hcju/isabelleps huggingface dataset:
subst_subst2: fixes a :: "coname"   and b :: "coname"   and P :: "trm"   and N :: "trm"   and x :: "name"   and y :: "name"   and M :: "trm" assumes "a \<sharp> (b, P, N)"   and "x \<sharp> (y, P, M)"   and "b \<sharp> (M, N)"   and "y \<sharp> P" shows "M{a:=x.N}{y:=<b>.P} = M{y:=<b>.P}{a:=x.N{y:=<b>.P}}"
In1_def: fixes M :: "('c, 'd) node set" shows "In1 M \<equiv> Scons (Numb 1) M"
filterlim_inverse_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> 0) F"   and "\<forall>\<^sub>F x in F. 0 < f x" shows "LIM x F. inverse (f x) :> at_top"
ba_top_def: fixes z :: "'a" shows "top = - - (z \<squnion> - z)"
IH: shows "map (n2l \<circ> l2n) zs = zs"
iso_char: fixes \<phi> :: "'a" assumes "Left.arr \<phi>" shows "Left.iso \<phi> = local.iso \<phi>"
bounded_byI2: fixes P :: "'b \<Rightarrow> real"   and b :: "real" assumes "P \<tturnstile> \<lambda>s. b" shows "bounded_by b P"
replace3: fixes f' :: "'a"   and ls :: "'a list"   and f :: "'a"   and fs :: "'a list" assumes "f' \<in> set ls"   and "f \<in> set fs" shows "f \<in> set (replace f' fs ls)"
simps_4: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas_inner (\<phi> U\<^sub>n \<psi>) = subformulas\<^sub>\<nu> (\<phi> U\<^sub>n \<psi>) \<union> subformulas\<^sub>\<mu> (\<phi> U\<^sub>n \<psi>)"
second_vertex_def: shows "Q.second_vertex \<equiv> \<lambda>xs. hd (tl xs)"
map_key_map: fixes g :: "'e \<Rightarrow> 'd"   and p :: "'b \<Rightarrow>\<^sub>0 'e" shows "map_key f (Poly_Mapping.map g p) = Poly_Mapping.map g (map_key f p)"
2: shows "w\<^sup>\<star> \<le> forest_components (w \<sqinter> - q \<squnion> (w \<sqinter> q)\<^sup>T)"
inject: fixes x :: "((common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>h"   and ya :: "((common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>h" shows "(Transaction\<^sub>h x = Transaction\<^sub>h ya) = (x = ya)"
lexord_Nil_left: fixes y :: "'a list"   and r :: "('a \<times> 'a) set" shows "(([], y) \<in> lexord r) = (\<exists>a x. y = a # x)"
module_hom_zero: shows "clinear (\<lambda>x. 0::'e)"
KBasic2_9: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<diamond>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>) in v]"
permutation_insert_expand: fixes i :: "'b"   and j :: "nat"   and p :: "'b \<Rightarrow> nat" shows "permutation_insert i j p \<equiv> \<lambda>i'. if i' < i then if p i' < j then p i' else Suc (p i') else if i' = i then j else if p (i' - (1::'b)) < j then p (i' - (1::'b)) else Suc (p (i' - (1::'b)))"
simps_1: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "lock_action_ok l t Lock = may_lock l t"
select_convs_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
a: shows "\<forall>i<length cms\<^sub>1. \<forall>x\<in>differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i. doesnt_read (fst (cms\<^sub>1 ! i)) x \<and> doesnt_read (fst (cms\<^sub>2 ! i)) x"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "Group.comm_monoid_axioms G" shows "Group.comm_monoid G"
relcompp_apply: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'d"   and c :: "'f" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
ipresIGWlsAbs_def: fixes hA :: "'o \<Rightarrow> 'p"   and MOD :: "('q, 'r, 's, 't, 'u, 'v, 'w, 'o, 'x) model_scheme"   and MOD' :: "('y, 'z, 's, 't, 'aa, 'ab, 'ac, 'p, 'ad) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
assms: shows "Lower_Semicontinuous.convex_on UNIV f"
le_5: shows "j < (if j = mm then Suc j else Suc j)"
relcomp_mono: fixes r' :: "('a \<times> 'c) set"   and r :: "('a \<times> 'c) set"   and s' :: "('c \<times> 'd) set"   and s :: "('c \<times> 'd) set" assumes "r' \<subseteq> r"   and "s' \<subseteq> s" shows "r' O s' \<subseteq> r O s"
abs_low_mds_eq_dma\<^sub>C_eq: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "abs.low_mds_eq (mds\<^sub>A_of mds) (mem\<^sub>A_of mem\<^sub>1\<^sub>C) (mem\<^sub>A_of mem\<^sub>2\<^sub>C)" shows "dma\<^sub>C mem\<^sub>1\<^sub>C = dma\<^sub>C mem\<^sub>2\<^sub>C"
degree_map_poly_hom: fixes hom :: "'c \<Rightarrow> 'd"   and p :: "'c poly" assumes "zero_hom_0 hom" shows "degree (map_poly hom p) = degree p"
eq_lex_refl: fixes x :: "real \<times> real"   and y :: "real \<times> real" assumes "x = y" shows "Counterclockwise_2D_Arbitrary.lex x y"
wf_eq_minimal2: fixes r :: "('d \<times> 'd) set" shows "wf r = (\<forall>A. A \<subseteq> Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r))"
induct: fixes P :: "'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b up\<^sub>d"   and a1 :: "'b"   and a2 :: "'b tree234"   and a3 :: "'b"   and a4 :: "'b tree234"   and a5 :: "'b"   and a6 :: "'b tree234" assumes "\<And>t1 a t2 b t3 c t4. P (T\<^sub>d t1) a t2 b t3 c t4"   and "\<And>t1 a t2 b t3 c t4 d t5. P (Up\<^sub>d t1) a \<langle>t2, b, t3\<rangle> c t4 d t5"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P (Up\<^sub>d t1) a \<langle>t2, b, t3, c, t4\<rangle> d t5 e t6"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6 f t7. P (Up\<^sub>d t1) a \<langle>t2, b, t3, c, t4, d, t5\<rangle> e t6 f t7"   and "\<And>v b d e f g. P (Up\<^sub>d v) b \<langle>\<rangle> d e f g" shows "P a0 a1 a2 a3 a4 a5 a6"
Prim2_def: shows "Prim2 \<equiv> {\<lambda>x y. eval r [x, y] |r. prim_recfn 2 r}"
StepMsg: fixes cfg1 :: "('p, 'v, 's) configuration"   and p :: "'p"   and v :: "'v"   and cfg2 :: "('p, 'v, 's) configuration" shows "cfg1 \<turnstile> <p, v> \<mapsto> cfg2 = ((\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Value v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, v> \<and> msgs cfg2 = sends p (states cfg1 p) (Value v) \<union># msgs cfg1 -# <p, v>)"
edge_from_def: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "Solved_Task.edge_from freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is = ((c, 0 # is), Reg (Abstract_Formulas.iOutPort anyP (tree_at (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c))) is)))"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set A)"
lm073: fixes x :: "'c set"   and X :: "'c set set" assumes "x \<in> X" shows "finestpart x \<subseteq> finestpart (\<Union> X)"
Bk_no_Oc_repeatE: fixes list :: "cell list"   and t :: "nat"   and RR :: "bool" assumes "Bk # list = Oc \<up> t" shows "RR"
enat_unfold_stop: fixes stop :: "'a \<Rightarrow> bool"   and a :: "'a"   and "next" :: "'a \<Rightarrow> 'a" assumes "stop a" shows "enat_unfold stop next a = 0"
not_bet_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Bet A B C" shows "A \<noteq> B \<and> B \<noteq> C"
distinct_7: fixes x1 :: "'e"   and x5 :: "'e" shows "broadcast x1 \<noteq> send x5"
s_2: shows "(- w) `\<rightarrow> C = s `\<rightarrow> C0"
intro: fixes R :: "'c set"   and I :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a b. \<lbrakk>a \<in> R; b \<in> I\<rbrakk> \<Longrightarrow> multiplication a b \<in> I"   and "\<And>a b. \<lbrakk>a \<in> R; b \<in> I\<rbrakk> \<Longrightarrow> multiplication b a \<in> I" shows "ideal_axioms I R multiplication"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "\<lbrace>t \<^bold>\<star> u\<rbrace> = \<lbrace>t\<rbrace> \<star> \<lbrace>u\<rbrace>"
cut_bound_mono: fixes a :: "nat"   and c :: "nat"   and b :: "nat"   and d :: "nat"   and F :: "'b formula" assumes "a \<le> c"   and "b \<le> d" shows "cut_bound a b F \<le> cut_bound c d F"
size_6: fixes x3 :: "'c" shows "size (In3 x3) = Suc 0"
DeMoivre: fixes a :: "real"   and n :: "nat" shows "cis a ^ n = cis (real n * a)"
TC_sup_lift: fixes x :: "V"   and y :: "V" shows "TC x \<sqinter> lift x y = 0"
mono_oInv: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "mono (oInv F)"
cart2_append1_proj2: fixes v :: "(real, 2) vec" shows "cart2_append1 (proj2_pt v) = vector2_append1 v"
from_index_geq_vector_simplex: fixes i :: "nat"   and v :: "rat vec"   and x :: "nat \<Rightarrow> rat"   and j :: "nat" assumes "x \<Turnstile>\<^sub>c\<^sub>s set (from_index_geq0_vector i v)"   and "j < dim_vec v" shows "v $ j \<le> x (i + j)"
is_class_def: fixes P :: "'b prog"   and C :: "char list" shows "is_class P C \<equiv> class P C \<noteq> None"
cases: fixes a :: "exp" assumes "isval a" obtains n :: "nat" where "a = ENat n"   | x :: "nat" and e :: "exp" where "a = ELam x e"
subset_not_subset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subset> B) = strict (\<subseteq>) A B"
keep_empty_if_no_events: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "msgs (S t i) cid = []"   and "ps (S t i) p \<noteq> None"   and "channel cid = Some (p, q)"   and "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p" shows "msgs (S t j) cid = []"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "category B"
sum_sound: fixes S :: "'b set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>x\<in>S. sound (P x)" shows "sound (\<lambda>s. \<Sum>x\<in>S. P x s)"
lcl_def: shows "lcl \<equiv> fst \<circ> snd"
gpv'''_def: shows "gpv''' \<equiv> Generative_Probabilistic_Value.Done y"
cx_leq_bx_in_creating_opt: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "c \<bullet> x \<le> b \<bullet> y"
length_list_updates: fixes xs :: "nat list"   and ys :: "'a list"   and vs :: "'a list" shows "length (list_updates xs ys vs) = length vs"
exi_elim: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "exi F \<pi>l" obtains \<pi> :: "('state \<times> 'aprop set) stream" where "wfp AP' \<pi>" and "\<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l)" and "\<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0" and "F \<pi> \<pi>l"
cinner_ge_zero: fixes x :: "'b" shows "0 \<le> x \<bullet>\<^sub>C x"
phi_equiv_mappingE': fixes g :: "'g"   and m\<^sub>1 :: "('val, ('node \<times> 'val) set) mapping"   and x :: "'node \<times> 'val"   and vs :: "'val list"   and b :: "'val" assumes "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "Mapping.lookup (phis g) x = Some vs"   and "b \<in> set vs"   and "b \<in> snd ` Mapping.keys (phis g)" obtains "Mapping.lookup m\<^sub>1 b = Some {n \<in> Mapping.keys (phis g). b \<in> set (the (Mapping.lookup (phis g) n))}"
Rxa_one: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R" shows "R \<diamondsuit>\<^sub>p 1\<^sub>r\<^bsub>R\<^esub> = carrier R"
Times_infinite_bij_betw: fixes A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "infinite A"   and "B \<noteq> {}"   and "inj_on g B \<and> g ` B \<subseteq> A" shows "(\<exists>f. bij_betw f (A \<times> B) A) \<and> (\<exists>h. bij_betw h (B \<times> A) A)"
cnf_of_encode_interfering_operator_exclusion_contains_clause_if: fixes k :: "nat"   and t :: "nat"   and op\<^sub>1 :: "'b strips_operator"   and \<Pi> :: "'b strips_problem"   and op\<^sub>2 :: "'b strips_operator" assumes "k < t"   and "op\<^sub>1 \<in> set (\<Pi>\<^sub>\<O>)"   and "op\<^sub>2 \<in> set (\<Pi>\<^sub>\<O>)"   and "index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2"   and "are_operators_interfering op\<^sub>1 op\<^sub>2" shows "{(Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>1))\<inverse>, (Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>2))\<inverse>} \<in> cnf (encode_interfering_operator_exclusion \<Pi> t)"
inj_on_image_csets: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "inj_on ((`\<^sub>c) f) (csets A)"
comp: fixes f :: "'l \<Rightarrow> 'j"   and g :: "'m \<Rightarrow> 'k"   and h :: "'h \<Rightarrow> 'l"   and i :: "'i \<Rightarrow> 'm" shows "map_sum f g \<circ> map_sum h i = map_sum (f \<circ> h) (g \<circ> i)"
nat_into_Ord: fixes n :: "V" assumes "n \<in> elts \<omega>" shows "Ord n"
assms: shows "\<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q)"
cancel: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point"   and p\<^sub>3 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.nonsingular R a b"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2"   and "ell_field.on_curve R a b p\<^sub>3"   and "ell_field.add R a p\<^sub>1 p\<^sub>2 = ell_field.add R a p\<^sub>1 p\<^sub>3" shows "p\<^sub>2 = p\<^sub>3"
monoI_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "mono f"   and "mono g" shows "mono (f \<circ> g)"
not_less: fixes y :: "'a list"   and x :: "'a list" shows "(\<not> y <lex x) = (x \<le>lex y)"
Amortized2_axioms: shows "Amortized2 arity exec inv cost \<Phi> U exec' inv' cost' U' hom"
forall_\<Pi>\<^sub>3_def: shows "forall \<equiv> forall\<^sub>3"
assms_5: shows "A' B' C' G H I SumA K L M"
assms: shows "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> f \<circ>\<^sub>\<bullet> g"
l_identity: fixes identity :: "'a \<Rightarrow> 'a \<Rightarrow> \<o>"   and w :: "i"   and x :: "'a"   and y :: "'a"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "class.identifiable identity"   and "[identity x y in w]"   and "[\<phi> x in w]" shows "[\<phi> y in w]"
assms_2: fixes a :: "'a" assumes "x = Some a" shows "f a h = f' a h"
path_edges_remove_prefix: fixes xs :: "'a list"   and x :: "'a"   and xs' :: "'a list" assumes "Q.path (xs @ x # xs')" shows "H.edges_of_walk (xs @ [x]) = H.edges_of_walk (xs @ x # xs') - H.edges_of_walk (x # xs')"
size1_height_if_incomplete: fixes t :: "'b tree" assumes "\<not> Tree.complete t" shows "size1 t < 2 ^ height t"
distinct_1: fixes x2 :: "'e" shows "None \<noteq> Some x2"
intros: fixes uu :: "nat" shows "Regions.isGreater (Regions.intv.Greater uu)"
IH_2: fixes d :: "nat"   and t' :: "'a tree" assumes "ins3 n d x r = Bal2 t'"   and "bal_i n (height r + d)" shows "real (T_ins3 n d x r) + \<Phi> t' - \<Phi> r \<le> (6 * e + 2) * real (height r + 1)"
simps_1: fixes x :: "('a, 'b) strand_step" shows "trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p (Step x) = trms\<^sub>s\<^sub>t\<^sub>p x"
\<gamma>\<^sub>f: shows "\<guillemotleft>\<gamma>\<^sub>f : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma>\<^sub>f \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>\<^sub>f)"
Nmlize_Hcomp: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Arr t"   and "E.Arr u" shows "\<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor>\<^bold>\<rfloor>"
transpose_transpose: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
Rep_splitres_inverse: fixes x :: "('a, 'b) splitres" shows "Abs_splitres (Rep_splitres x) = x"
csubspace_scaleC_invariant: fixes a :: "complex"   and S :: "'d set" assumes "a \<noteq> 0"   and "csubspace S" shows "(*\<^sub>C) a ` S = S"
IH_1: assumes "inv_max_hi l"   and "l \<noteq> \<langle>\<rangle>" shows "\<exists>a\<in>Tree2.set_tree l. high a = max_hi l"
Fubini_integral: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f x y)" shows "LBINT y. LBINT x. f x y = LBINT x. integral\<^sup>L lborel (f x)"
i_join_i_shrink_iMOD: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat" assumes "0 < k" shows "f \<longmapsto> k \<Join>\<^sub> [ n * k + (k - 1), mod k ] = f \<div> k \<Join>\<^sub> [n\<dots>]"
shiftr_word_eq: fixes w :: "'b word"   and n :: "nat" shows "w >> n = drop_bit n w"
postdominate_inner_path_targetnode: fixes n' :: "'node"   and n :: "'node"   and as :: "'edge list"   and n'' :: "'node" assumes "n' postdominates n"   and "n -as\<rightarrow>\<^sub>\<iota>* n''"   and "n' \<notin> set (sourcenodes as)" shows "n' postdominates n''"
minimal: fixes x :: "'a" shows "\<bottom> \<sqsubseteq> x"
c_guard: shows "opt_mru_guard (process_mru (votes s)) Q v"
ex_un_null: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "partial_magma C" shows "\<exists>!n. \<forall>f. C n f = n \<and> C f n = n"
subst_not_free: fixes s :: "db"   and i :: "nat"   and t :: "db"   and u :: "db" assumes "\<not> freedb s i" shows "s<t/i> = s<u/i>"
max_bot: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order_bot bot less_eq less" shows "ord.max less_eq bot x = x"
leI: fixes x :: "'a"   and y :: "'a" assumes "\<not> x \<prec> y" shows "y \<preceq> x"
connectedI_interval: fixes U :: "'b set" assumes "\<And>x y z. \<lbrakk>x \<in> U; y \<in> U; x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> z \<in> U" shows "connected U"
Dom_rsA: fixes x :: "'b set" shows "Domainp (rel_set A) x = (x \<subseteq> U\<^sub>V\<^sub>S)"
ltake_ldrop: fixes n :: "enat"   and m :: "enat"   and xs :: "'a llist" shows "ltake n (ldrop m xs) = ldrop m (ltake (n + m) xs)"
simps_1: fixes x :: "nat" shows "free_vars (Var x) = {x}"
right_congruent: shows "(\<lambda>U. msgrel `` {freeright U}) respects msgrel"
min: shows "\<forall>(e1, e2)\<in>F. \<not> sinvar \<lparr>nodes = nodes G, edges = edges G - F \<union> {(e1, e2)}\<rparr> nP"
left_cancel: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "group f z inverse" shows "(f a b = f a c) = (b = c)"
ax_free_wt: fixes G :: "prog"   and t :: "Term.term"   and A :: "'b triple set"   and P :: "'b assn"   and Q :: "'b assn" assumes "(\<exists>T L C. \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow> G,A\<turnstile>{Normal P} t\<succ> {Q}" shows "G,A\<turnstile>{Normal P} t\<succ> {Q}"
x01_elim: fixes emBits :: "nat"   and M :: "bit list" assumes "176 + sLen \<le> emBits"   and "roundup emBits 8 * 8 - emBits \<le> 7" shows "take 8 (drop (roundup emBits 8 * 8 - (length (sha1 M) + sLen + 16)) (maskedDB_zero (generate_DB (generate_PS emBits 160)) emBits)) = [\<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<one>]"
closed_subset_Ints: fixes A :: "'b set" assumes "A \<subseteq> \<int>" shows "closed A"
no_overlapsI: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('b, 'd) flow_entry_match list" assumes "check_no_overlap \<gamma> t"   and "distinct t" shows "no_overlaps \<gamma> t"
d: shows "nat_of_ipv4addr (ipv4addr_of_nat (d + 256 * c + 65536 * b + 16777216 * a) && mask 8) = d"
H_edge_succ_tail_eqD: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and a :: "'d"   and b :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"   and "edge_succ HM a = b" shows "tail H a = tail H b"
assms_2: shows "\<not> is_zero_row_upt_k i k A"
\<tau>mredT_thread_preserved: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "\<tau>mredT s s'" shows "(thr s t = None) = (thr s' t = None)"
rga_ops_def: fixes list :: "('a \<times> 'a option) list" shows "rga_ops list \<equiv> crdt_ops list set_option"
empty_prob_dom_finite: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "prob_dom PROB = {}" shows "finite PROB"
rtrancl_refl: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" shows "r\<^sup>*\<^sup>* a a"
insertAssertionStatImp: fixes F :: "'b frame"   and \<Psi> :: "'b"   and G :: "'b frame"   and \<Psi>' :: "'b" assumes "insertAssertion F \<Psi> \<hookrightarrow>\<^sub>F insertAssertion G \<Psi>"   and "\<Psi> \<simeq> \<Psi>'" shows "insertAssertion F \<Psi>' \<hookrightarrow>\<^sub>F insertAssertion G \<Psi>'"
res_alt_def: fixes \<eta> :: "nat" shows "CIPHER.res \<eta> = (1\<^sub>C |\<^sub>= CIPHER.enc \<eta> |\<^sub>= CIPHER.dec \<eta>) \<odot> (1\<^sub>C |\<^sub>= parallel_wiring) \<rhd> parallel_resource1_wiring \<rhd> CIPHER.KEY.res \<eta> \<parallel> AUTH.res"
assms_1: shows "\<bar>Inf X\<bar> \<noteq> \<infinity>"
swapidseq_ext_backwards: fixes A :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext A (Suc n) p" shows "\<exists>a b A' p'. a \<noteq> b \<and> A = insert a (insert b A') \<and> swapidseq_ext A' n p' \<and> p = Fun.swap a b id \<circ> p'"
hyp2: shows "snd (Gauss_Jordan_upt_k_det_P A k) = snd (Gauss_Jordan_upt_k_PA A k)"
fmap_fmap: fixes f :: "'c \<rightarrow> 'a"   and g :: "'d \<rightarrow> 'c"   and xs :: "'d\<cdot>'b" shows "fmap\<cdot>f\<cdot>(fmap\<cdot>g\<cdot>xs) = fmap\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
card_of_min_algs: fixes i :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set"   and s1 :: "('e, 'd, 'f) F1 \<Rightarrow> 'd"   and s2 :: "('e, 'd, 'f) F2 \<Rightarrow> 'f" shows "i \<in> Field SucFbd \<longrightarrow> |fst (min_algs s1 s2 i)| \<le>o ASucFbd \<and> |snd (min_algs s1 s2 i)| \<le>o ASucFbd"
surj_pair: fixes p :: "'a \<times> 'b" shows "\<exists>x y. p = (x, y)"
existence_ivl_def: fixes t0 :: "real"   and x0 :: "'a" shows "existence_ivl t0 x0 = (\<Union>(x, t1)\<in>csols t0 x0. {t0--t1})"