hcju/isabelleps huggingface dataset:
rel_S_witness2: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok :: "('o \<times> ('o \<times> 'p) \<times> 'p \<times> 'q \<times> ('q \<times> 'r) \<times> 'r \<times> 's) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "('o, 'q, 'b, 'f, 'i, 'l, 's) S"   and y :: "('p, 'r, 'e, 'h, 'k, 'n, 's) S" assumes "rel_S_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_S L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_S (\<lambda>(x, y') y. y' = y \<and> L1 x y) (\<lambda>(x, y') y. y' = y \<and> L2 x y) Co1' Co2' Contra1' Contra2' (rel_S_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y)) y"
transition_system_interpreted_traces_axioms: shows "transition_system_interpreted_traces ex en int ind"
unify_complete: fixes E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and bs :: "('a \<times> ('b, 'a) Term.term) list" assumes "unify E bs = None" shows "unifiers (set E) = {}"
o_def: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" shows "f \<circ> g = (\<lambda>x. f (g x))"
assms: shows "prob0_fst_qubits n jozsa_algo = 0"
take_map: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "take n (map f xs) = map f (take n xs)"
permutation_mat_def: fixes n :: "nat"   and p :: "nat \<Rightarrow> nat" shows "permutation_mat n p = Matrix.mat n n (\<lambda>(i, j). if i = p j then 1::'b else (0::'b))"
l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<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" shows "l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs \<and> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
mojmir_accept_token_set_def2: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F)"   and "finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i)"   and "infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)" shows "mojmir_def.accept \<delta> q\<^sub>0 w F"
in_arcs_add_arc_iff: fixes a :: "'a \<times> 'a"   and u :: "'a" shows "in_arcs (G.add_arc a) u = (if snd a = u then insert a (in_arcs (with_proj G) u) else in_arcs (with_proj G) u)"
Lxyinitxy: shows "Lxy init {x, y} \<in> {[x, y], [y, x]}"
ksub: shows "k ` ({0..1} \<times> T) \<subseteq> T"
elims: fixes x :: "('d, 'e) Term.term"   and y :: "('d \<times> nat) option" assumes "root x = y" obtains xa :: "'e" where "x = Var xa" and "y = None"   | f :: "'d" and ts :: "('d, 'e) Term.term list" where "x = Fun f ts" and "y = Some (f, length ts)"
eWlsInp_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym" shows "\<not> eWlsInp MOD delta (\<lambda>i. Some ERR)"
FD_D: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>D Q"
rsp: shows "rel_fun (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b))) (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b))) uminus uminus"
finite_sets_supp: fixes S :: "'b set" assumes "finite S" shows "finite (supp S)"
no_Trace_implies_no_Failure: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<notin> T P" shows "(s, {}) \<notin> F P"
InitRInit: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and C' :: "char list" assumes "P \<turnstile> \<langle>RI (C,C\<bullet>\<^sub>sclinit([])) ; Cs \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
wf_asym: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and x :: "'b" assumes "wf r"   and "(a, x) \<in> r" obtains "(x, a) \<notin> r"
measurable_vimage_algebra2: fixes g :: "'b \<Rightarrow> 'c"   and N :: "'b measure"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'d measure" assumes "g \<in> space N \<rightarrow> X"   and "(\<lambda>x. f (g x)) \<in> N \<rightarrow>\<^sub>M M" shows "g \<in> N \<rightarrow>\<^sub>M vimage_algebra X f M"
dgrad_p_set_le_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dgrad_p_set_le d F G"   and "G \<subseteq> dgrad_p_set d m" shows "F \<subseteq> dgrad_p_set d m"
nat_to_set_srj1: fixes D :: "nat set" assumes "finite D" shows "\<exists>u. nat_to_set u = D"
emeasure_pair_measure_alt: fixes M :: "'c measure"   and X :: "('d \<times> 'c) set"   and N :: "'d measure" assumes "sigma_finite_measure M"   and "X \<in> sets (N \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M M) X = \<integral>\<^sup>+ x. emeasure M (Pair x -` X) \<partial>N"
f9: shows "module.lincomb (module_cpx_vec n) a A $ i = (\<Sum>u\<in>A. a u * u $ i)"
swap_id_idempotent: fixes a :: "'b"   and b :: "'b" shows "Fun.swap a b id \<circ> Fun.swap a b id = id"
is_ord_delete_max: fixes t :: "'b tree" assumes "avl t"   and "is_ord t"   and "t \<noteq> ET" shows "is_ord (snd (delete_max t))"
sound_isTrue: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "isTrue \<phi>"
npow_exp_minusTr1: fixes x :: "'a"   and i :: "int"   and j :: "nat" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>"   and "0 \<le> i" shows "0 \<le> i - int j \<longrightarrow> x\<^bsub>K\<^esub>\<^bsup>(i - int j)\<^esup> = x^\<^bsup>K nat i\<^esup> \<cdot>\<^sub>r x\<^bsup>\<hyphen> K\<^esup>^\<^bsup>K j\<^esup>"
tabulate_Mapping: fixes ks :: "'b list"   and f :: "'b \<Rightarrow> 'c" shows "Mapping.tabulate ks f = RBT_Mapping.Mapping (RBT.bulkload (map (\<lambda>k. (k, f k)) ks))"
mat_of_rows_list_def: fixes nc :: "nat"   and rs :: "'c list list" shows "mat_of_rows_list nc rs = mat (length rs) nc (\<lambda>(i, j). rs ! i ! j)"
simps_2: fixes t :: "'b Trace"   and s :: "'b" shows "tLength (t \<leadsto> s) = 1 + tLength t"
strip_guards_simps_10: fixes F :: "'f set" shows "strip_guards F Throw = Throw"
assms: fixes A :: "'b set" shows "times_abc a (Sup_b A) = Sup_c (times_abc a ` A)"
arr_char: fixes \<mu> :: "'a arrow_of_spans_data" shows "V.arr \<mu> = arrow_of_spans (\<cdot>) \<mu>"
M_def: fixes x :: "int" shows "M x = x mod p ^ n"
simps_1: fixes Q :: "('a \<times> 'b, unit) HashMap.hashmap"   and W :: "('a \<times> 'b) list" shows "pa_upd_rule Q W [] = (Q, W)"
b_e_type_empty: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" shows "\<C> \<turnstile> [] : ts _> ts' = (ts = ts')"
lipschitz_area: fixes t :: "real" assumes "t \<in> T" shows "cball (solution t) \<bar>r\<bar> \<subseteq> X'"
P9: shows "B Midpoint C D''__ \<and> Cong B' C B' D"
shows_prec_char_def: fixes p :: "nat"   and c :: "char" shows "showsp_char p c = (#) c"
HFinite_HFinite_hcomplex_of_hypreal: fixes z :: "real star" assumes "z \<in> HFinite" shows "hcomplex_of_hypreal z \<in> HFinite"
Entry_reaches: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" shows "\<exists>ns. g \<turnstile> Entry g-ns\<rightarrow>n"
lte: fixes x :: "'a"   and y :: "'b" assumes "b / 2 \<le> norm (f x y)"   and "norm (f x y - l y) < e * b\<^sup>2 / 2"   and "y \<in> S" shows "dist (inverse (f x y)) ((inverse \<circ> l) y) < e"
function_set_finite: fixes R :: "'c set"   and A :: "'d set"   and c :: "'d" assumes "finite R"   and "finite A" shows "finite {f. (\<forall>x. x \<notin> R \<longrightarrow> f x = c) \<and> (\<forall>x. x \<in> R \<longrightarrow> f x \<in> A)}"
in_disconnected_nodes_no_parent: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and h :: "(_) heap"   and node_ptr :: "(_) node_ptr"   and owner_document :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes get_owner_document get_parent"   and "heap_is_wellformed h"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"   and "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r owner_document"   and "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"   and "known_ptrs h"   and "type_wf h" shows "node_ptr \<in> set disc_nodes"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>L 'c" shows "blinfun_apply (x o\<^sub>L xa) = blinfun_apply x \<circ> blinfun_apply xa"
Ass: fixes G :: "prog"   and s0 :: "st"   and va :: "Term.var"   and n :: "nat"   and w :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state"   and e :: "expr"   and v :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w, f)\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<midarrow>n\<rightarrow> assign f v s2"
sqrt_1_plus_square: fixes a :: "real" shows "sqrt (1 + a\<^sup>2) \<noteq> 0"
PO_rhoare_defs_1: fixes pre :: "('e \<times> 'f) set"   and Ra :: "('e \<times> 'e) set"   and Rc :: "('f \<times> 'f) set"   and post :: "('e \<times> 'f) set" shows "{pre} Ra, Rc {> post} \<equiv> pre O Rc \<subseteq> Ra O post"
euclidean_size_of_nat: fixes divide :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f"   and modulo :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and euclidean_size :: "'f \<Rightarrow> nat"   and division_segment :: "'f \<Rightarrow> 'f"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (semiring_1.of_nat one plus zero n) = n"
acyclic_4a_4b: fixes x :: "'a" shows "pd_kleene_allegory.acyclic_4a (\<lambda>x. top) (\<le>) (\<sqinter>) x = pd_kleene_allegory.acyclic_4b (\<lambda>x. top) (\<le>) (\<sqinter>) x"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<And>C. Prop C \<Psi> P M N P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> N"   and "x \<sharp> C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M N (\<lparr>\<nu>x\<rparr>P')"
poly_of_const_def: fixes R :: "('d, 'e) ring_scheme" assumes "ring R" shows "ring.poly_of_const R = (\<lambda>k. ring.normalize R [k])"
is_lub_bin: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "{x, y} <<| y"
surj_on_conv: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "surj_on f A B = (\<forall>b\<in>B. \<exists>a\<in>A. b = f a)"
cancel_at_def: fixes i :: "nat"   and l :: "(bool \<times> 'c) list" shows "cancel_at i l = take i l @ drop (2 + i) l"
set_of_graph_rules_def: fixes Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set" shows "set_of_graph_rules Rs \<equiv> \<forall>R\<in>Rs. graph_rule R"
simps: fixes kv :: "'a \<times> 'c"   and xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "update_by_raw kv (xs, ko) = (update_by_pair ko kv xs, ko)"
l0: shows "2 * 2 / (1 * (min 2 2 - 1)) \<le> real 4"
cod_dom: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.cod (Cop.dom f) = Cop.dom f"
comp_cod_arr: fixes f :: "J.Arr"   and b :: "J.Arr" assumes "J.arr f"   and "J.cod f = b" shows "b \<cdot>\<^sub>J f = f"
x_nblks: shows "\<forall>xa. x \<notin> blocksOf s xa"
replace_coeff_def: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d mpoly" shows "replace_coeff f p = MPoly (Abs_poly_mapping (\<lambda>m. f (lookup (mapping_of p) m)))"
fset_simps_1: shows "fset_from_list [] = {||}"
Polys_closed_monom_mult: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "t \<in> .[X]"   and "p \<in> P[X]" shows "MPoly_Type_Class.punit.monom_mult c t p \<in> P[X]"
strict_implies_order: fixes b :: "'a"   and a :: "'a" assumes "lt b a" shows "le b a"
separatedin_refl: fixes X :: "'c topology"   and S :: "'c set" shows "separatedin X S S = (S = {})"
greatest_def: fixes r :: "('d \<times> 'd) set"   and X :: "'d set" shows "greatest r X = {x \<in> X. \<forall>y\<in>X. (y, x) \<in> r}"
countable_Union: fixes X :: "'d set set"   and M :: "'d measure" assumes "countable X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
rec_pr_Suc_simp_rewrite: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" shows "rec_exec (Pr n f g) (xs @ [Suc x]) = rec_exec g (xs @ [x] @ [rec_exec (Pr n f g) (xs @ [x])])"
ex_path: shows "valid_graph.is_path (del_unEdge v e v' G) v ex_path v'"
com: shows "compact ({0..1} \<inter> {u. g u \<in> closure (- S)})"
src_dom: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.src (Span.dom \<mu>) = Span.src \<mu>"
vint_vsubset_vreal: shows "\<int>\<^sub>\<circ> \<subseteq>\<^sub>\<circ> \<real>\<^sub>\<circ>"
map2_val_compat_id: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator comp" shows "comparator.map2_val_compat comp id"
topspace_standard_simplex: fixes p :: "nat" shows "topspace (subtopology (powertop_real UNIV) (standard_simplex p)) = standard_simplex p"
reach_in_beh: fixes s :: "'e"   and T :: "('e, 'f) TS_scheme" assumes "s \<in> reach T" shows "\<exists>b\<in>beh T. s \<in> set b"
abs_summable_on_Sigma_project2: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f abs_summable_on Sigma A B"   and "x \<in> A" shows "(\<lambda>y. f (x, y)) abs_summable_on B x"
list_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>list_rel \<equiv> {(l, l'). list_all2 (\<lambda>x x'. (x, x') \<in> R) l l'}"
a_repr_independenceD: fixes y :: "'a"   and x :: "'a" assumes "y \<in> carrier R"   and "I +> x = I +> y" shows "y \<in> I +> x"
bisim1AAss3: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and a :: "'addr expr1"   and i :: "'addr expr1"   and v :: "'addr val"   and v' :: "'addr val" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Val v\<lfloor>Val v'\<rceil> := e', xs) \<leftrightarrow> (stk @ [v', v], loc, length (compE2 a) + length (compE2 i) + pc, xcp)"
mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "min a b \<le> min c d"
SReal_dense: fixes x :: "real star"   and y :: "real star" assumes "x \<in> \<real>"   and "y \<in> \<real>"   and "x < y" shows "\<exists>r\<in>\<real>. x < r \<and> r < y"
pedersen_def: fixes \<G> :: "'a cyclic_group" shows "pedersen \<G> \<equiv> pedersen_base \<G> \<and> Cyclic_Group.cyclic_group \<G>"
frontier_def: fixes S :: "'a set" shows "frontier S = closure S - interior S"
select_convs_1: fixes ta_initial :: "'a set"   and ta_rules :: "('a, 'b) ta_rule set"   and more :: "'c" shows "ta_initial \<lparr>ta_initial = ta_initial, ta_rules = ta_rules, \<dots> = more\<rparr> = ta_initial"
bound_compare_defs_5: fixes lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and b :: "'a option" shows "\<unrhd>\<^sub>l\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> True | Some b' \<Rightarrow> lt b' c \<or> b' = c"
lcm_least: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c" shows "dvd.dvd times (lcm a b) c"
axioms_1: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "strict_ordered_set A less" shows "irreflexive A less"
find_non0_3: fixes l :: "nat"   and A :: "'a mat"   and m :: "nat" assumes "det_selection_fun sel_fun"   and "local.find_non0 l A = Some m" shows "m < dim_row A"
UN_extend_simps_2: fixes A :: "'b \<Rightarrow> 'a set"   and C :: "'b set"   and B :: "'a set" shows "\<Union> (A ` C) \<union> B = (if C = {} then B else \<Union>x\<in>C. A x \<union> B)"
xor_def: fixes x :: "'a"   and y :: "'a" shows "boolean_algebra.xor (AND) (OR) NOT x y = x AND NOT y OR NOT x AND y"
NegPos_map_map_Pos: fixes C :: "'c \<Rightarrow> 'b"   and as :: "'c list" shows "NegPos_map C (map Pos as) = map Pos (map C as)"
cfs_minus: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "(p \<ominus>\<^bsub>UP R\<^esub> q) n = p n \<ominus>\<^bsub>R\<^esub> q n"
sig: shows "sigma_array dmus (dmus !! i) (dmus !! Suc j) (dmus !! j !! j) j = sig"
A1': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "tarski_first3 C" shows "C a b b a"
wf_until_aux_Cons1: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and t :: "nat"   and a1 :: "'a option list set"   and a2 :: "'a option list set"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ne :: "nat" assumes "wf_until_aux \<sigma> n R pos \<phi> I \<psi> ((t, a1, a2) # aux) ne" shows "t = \<tau> \<sigma> ne"
lp_monom_mult: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "punit.lt (local.punit.monom_mult c t p) = t + punit.lt p"
h0: shows "independent ((+) (- a) ` (S - {a}))"
P1_security_inf_the: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool" shows "OT_12.perfect_sec_P1 n m1 m2"
trace_parallel_plan_strips_length_gt_one_if: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "1 < length (trace_parallel_plan_strips I \<pi>)"
wellBehavedComposition_def: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" shows "wellBehavedComposition ES1 ES2 \<V> \<V>1 \<V>2 \<equiv> N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {} \<or> (\<exists>\<rho>1. N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>) \<and> BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>) \<or> (\<exists>\<rho>2. N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {} \<and> EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>) \<and> BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>) \<or> (\<exists>\<rho>1 \<rho>2 \<Gamma>1 \<Gamma>2. \<nabla>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub> \<and> \<nabla>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub> \<and> BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub> \<and> BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub> \<and> EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>) \<and> EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>) \<and> FCIA \<rho>1 \<Gamma>1 \<V>1 Tr\<^bsub>ES1\<^esub> \<and> FCIA \<rho>2 \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub> \<and> V\<^bsub>\<V>1\<^esub> \<inter> V\<^bsub>\<V>2\<^esub> \<subseteq> \<nabla>\<^bsub>\<Gamma>1\<^esub> \<union> \<nabla>\<^bsub>\<Gamma>2\<^esub> \<and> C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<and> C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<and> N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {} \<and> N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {})"
inv_comp_right_1: fixes g :: "'a"   and f :: "'a" assumes "C.iso (g \<cdot> f)"   and "C.iso f" shows "C.inv (g \<cdot> f) = C.inv f \<cdot> C.inv g"
hull_antimono: fixes S :: "'d set \<Rightarrow> bool"   and T :: "'d set \<Rightarrow> bool"   and s :: "'d set" assumes "\<forall>x. S x \<longrightarrow> T x" shows "T hull s \<subseteq> S hull s"
permutation_self: fixes p :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "permutation p" shows "\<exists>n>0. (p ^^ n) x = x"
assum_def: shows "assum \<equiv> \<lambda>(pre, rely). {c. snd (c ! 0) \<in> pre \<and> (\<forall>i. Suc i < length c \<longrightarrow> c ! i -e\<rightarrow> c ! Suc i \<longrightarrow> (snd (c ! i), snd (c ! Suc i)) \<in> rely)}"
schur: shows "similar_mat_wit A B P (adjoint P) \<and> diagonal_mat B \<and> diag_mat B = es \<and> unitary P"
funpow_prefix: fixes j :: "nat"   and k :: "nat" assumes "j < k" shows "((f ^^ j) a, (f ^^ k) a) \<in> leq"