> {}"
F: shows "F \<in> restrict1 opp_fold_A (\<Union> A) \<turnstile> ChamberComplex.chamber_system A"
subprob_space_measure_pmf: fixes x :: "'c pmf" shows "subprob_space (measure_pmf x)"
conj_disj_distribR: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "((P \<or> Q) \<and> R) = (P \<and> R \<or> Q \<and> R)"
stutter_sampler_mono: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "stutter_sampler f \<sigma>" shows "strict_mono f"
map_id: fixes p :: "'c spmf" shows "map_spmf id p = p"
axioms_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b"   and p0 :: "'b"   and p1 :: "'b" assumes "binary_product_cone C f0 f1 p0 p1" shows "binary_product_diagram C f0 f1"
assms_1: shows "monotone (\<le>) (\<le>) f"
nogt: shows "\<not> (\<exists>w>y. a * w\<^sup>2 + b * w + c = 0)"
ladj_Sup_pres: fixes f :: "'d \<Rightarrow> 'c" assumes "\<exists>g. f \<stileturn> g" shows "Sup_pres f"
simps_1: fixes OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and N :: "'a" shows "rexp_of_list OP N [] = N"
ord_ordered: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ord a b c"   and "a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c" shows "\<exists>f. TernaryOrdering.ordering f ord {a, b, c}"
select_convs_1: fixes g_V :: "'c set"   and g_E :: "('c \<times> 'c) set"   and g_V0 :: "'c set"   and more :: "'d" shows "g_V \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V"
induced_map_measure_preserving_aux: fixes A :: "'a set"   and W :: "'a set" assumes "A \<in> sets M"   and "W \<in> sets M"   and "W \<subseteq> A"   and "emeasure M W < \<infinity>" shows "emeasure M (induced_map A --` W) = emeasure M W"
iT_cut_le_4: fixes r :: "nat"   and m :: "nat"   and t :: "nat" shows "[ r, mod m ] \<down>\<le> t = (if t < r then {} else [ r, mod m, (t - r) div m ])"
can_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.can C f \<equiv> partial_magma.arr (free_monoidal_category.comp C) f \<and> (\<exists>t. t \<in> f \<and> monoidal_language.Ide C (monoidal_language.Diagonalize C t))"
ordering_top_axioms: shows "ordering_top (\<sqsubseteq>) (\<lambda>x y. y < x) false"