hcju/isabelleps huggingface dataset:
Un_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set"   and D :: "'b set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
is_red_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and a :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red F a = (\<not> relation.is_final (red F) a)"
map_simp: fixes f :: "'d" assumes "arr f" shows "\<rho>' f = local.inv (\<rho> (cod f)) \<cdot> local.map f"
less_sqrt_iff: fixes y :: "real"   and x :: "real" assumes "0 \<le> y" shows "(x < sqrt y) = (0 \<le> x \<longrightarrow> x\<^sup>2 < y)"
loc_Op1_def: shows "loc_Op1 = (\<lambda>f x y z. sgn1 (z - y) * y + sgn2 (z - y) * (sgn1 (f x z) * z + sgn2 (f x z) * Suc z))"
hom_char: fixes f :: "'a"   and b :: "'a"   and a :: "'a" shows "Cop.in_hom f b a = \<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>"
assms_2: shows "eq_\<I>_gpv A \<I> gpv gpv'"
ord_equiv_isom: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" assumes "Order D"   and "Order E"   and "ord_equiv D E" shows "\<exists>f. ord_isom D E f"
C_Proj: fixes E :: "trm \<Rightarrow> trm"   and l :: "char list" assumes "E \<in> ctxt" shows "(\<lambda>t. E t..l) \<in> ctxt"
WriteVolatile: fixes \<O>s :: "nat set list"   and i :: "nat"   and a :: "nat"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = _emptyset"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> \<O>s ! j = _emptyset"   and "a \<notin> read_only \<S>" shows "\<O>s,i\<turnstile> (Write True a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
exhaust: fixes y :: "transaction" obtains x1 :: "common_metadata" and x2 :: "participant_metadata" and x3 :: "view list" where "y = Transaction x1 x2 x3"
ma_times: fixes r1 :: "mini_alg"   and r2 :: "mini_alg" shows "real_of r1 * real_of r2 = (if ma_compatible r1 r2 then real_of (ma_times r1 r2) else Code.abort STR ''different base'' (\<lambda>_. real_of r1 * real_of r2))"
tableauproof_def: fixes ps :: "('a, 'b) form list"   and p :: "('a, 'b) form" shows "tableauproof ps p \<equiv> (\<stileturn> Neg p # ps)"
eval_\<theta>: shows "\<theta> = (\<lambda>x. \<Sum>p\<leftarrow>map fst (filter snd (List.enumerate 2 (sieve 1 (replicate (nat \<lfloor>x\<rfloor> - 1) True)))). ln (real p))"
M: shows "M = add_mset (Min_mset M) (M - {#Min_mset M#})"
graph_homomorphism_nonempty: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set"   and e :: "'b allegorical_term" assumes "graph_homomorphism A B f"   and ":A:\<lbrakk>e\<rbrakk> \<noteq> {}" shows ":B:\<lbrakk>e\<rbrakk> \<noteq> {}"
simps_1: fixes A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat" shows "Smith_1xn_aux 0 A (S, Q) = (S, Q)"
sort_eqv_imp_normalize_sort_eq: fixes s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "sort_ex cs s1"   and "sort_ex cs s2"   and "sort_eqv cs s1 s2" shows "normalize_sort cs s1 = normalize_sort cs s2"
cc_restr_ccSquare: fixes S :: "Vars.var set"   and S' :: "Vars.var set" shows "(S'\<^sup>2) G|` S = (S' \<inter> S)\<^sup>2"
up_eq: fixes x :: "'a"   and y :: "'a" shows "(up\<cdot>x = up\<cdot>y) = (x = y)"
PScope: fixes \<pi>' :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option" assumes "\<pi>': (c, s) \<Rightarrow> t" shows "\<pi>: (PScope \<pi>' c, s) \<Rightarrow> t"
Transset_hinsert: fixes x :: "hf"   and y :: "hf" assumes "Transset x"   and "y \<le> x" shows "Transset (x \<triangleleft> y)"
IdE: fixes p :: "'d \<times> 'd" assumes "p \<in> Id" obtains x :: "'d" where "p = (x, x)"
offs_enum_zero: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and i :: "nat"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and k :: "nat" assumes "index_less index key"   and "i < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "offs_num n xs index key mi ma i = 0" shows "offs (enum xs index key n mi ma) k ! i = offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i"
composites_are_identities_1: shows "ide (\<mu> \<star> \<nu> \<star> \<pi>)"
of_int_le_numeral_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "int"   and x :: "num"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus a) (power.power one times (numeral.numeral one plus x) n) = (a \<le> numeral x ^ n)"
generated_ringI: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and C :: "'c set set"   and a :: "'c set" assumes "semiring_of_sets \<Omega> M"   and "finite C"   and "disjoint C"   and "C \<subseteq> M"   and "a = \<Union> C" shows "a \<in> semiring_of_sets.generated_ring M"
A4b: fixes x :: "i"   and y :: "i" shows "cod x \<cdot> y \<cong> cod x \<cdot> cod y"
axioms_3: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs set_child_nodes set_child_nodes_locs remove heap_is_wellformed parent_child_rel" shows "l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes get_owner_document get_parent get_child_nodes"
apsnd_conv: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'b"   and y :: "'d" shows "apsnd f (x, y) = (x, f y)"
onorm: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "bounded_linear f" shows "norm (f x) \<le> onorm f * norm x"
Melem_subseteq_Union_mset: fixes x :: "'a multiset"   and T :: "'a multiset multiset" assumes "x \<in># T" shows "x \<subseteq># \<Sum>\<^sub># T"
subset_Pow_Union: fixes A :: "'d set set" shows "A \<subseteq> Pow (\<Union> A)"
absolutely_integrable_mult_Fejer_kernel_reflected_part3: fixes d :: "real"   and n :: "nat"   and c :: "real" assumes "d \<le> pi" shows "(\<lambda>x. Fejer_kernel n x * c) absolutely_integrable_on {0..d}"
uint64_set_bit_def: fixes x :: "uint64"   and n :: "integer"   and b :: "bool" shows "uint64_set_bit x n b = (if n < 0 \<or> 63 < n then undefined set_bit_class.set_bit x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
Suc_2: shows "\<forall>A\<in>F. finite A \<and> card A = Suc k"
space_count_space: fixes \<Omega> :: "'b set" shows "space (count_space \<Omega>) = \<Omega>"
cnf2bhv_bhv2cnf: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 \<le> n" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>(\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n) = n"
isometries_subspaces: fixes S :: "'c set"   and T :: "'d set" assumes "subspace S"   and "subspace T"   and "dim S = dim T" obtains f :: "'c \<Rightarrow> 'd" and g :: "'d \<Rightarrow> 'c" where "linear f" and "linear g" and "f ` S = T" and "g ` T = S" and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x" and "\<And>x. x \<in> T \<Longrightarrow> norm (g x) = norm x" and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" and "\<And>x. x \<in> T \<Longrightarrow> f (g x) = x"
bind_split: fixes P :: "'a option \<Rightarrow> bool"   and m :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
Functional: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and \<phi> :: "('a, 'b) Language"   and i1 :: "('c, 'd) IType"   and i2 :: "('c, 'd) IType" assumes "Interpretation I"   and "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i1"   and "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i2" shows "i1 = i2"
left_total_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "left_total R = (\<forall>x. \<exists>y. R x y)"
LB_Some: shows "LB dir s' x\<^sub>i = Some l\<^sub>i"
subset: shows "inv_into A g ` g ` A \<subseteq> A"
parts_init: shows "parts (used s\<^sub>0) = used s\<^sub>0"
snd_map_help_2: fixes x :: "nat"   and tsi :: "('a \<times> 'b) list" assumes "x < length tsi" shows "snd (tsi ! x) = separators tsi ! x"
mem_val_alt_8_mem_equal_0: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 (pa AND 68719476732) s1 = mem_val_alt 8 (pa AND 68719476732) s2"
l_def: shows "l = (\<lambda>s. x + (s / norm (f x)) *\<^sub>R rot (f x))"
mult: fixes c :: "real"   and u :: "'a \<Rightarrow> real" assumes "0 \<le> c"   and "u \<in> borel_measurable M"   and "\<And>x. 0 \<le> u x"   and "P u" shows "P (\<lambda>x. c * u x)"
symKey_secrecy: fixes i :: "nat"   and K :: "nat"   and evs :: "event list"   and c :: "nat"   and X :: "msg" assumes "CA i \<notin> bad"   and "K \<in> symKeys"   and "evs \<in> set_cr"   and "Says (Cardholder c) (CA i) X \<in> set evs"   and "Key K \<in> parts {X}"   and "Cardholder c \<notin> bad" shows "Key K \<notin> analz (knows Spy evs)"
exists_connected_component_of_superset: fixes X :: "'a topology"   and S :: "'a set" assumes "connectedin X S"   and "topspace X \<noteq> {}" shows "\<exists>C. C \<in> connected_components_of X \<and> S \<subseteq> C"
igConsIPresIGWlsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igConsIPresIGWlsSTR MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWlsSTR MOD \<and> igOpIPresIGWlsSTR MOD"
splitTree_aux_def: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and t :: "('a, 'b) FingerTree" shows "FingerTree.splitTree_aux p i t \<equiv> if \<not> p i \<and> p (i + FingerTree.annot t) then Abs_splitres (FingerTreeStruc.splitTree p i (Rep_FingerTree t)) else Abs_splitres (Empty, undefined, Empty)"
add_point_design: fixes p :: "'a" shows "design (add_point p) \<B>"
assms_2: shows "B \<Longrightarrow>\<^sub>A B' * true"
msort_monoms_impl_def: shows "msort_monoms_impl = msort (\<le>)"
transitive_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "transitive A less_eq \<equiv> \<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> z \<in> A \<longrightarrow> less_eq x z"
d_mu_Ints: fixes l :: "nat"   and k :: "nat" assumes "l \<le> k"   and "k < length (map of_int_hom.vec_hom fs)" shows "gs.d (Suc l) * gs.\<mu> k l \<in> \<int>"
wf_dickson_less_v: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfP (punit.dickson_less_v d m)"
distinct_6: fixes x1 :: "nat" shows "Em \<noteq> At x1"
dbm_int_guard_abstr: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and X :: "'f set"   and k :: "'f \<Rightarrow> 'g"   and l :: "'e"   and g :: "('f, 'g) cconstraint"   and a :: "'h"   and r :: "'f list"   and l' :: "'e"   and n :: "nat"   and v :: "'f \<Rightarrow> nat" assumes "valid_abstraction A X k"   and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "dbm_int (abstr g (\<lambda>i j. \<infinity>) v) n"
at_within_open: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and S :: "'c set" assumes "class.topological_space open"   and "a \<in> S"   and "open S" shows "topological_space.at_within open a S = topological_space.at_within open a UNIV"
big_unclocked0_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3 :: "'b state \<times> (v, v) result" assumes "evaluate ck env s e r3"   and "ck = False" shows "snd r3 \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock (fst r3)"
map_sel_2: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d rule" shows "get_action (map_rule f a) = get_action a"
gauss_0_1: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "(B *\<^sub>v x = 0\<^sub>v n) = (C *\<^sub>v x = 0\<^sub>v n)"
FreeGroupI_transfer: fixes xs :: "('a \<times> bool) list"   and S :: "'a set" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "Abs_freeword xs \<in> FreeGroup S"
alpha_Tree_induct': fixes P :: "('a, 'b, 'c) Tree \<Rightarrow> ('a, 'b, 'c) Tree \<Rightarrow> bool"   and a0 :: "('a, 'b, 'c) Tree"   and a1 :: "('a, 'b, 'c) Tree" assumes "\<And>tset1 tset2. (\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2\<rbrakk> \<Longrightarrow> P a b) \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. P t1 t2 \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>\<phi>1 \<phi>2. P (tPred \<phi>1) (tPred \<phi>2)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>x. P (x \<bullet> t1) t2; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>x a b. P ((a \<rightleftharpoons> b) \<bullet> t2) t2\<rbrakk> \<Longrightarrow> P (tAct \<alpha>1 t1) (tAct \<alpha>2 t2)"   and "\<And>v va. P (tNot v) (tConj va)"   and "\<And>v va. P (tNot v) (tPred va)"   and "\<And>v va vb. P (tNot v) (tAct va vb)"   and "\<And>v va. P (tPred v) (tConj va)"   and "\<And>v va. P (tPred v) (tNot va)"   and "\<And>v va vb. P (tPred v) (tAct va vb)"   and "\<And>v va vb. P (tAct v va) (tConj vb)"   and "\<And>v va vb. P (tAct v va) (tNot vb)"   and "\<And>v va vb. P (tAct v va) (tPred vb)"   and "\<And>va v. P (tConj va) (tNot v)"   and "\<And>va v. P (tPred va) (tNot v)"   and "\<And>va vb v. P (tAct va vb) (tNot v)"   and "\<And>va v. P (tConj va) (tPred v)"   and "\<And>va v. P (tNot va) (tPred v)"   and "\<And>va vb v. P (tAct va vb) (tPred v)"   and "\<And>vb v va. P (tConj vb) (tAct v va)"   and "\<And>vb v va. P (tNot vb) (tAct v va)"   and "\<And>vb v va. P (tPred vb) (tAct v va)" shows "P a0 a1"
assms_2: fixes K :: "nat" assumes "P K"   and "\<And>k. k < K \<Longrightarrow> \<not> P k" shows "Q"
gmn_gmd: fixes nd :: "('b, 'c) Node" assumes "is_measured_node nd" shows "gmd (nodeToDigit nd) = gmn nd"
except_eq_zeroE: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" assumes "except p S = 0" shows "keys p \<subseteq> S"
fps_divide_by_mult: fixes g :: "'a fps"   and f :: "'a fps"   and h :: "'a fps" assumes "subdegree g \<le> subdegree f" shows "f / (g * h) = f / g / h"
IN_f_outside: fixes x :: "'v"   and n :: "nat" assumes "x \<notin> \<^bold>V" shows "d_IN (f n) \<langle>x\<rangle> = 0"
M_def: shows "M = Tensor.mat_of_cols_list (dim_row A) (mat_mult (mult.row_length (mat_to_cols_list A)) (mat_to_cols_list A) (mat_to_cols_list B))"
simps_3: fixes v :: "'b" shows "Some v \<squnion> None = Some v"
cfun_cong: fixes f :: "'r \<rightarrow> 's"   and g :: "'r \<rightarrow> 's"   and x :: "'r"   and y :: "'r" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
assms_3: shows "v \<leadsto> w = ps @ u # ps'"
subset_not_subset_eq: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subset> B) = (A \<subseteq> B \<and> \<not> B \<subseteq> A)"
pr_simps_6: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "cod \<p>\<^sub>1[a0, a1] = a1"
complex_poly_cong: fixes p :: "complex poly"   and q :: "complex poly" shows "(map_poly Re p = map_poly Re q \<and> map_poly Im p = map_poly Im q) = (p = q)"
unfold_lesub_err: fixes e1 :: "'b err"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e2 :: "'b err" shows "(e1 \<sqsubseteq>\<^bsub>Err.le r\<^esub> e2) = Err.le r e1 e2"
DBM_reset_complete: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and c :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "v c \<le> n"   and "DBM_reset M n (v c) d M'"   and "u \<turnstile>\<^bsub>v,n\<^esub> M" shows "u(c := d) \<turnstile>\<^bsub>v,n\<^esub> M'"
well_valued0_rchild_if_well_valued0: fixes t :: "'a tree" assumes "local.well_valued0 t" shows "local.well_valued0 (rchild t)"
FFreshBottom_1: fixes xvec :: "name list" shows "xvec \<sharp>* \<bottom>\<^sub>F"
is_Vabs_def: fixes "value" :: "value" shows "is_Vabs value = (\<exists>x21 x22. value = Vabs x21 x22)"
mem_in_image3: fixes b :: "'h"   and f :: "'i \<Rightarrow> 'h"   and A :: "'i set" assumes "b \<in> f ` A" shows "\<exists>a\<in>A. b = f a"
uconnectedV_sym: fixes E' :: "'a uprod set" shows "sym (Restr (uconnected E') verts)"
induced_arrow_eqI: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and a' :: "'b"   and \<chi>' :: "'a \<Rightarrow> 'b"   and f :: "'b" assumes "limit_cone J C D a \<chi>"   and "Limit.cone J C D a' \<chi>'"   and "partial_magma.in_hom C f a' a"   and "(\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D (partial_magma.cod C f) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) f else partial_magma.null C) \<chi> = \<chi>'" shows "limit_cone.induced_arrow J C D a \<chi> a' \<chi>' = f"
llcp_ltake1: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'c llist" shows "llcp (ltake n xs) ys = min n (llcp xs ys)"
rel_def_internal: fixes Rv :: "('V \<times> 'c) set"   and Rw :: "('W \<times> 'd) set" shows "rel Rv Rw \<equiv> br \<alpha> invar O \<langle>Rv, Rw\<rangle>graph_rel"
simps: fixes a :: "('test, 'atom, 'choice) cmd" shows "discr a = (\<exists>c. a = c \<and> (\<forall>x xa. xa < brn c \<longrightarrow> x \<approx> eff c x xa \<and> discr (local.cont c x xa)))"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('a, 'b) atm" shows "U.satL \<xi> (Pos a) = U.satA \<xi> a"
not_le_imp_less: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> y \<preceq> x" shows "x \<prec> y"
fresh_2: fixes a :: "name"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2 \<rightarrow> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
leq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x + y = y)"
cardEx: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "card B < card A" shows "\<exists>x\<in>A. x \<notin> B"
yg: shows "y = map_default b ((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b})"
shortestPath_is_simple: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "Graph.isShortestPath c s p t" shows "Graph.isSimplePath c s p t"
minus_ant_def: fixes z :: "ant" shows "- z \<equiv> Abs_Ainteg (- fst (Rep_Ainteg z), - snd (Rep_Ainteg z))"
reach_trans: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set" assumes "reach T \<subseteq> I" shows "{reach T} TS.trans T {> I}"
iprev_nth_imirror_inext_nth_conv2: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "mirror_elem (imirror I \<rightarrow> n) I = I \<leftarrow> n"
integral_diff: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x - g x = integral\<^sup>L M f - integral\<^sup>L M g"
equivalence_map_def: fixes f :: "'a" shows "B.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. B.internal_equivalence f g \<eta> \<epsilon>"
ASSUME_leI: fixes \<Phi> :: "bool"   and M :: "'b nres"   and M' :: "'b nres" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "ASSUME \<Phi> \<bind> (\<lambda>_. M) \<le> M'"
update_arg_wf_tuples_2: fixes n :: "nat"   and hops :: "nat"   and nhip :: "nat" shows "update_arg_wf (Suc n, kno, val, hops, nhip)"
keys_fold_combine: fixes A :: "'c set"   and g :: "'c \<Rightarrow> ('d, real) mapping" assumes "finite A" shows "Mapping.keys (fold_combine_plus.combine.F g A) = (\<Union>x\<in>A. Mapping.keys (g x))"
encode_Suc: fixes e :: "nat"   and n :: "nat" shows "local.encode e (Suc n) = of_nat (Suc n mod base) * \<omega>^ (local.encode 0 e) + local.encode (e + 1) (Suc n div base)"
i_1: shows "normal_mantissa x = mantissa f * 2 ^ i"
ucast_up_neq: fixes x :: "'c word"   and y :: "'d word" assumes "UCAST('c \<rightarrow> 'b) x \<noteq> UCAST('d \<rightarrow> 'b) y"   and "LENGTH('b) \<le> LENGTH('e)" shows "UCAST('c \<rightarrow> 'e) x \<noteq> UCAST('d \<rightarrow> 'e) y"
subst_psubst: fixes \<rho> :: "(nat \<times> exp) list"   and v :: "exp"   and x :: "nat"   and e :: "exp" assumes "closed_env \<rho>"   and "FV v = {}" shows "subst x v (psubst ((x, EVar x) # \<rho>) e) = psubst ((x, v) # \<rho>) e"
conversep_conversep: fixes r :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
fv_list_is_fv: fixes t :: "('b, 'a) Term.term" shows "fv t = set (fv_list t)"
simps_10: fixes y :: "name"   and xvec :: "name list"   and Tvec :: "'a list"   and I :: "('a, 'b, 'c) input" assumes "y \<sharp> xvec"   and "y \<sharp> Tvec" shows "subs' (\<nu>yI) xvec Tvec = \<nu>ysubs' I xvec Tvec"
runit'_simps_1: fixes f :: "'d" assumes "D.ide f" shows "D.arr \<r>\<^sub>D\<^sup>-\<^sup>1[f]"
f0: shows "snd (execute_instruction () s1) = False \<and> snd (execute_instruction () s2) = False"