hcju/isabelleps huggingface dataset:
justified_action_committedD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('b, 'a) justifying_execution"   and a :: "nat" assumes "P \<turnstile> (E, ws) weakly_justified_by J"   and "a \<in> actions E" obtains n :: "nat" and a' :: "nat" where "a = action_translation (J n) a'" and "a' \<in> committed (J n)"
filter_eq_replicate_mset: fixes x :: "'a"   and D :: "'a multiset" shows "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
max_def: fixes a :: "'c list"   and b :: "'c list" shows "prefix_order.max a b = (if prefix a b then b else a)"
simps_2: fixes sel :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and Rs :: "'f"   and init :: "nat \<times> ('d \<times> nat \<times> nat) set"   and n :: "nat" shows "mk_chain sel Rs init (Suc n) = mk_chain sel Rs (make_step sel init) n"
Lambert_W'_le_minus_one: fixes x :: "real" assumes "x \<le> - exp (- 1)" shows "Lambert_W' x = - 1"
oghoare_steps_Stuck: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and x :: "'d"   and c' :: "('d, 'e, 'f) com"   and P :: "('d, 'e, 'f) ann"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set"   and P' :: "bool" assumes "\<Gamma> \<turnstile> (c, Normal x) \<rightarrow>\<^sup>* (c', Stuck)"   and "x \<in> pre P"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A" shows "P'"
in_measure_of: fixes M :: "'b set set"   and \<Omega> :: "'b set"   and A :: "'b set"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "M \<subseteq> Pow \<Omega>"   and "A \<in> M" shows "A \<in> sets (measure_of \<Omega> M \<mu>)"
insert_diff: fixes x :: "'c"   and S :: "\<langle>'c\<rangle>\<^sub>\<bottom> set" shows "insert \<lfloor>x\<rfloor> (S - {None}) = insert \<lfloor>x\<rfloor> S - {None}"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "fw m n 0 0 0 = fw_upd m 0 0 0"
Inf_to_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf = dual \<circ> Sup \<circ> (`) dual"
interior_Int: fixes S :: "'c set"   and T :: "'c set" shows "interior (S \<inter> T) = interior S \<inter> interior T"
class_def: fixes a :: "'d" shows "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"
linear_imp_scale: fixes D :: "complex \<Rightarrow> 'b" assumes "clinear D" obtains d :: "'b" where "D = (\<lambda>x. x *\<^sub>C d)"
lmap_fst_lzip_conv_ltake: fixes xs :: "'c llist"   and ys :: "'d llist" shows "lmap fst (lzip xs ys) = ltake (min (llength xs) (llength ys)) xs"
nat_of_fatom_bij: shows "bij_betw nat_of_fatom ground_fatoms UNIV"
eq_fold: fixes A :: "'b sat set" shows "Inf A = Finite_Set.fold min top A"
ideal_points_line_unique: fixes H :: "circline"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "is_poincare_line H"   and "ideal_points H = {i1, i2}" shows "H = poincare_line i1 i2"
ssubst: fixes t :: "'f"   and s :: "'f"   and P :: "'f \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
R\<^sub>C_of_strong_low_bisim_mm: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" assumes "abs.strong_low_bisim_mm \<R>\<^sub>A"   and "gen_refine.secure_refinement \<R>\<^sub>A \<R> P"   and "sym P" shows "conc.strong_low_bisim_mm (gen_refine.R\<^sub>C_of \<R>\<^sub>A \<R> P)"
dfs_subset_reachable_visit_nodes: fixes g :: "(node \<times> node) list"   and xs :: "node list"   and ys :: "node list" shows "set (dfs g xs ys) \<subseteq> reachable g xs \<union> set ys"
induct: fixes P :: "('b \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> 'b match_expr \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> bool"   and a1 :: "bool"   and a2 :: "'b match_expr" assumes "\<And>uu_ uv_. P uu_ uv_ MatchAny"   and "\<And>disc neg a. P disc neg (Match a)"   and "\<And>disc neg m. P disc (\<not> neg) m \<Longrightarrow> P disc neg (MatchNot m)"   and "\<And>disc neg m1 m2. \<lbrakk>P disc neg m1; P disc neg m2\<rbrakk> \<Longrightarrow> P disc neg (MatchAnd m1 m2)" shows "P a0 a1 a2"
assms_3: shows "A, n \<turnstile> (q # ps, a) # branch"
cdidi: shows "d\<inverse> O d\<inverse> \<subseteq> d\<inverse>"
single_numeral: fixes n :: "num" shows "single (0::'b) (numeral n) = numeral n"
is_invarI_full: fixes I :: "('v, 'es) state_scheme \<Rightarrow> bool" assumes "\<And>e. I (empty_state e)"   and "\<And>s s' v0 e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s = []; v0 \<notin> dom (discovered s); v0 \<in> V0; s' = new_root v0 s\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; pending s `` {u} = {}; u = hd (stack s); s' = finish u s\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u v e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; v \<in> pending s `` {u}; v \<in> dom (discovered s); v \<in> dom (finished s); u = hd (stack s); s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u v e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; v \<in> pending s `` {u}; v \<in> dom (discovered s); v \<notin> dom (finished s); u = hd (stack s); s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s s' u v e. \<lbrakk>I s; cond s; DFS_invar G param s; DFS_invar G param s'; stack s \<noteq> []; v \<in> pending s `` {u}; v \<notin> dom (discovered s); u = hd (stack s); s' = discover u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<lparr>state.more := e\<rparr>\<rbrakk> \<Longrightarrow> I s'" shows "is_invar I"
fresh_psubst_simp: fixes x :: "name"   and t :: "trm"   and u :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "x \<sharp> t" shows "((x, u) # \<theta>)<t> = \<theta><t>"
a_redu_AndR_elim: fixes a :: "coname"   and M :: "trm"   and b :: "coname"   and N :: "trm"   and c :: "coname"   and R :: "trm" assumes "AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a R" shows "(\<exists>M'. R = AndR <a>.M' <b>.N c \<and> M \<longrightarrow>\<^sub>a M') \<or> (\<exists>N'. R = AndR <a>.M <b>.N' c \<and> N \<longrightarrow>\<^sub>a N')"
handle_Cons: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat"   and vs :: "val list"   and pc :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and xa :: "nat"   and v :: "val"   and ls :: "val list"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P,C,M \<rhd> xt / I,d"   and "d \<le> length vs"   and "pc \<in> I"   and "\<forall>x\<in>set xt. \<not> matches_ex_entry P (cname_of h xa) pc x" shows "handle P C M xa h (v # vs) ls pc frs = handle P C M xa h vs ls pc frs"
one_more_paths_exists_nontrivial: assumes "new_last \<noteq> v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
le_add_same_cancel2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b + a) = (\<Empt> \<subseteq># b)"
proots_within_def: fixes p :: "'b poly"   and s :: "'b set" shows "proots_within p s = {x \<in> s. poly p x = (0::'b)}"
coeffs_uminus: fixes p :: "'b poly" shows "coeffs (- p) = map uminus (coeffs p)"
hom_uminus: fixes x :: "'b poly" shows "pCons (0::'b) (- x) = - pCons (0::'b) x"
ksingle: fixes x :: "'a \<times> bool" shows "k [x] = g [x]"
aext_and: fixes P :: "'c upred"   and Q :: "'c upred"   and x :: "'c \<Longrightarrow> 'b" shows "(P \<and> Q) \<oplus>\<^sub>p x = (P \<oplus>\<^sub>p x \<and> Q \<oplus>\<^sub>p x)"
rotation_matrix_def: fixes Q :: "(('a, 'b) vec, 'b) vec" shows "rotation_matrix Q = (orthogonal_matrix Q \<and> det Q = (1::'a))"
is_span: shows "C.span trg_\<mu>.cod.leg0 trg_\<mu>.cod.leg1"
eq2: shows "update_by_fun_pair k f xs = xs @ (if f (0::'b) = (0::'b) then [] else [(k, f (0::'b))])"
rcosetsI: fixes H :: "'a set set"   and x :: "'a set" assumes "H \<subseteq> carrier (G Mod N)"   and "x \<in> carrier (G Mod N)" shows "H #>\<^bsub>G Mod N\<^esub> x \<in> rcosets\<^bsub>G Mod N\<^esub> H"
subst_respects_ltl_prop_entailment_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "subst \<phi> m \<sim>\<^sub>P subst \<psi> m"
prob_space_distr: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "prob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "prob_space (distr M M' f)"
\<R>\<^sub>1_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "closed_glob_consistent (\<R>\<^sub>1 \<Gamma>')"
Mem_Eats_I1: fixes H :: "fm set"   and u :: "tm"   and t :: "tm"   and z :: "tm" assumes "H \<turnstile> u IN t" shows "H \<turnstile> u IN Eats t z"
empty: shows "\<Prod>\<^sub># {#} = (1::'a)"
list_all2_appendI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c list"   and b :: "'d list"   and c :: "'c list"   and d :: "'d list" assumes "list_all2 P a b"   and "list_all2 P c d" shows "list_all2 P (a @ c) (b @ d)"
fdia_dom: fixes x :: "('a \<times> 'a) set" shows "relfdia x Id = rel_ad (rel_ad x)"
integrable_measure_pmf_finite: fixes M :: "'e pmf"   and f :: "'e \<Rightarrow> 'f" assumes "finite (set_pmf M)" shows "integrable (measure_pmf M) f"
locally_full: fixes f :: "'d"   and f' :: "'d"   and \<nu> :: "'d" assumes "D.ide f"   and "D.ide f'"   and "src\<^sub>D f = src\<^sub>D f'"   and "trg\<^sub>D f = trg\<^sub>D f'"   and "\<guillemotleft>\<nu> : GF.map f \<Rightarrow>\<^sub>D GF.map f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D f'\<guillemotright> \<and> GF.map \<mu> = \<nu>"
lfinite_lSupD: fixes A :: "'c llist set" assumes "lfinite (lSup A)" shows "\<forall>xs\<in>A. lfinite xs"
C_App2: fixes v :: "trm"   and E :: "trm \<Rightarrow> trm" assumes "v \<in> value"   and "E \<in> ctxt" shows "(\<lambda>t. v \<bullet> E t) \<in> ctxt"
standard_redundancy_criterion_counterex_reducing_def: fixes \<Gamma> :: "'b inference set"   and I_of :: "'b literal multiset set \<Rightarrow> 'b set" shows "standard_redundancy_criterion_counterex_reducing \<Gamma> I_of \<equiv> counterex_reducing_inference_system \<Gamma> I_of"
compare_Tuple2_def: shows "compare = (\<Lambda> \<langle>x1, y1\<rangle> \<langle>x2, y2\<rangle>. thenOrdering\<cdot>(compare\<cdot>x1\<cdot>x2)\<cdot>(compare\<cdot>y1\<cdot>y2))"
LEAST: fixes x :: "'a" assumes "f x = x" shows "a \<le> x"
elts_\<omega>: shows "elts \<omega> = {\<alpha>. \<exists>n. \<alpha> = n\<^sub>\<nat>}"
right_inverse: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "group f z inverse" shows "f a (inverse a) = z"
ordinal_exp_oLimit: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "0 < x" shows "x ** oLimit f = oLimit (\<lambda>n. x ** f n)"
cases: fixes a :: "lam" assumes "SN a" obtains t :: "lam" where "a = t" and "\<And>t'.  t \<longrightarrow>\<^sub>\<beta> t' \<Longrightarrow> SN t'"
sum_list_replicate_length: fixes xs :: "'a list" shows "sum_list (replicate (length xs) (Suc 0)) = length xs"
prems_4: shows "P,E \<turnstile> es [::] Ts'"
domain_1'': fixes y :: "'a"   and x :: "'a" shows "ar ar (y \<cdot> x) \<le> ar ar x"
Un_Int_distrib: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> B \<inter> C = (A \<union> B) \<inter> (A \<union> C)"
exec_move_SeqI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e' :: "'addr expr1" assumes "exec_move ci P t e h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e';; e) h (stk, loc, Suc (length (compE2 e') + pc), xcp) ta h' (stk', loc', Suc (length (compE2 e') + pc'), xcp')"
sum_list_map_num_funs_subst: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term"   and ts :: "('a, 'b) Term.term list" assumes "sum_list (map (num_funs \<circ> (\<lambda>t. t \<cdot> \<sigma>)) ts) = sum_list (map num_funs ts)" shows "\<forall>i<length ts. num_funs (ts ! i \<cdot> \<sigma>) = num_funs (ts ! i)"
leg_simps_2: shows "C.dom \<nu>\<mu>.dom.leg0 = \<nu>\<mu>.dom.apex"
closure_interior: fixes S :: "'d set" shows "closure S = - interior (- S)"
rsquarefree_card_degree: fixes p :: "complex poly" assumes "p \<noteq> 0" shows "rsquarefree p = (card {x. poly p x = 0} = Polynomial.degree p)"
step: fixes x :: "'a"   and y :: "'b"   and xm :: "'c \<rightharpoonup>\<^sub>f 'a"   and ym :: "'c \<rightharpoonup>\<^sub>f 'b"   and i :: "'c" assumes "R x y"   and "fmrel R xm ym"   and "fmdom xm = fmdom ym"   and "P xm ym" shows "P (xm(i \<mapsto>\<^sub>f x)) (ym(i \<mapsto>\<^sub>f y))"
card_singleton_set: fixes A :: "'c set set" assumes "\<forall>x\<in>A. card x = 1" shows "card (\<Union> A) = card A"
diff_fresh_vsubst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y \<noteq> y1" shows "fresh ys y (X #[y1 // y]_ys)"
assms_1: shows "x < e_length (block i t) \<and> x \<noteq> gap i t"
negligible_Un_eq: fixes s :: "'b set"   and t :: "'b set" shows "negligible (s \<union> t) = (negligible s \<and> negligible t)"
confs_Cons2: fixes P :: "'a prog"   and h :: "heap"   and xs :: "val list"   and y :: "ty"   and ys :: "ty list" shows "(P,h \<turnstile> xs [:\<le>] y # ys) = (\<exists>z zs. xs = z # zs \<and> P,h \<turnstile> z :\<le> y \<and> P,h \<turnstile> zs [:\<le>] ys)"
FSwImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b, 'c) model_scheme" shows "FSwImorph h hA termMOD MOD = termFSwImorph h hA MOD"
poly_pos_between_leq_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> 0 < poly p x) = (b < a \<or> p \<noteq> 0 \<and> 0 < poly p a \<and> count_roots_between p a b = 0)"
distinct_sorted_wrt_irrefl: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "irreflp rel"   and "transp rel"   and "sorted_wrt rel xs" shows "distinct xs"
path_components_of_subset_singleton: fixes X :: "'c topology"   and S :: "'c set" shows "(path_components_of X \<subseteq> {S}) = (path_connected_space X \<and> (topspace X = {} \<or> topspace X = S))"
simps: fixes s :: "nat"   and p :: "abc_inst list" shows "abc_fetch s p = (if s < length p then Some (p ! s) else None)"
m_comm: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "x \<oplus>\<^bsub>poly_ring R\<^esub> y = y \<oplus>\<^bsub>poly_ring R\<^esub> x"
select_convs_1: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getF \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = getF"
measure_lub_spmf: fixes Y :: "'c spmf set"   and A :: "'c set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "Sigma_Algebra.measure (measure_spmf (lub_spmf Y)) A = (\<Squnion>y\<in>Y. Sigma_Algebra.measure (measure_spmf y) A)"
fps_poly_swap1_def: fixes p :: "'b fps poly" shows "fps_poly_swap1 p = Abs_fps (\<lambda>m. Abs_poly (\<lambda>n. coeff p n $ m))"
3: shows "\<sigma>\<^sup>\<leftarrow>(A,\<sigma>(A,a)) = set_func (F \<^bsub>\<a>\<^esub> (Id A)) a"
node3: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))"
currying_axioms: shows "currying (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B)"
index_ordered_less: fixes p :: "nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "infinitely_many p"   and "i < j" shows "infinitely_many.index p i < infinitely_many.index p j"
notub: shows "OF_priority_match OF_match_fields_safe oft p \<noteq> Undefined"
Max_wrt_among_preorder: fixes A :: "'b set" shows "Max_wrt_among (map_relation (inv f) le) A = {x \<in> f ` carrier \<inter> A. \<forall>y\<in>f ` carrier \<inter> A. map_relation (inv f) le x y \<longrightarrow> map_relation (inv f) le y x}"
dual_inv_ball: fixes X :: "'b dual set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>X. P (\<partial>\<^sup>- x)) = (\<forall>y\<in>\<partial>\<^sup>- ` X. P y)"
lexs_lex_trans: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "lexs x y"   and "Counterclockwise_2D_Arbitrary.lex y z" shows "lexs x z"
adjoint_pair_antipar_2: fixes f :: "'a"   and g :: "'a" assumes "B.adjoint_pair f g" shows "B.ide g"
set_s_mtf: fixes n :: "nat" shows "set (s_mtf n) = set s0"
dimension_sum_space: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and m :: "nat"   and F :: "'a set"   and k :: "nat" assumes "subfield K R"   and "dimension n K E"   and "dimension m K F"   and "dimension k K (E \<inter> F)" shows "dimension (n + m - k) K (E <+> F)"
SD_pref_profileI: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and p :: "'b pmf"   and q :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "i \<in> agents"   and "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "\<And>x. x \<in> alts \<Longrightarrow> measure_pmf.prob q (preferred_alts (R i) x) \<le> measure_pmf.prob p (preferred_alts (R i) x)" shows "SD (R i) q p"
reachable_target_elim: fixes q :: "'c"   and A :: "('d, 'c) nba"   and p :: "'c" assumes "q \<in> NBA.reachable A p" obtains r :: "('d \<times> 'c) list" where "NBA.path A r p" and "q = target r p"
distinct_disjoint_shuffles: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "distinct xs"   and "distinct ys"   and "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "distinct zs"
simps_1: fixes xs :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and a :: "'f" shows "update xs opt (Val a) = (if xs = [] then case opt of None \<Rightarrow> Val a | Some e \<Rightarrow> e else Val a)"
intros_2: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f"   and s' :: "'f" assumes "g s = Skip s'"   and "s' \<in> terminates_on g" shows "s \<in> terminates_on g"
set_rel_witness_option: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option"   and a :: "'c"   and b :: "'d" assumes "rel_option A x y"   and "(a, b) \<in> set_option (rel_witness_option (x, y))" shows "A a b"
pmf_integral_code_unfold: fixes p :: "'b pmf" shows "pmf_integral p = measure_pmf.expectation p"
simps_3: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" shows "used_tmps (Tmp sop) = 0"
Infinitesimal_add_approx: fixes y :: "'b star"   and x :: "'b star"   and z :: "'b star" assumes "y \<in> Infinitesimal"   and "x + y = z" shows "x \<approx> z"
CD_on_Xd: fixes x :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "x \<in> Pdds'.CD_on ds A" shows "Xd x \<in> ds"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "C.ide (F' a)"
subspace_hyperplane: fixes a :: "'a" shows "subspace {x. a \<bullet> x = 0}"
bdia_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "relfdia x y = rel_ad (rel_ad (x ; y))"
pvalid_left_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and p :: "(symbol \<times> character list) list"   and x :: "item" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.pvalid_left \<NN> \<TT> \<RR> \<SS> Doc p x = (\<exists>u \<gamma>. LocalLexing.wellformed_tokens \<TT> p \<and> LocalLexing.wellformed_item \<RR> Doc x \<and> u \<le> length p \<and> charslength p = item_end x \<and> charslength (take u p) = item_origin x \<and> CFG.is_leftderivation \<NN> \<TT> \<RR> \<SS> (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> CFG.leftderives \<NN> \<TT> \<RR> (item_\<alpha> x) (terminals (drop u p)))"
emeasure_pair_measure_Times: fixes M :: "'c measure"   and A :: "'d set"   and N :: "'d measure"   and B :: "'c set" assumes "sigma_finite_measure M"   and "A \<in> sets N"   and "B \<in> sets M" shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
HT'setI: assumes "wf R"   and "\<And>f P c Q s\<^sub>0. \<lbrakk>HT'set_r (\<lambda>f' s'. (f' s', f s\<^sub>0) \<in> R) \<pi> \<Theta>; (f, P, c, Q) \<in> \<Theta>; P s\<^sub>0\<rbrakk> \<Longrightarrow> wp \<pi> (c s\<^sub>0) (Q s\<^sub>0) s\<^sub>0" shows "HT'set \<pi> \<Theta>"
extf_ext_hd_or_tl: fixes f :: "'s" shows "ext_hd_or_tl (extf f)"
intros_4: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and tf :: "tf" shows "\<S>\<bullet>\<C> \<turnstile> [Trap] : tf"
cases: fixes a1 :: "'v vertex"   and a2 :: "'v vertex" assumes "edg a1 a2" obtains x :: "'v" and y :: "'v" where "a1 = Vertex x__" and "a2 = Edge x__ y__" and "edge \<Delta> x__ y__"   | x :: "'v" and y :: "'v" where "a1 = Edge x y" and "a2 = Vertex y" and "edge \<Delta> x y"
LLq_pls: fixes t1 :: "'trm"   and t2 :: "'trm"   and z :: "'var" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "z \<in> var"   and "z \<notin> FvarsT t1"   and "z \<notin> FvarsT t2" shows "LLq t1 t2 = exi z (eql t2 (pls (Var z) t1))"
normalize_sort_eq_imp_sort_eqv: fixes s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "sort_ex cs s1"   and "sort_ex cs s2"   and "finite s1"   and "finite s2"   and "normalize_sort cs s1 = normalize_sort cs s2" shows "sort_eqv cs s1 s2"