hcju/isabelleps huggingface dataset:
bconf_SFAss: fixes e2 :: "'a exp"   and P :: "'b prog"   and sh :: "sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and b :: "bool" assumes "val_of e2 = None" shows "(P,sh \<turnstile>\<^sub>b (C\<bullet>\<^sub>sF{D} := e2,b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e2,b) \<surd>)"
resultant_sub_def: fixes m :: "nat"   and n :: "nat"   and p :: "'a poly"   and q :: "'a poly" shows "resultant_sub m n p q = det (sylvester_mat_sub m n p q)"
abs_eq: fixes xb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'c \<times> 'c"   and x :: "'d \<times> 'd" shows "rel_uprod xb (abs_uprod xa) (abs_uprod x) = (case xa of (a, b) \<Rightarrow> \<lambda>(c, d). xb a c \<and> xb b d \<or> xb a d \<and> xb b c) x"
fmfilter_alt_defs_5: fixes B :: "'c fset" shows "fmrestrict_fset B = fmfilter (\<lambda>a. a |\<in>| B)"
ide_def: fixes a :: "'s" shows "ide a \<equiv> a \<cdot> a \<noteq> null \<and> (\<forall>f. (f \<cdot> a \<noteq> null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> null \<longrightarrow> a \<cdot> f = f))"
FtorCompId: fixes X :: "'a"   and F :: "('a, 'd, 'b, 'e, 'c, 'f, 'g) Functor_scheme"   and G :: "('d, 'h, 'e, 'i, 'f, 'j, 'k) Functor_scheme" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>"   and "F \<approx>>;;; G" shows "G ## F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod G\<^esub> (G @@ F @@ X) \<and> G @@ F @@ X \<in> obj\<^bsub>CatCod G\<^esub>"
has_parametrized_subprob_densityD_2: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and x :: "'c" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M" shows "subprob_space (N x)"
simps_6: fixes \<alpha> :: "('e, 'f + 'h, 'g) hp"   and \<phi> :: "('e, 'f + 'h, 'g) formula"   and \<sigma> :: "'h \<Rightarrow> ('e, 'f, 'g) formula" shows "PFsubst (\<langle> \<alpha> \<rangle> \<phi>) \<sigma> = (\<langle> PPsubst \<alpha> \<sigma> \<rangle> PFsubst \<phi> \<sigma>)"
inf2: fixes B :: "'d"   and k :: "'d"   and b :: "'d"   and a :: "'d" assumes "B \<equiv> k \<sqinter> b" shows "a \<sqinter> B \<equiv> k \<sqinter> (a \<sqinter> b)"
lprod_subset_elem: fixes as :: "'b list"   and bs :: "'b list"   and S :: "('b \<times> 'b) set"   and R :: "('b \<times> 'b) set" assumes "(as, bs) \<in> lprod S"   and "S \<subseteq> R" shows "(as, bs) \<in> lprod R"
list_emb_set: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "list_emb P xs ys"   and "x \<in> set xs" obtains y :: "'c" where "y \<in> set ys" and "P x y"
fds_nth_const: fixes c :: "'b" shows "fds_nth (fds_const c) = (\<lambda>n. if n = 1 then c else (0::'b))"
ivl_disj_un_22: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m<..u} = {l..u}"
simps_5: fixes V :: "'c set"   and uu :: "('d, 'c) Term.term"   and uv :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>uu == uv\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
join_left: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> y"
pf: fixes t :: "'a" assumes "t \<in> S - U" shows "pf t \<in> R \<and> pf t t0 = 0 \<and> 0 < pf t t"
\<alpha>_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "\<alpha> f \<equiv> if CCC.arr f then (fst f \<otimes> fst (snd f) \<otimes> snd (snd f)) \<cdot> \<a>[local.dom (fst f), local.dom (fst (snd f)), local.dom (snd (snd f))] else null"
T_measure1_less: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set" assumes "Prim_Invar1_loc w g r A" shows "Prim.T_measure1 g r A < card (nodes (Prim.rg g r))"
sep_disj_addD2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y + z"   and "y ## z" shows "x ## z"
TS_non_containment_causes_2: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<notin> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "set xs \<subseteq> inputs M2"   and "0 < i" shows "\<not> ((\<exists>xr j. xr \<noteq> xs \<and> prefix xr xs \<and> j \<le> i \<and> vs @ xr \<in> RM M2 M1 \<Omega> V m j) \<and> (\<exists>xc. xc \<noteq> xs \<and> prefix xc xs \<and> vs @ xc \<in> C M2 M1 \<Omega> V m i - RM M2 M1 \<Omega> V m i))"
less_le_mult_nat': fixes w :: "nat"   and c :: "nat"   and b :: "nat" assumes "w * c < b * c"   and "0 \<le> c" shows "Suc w * c \<le> b * c"
assoc'_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>' f g h = B.\<a>' f g h"
simps: fixes a1 :: "'c DBMEntry"   and a2 :: "'c DBMEntry" shows "(a1 \<prec> a2) = ((\<exists>uu_. a1 = Lt uu_ \<and> a2 = \<infinity>) \<or> (\<exists>uv_. a1 = Le uv_ \<and> a2 = \<infinity>) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Le b \<and> a < b) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Lt b \<and> a < b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Le b \<and> a \<le> b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Lt b \<and> a < b))"
matrix_right_invertible_surjective: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. A ** B = mat (1::'a)) = surj ((*v) A)"
prob0_bell_fst: fixes v :: "complex mat" assumes "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>" shows "prob0 2 v 0 = 1 / 2"
epi_implies_arr: fixes f :: "'s" assumes "epi f" shows "arr f"
has_sort_Tv: fixes sub :: "(String.literal \<times> String.literal) set"   and S :: "String.literal set"   and S' :: "String.literal set"   and tcs :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option"   and a :: "variable" assumes "sort_leq sub S S'" shows "has_sort (sub, tcs) (Tv a S) S'"
hyps: shows "edges (rem_unPath xs G) = edges G - (set xs \<union> set (rev_path xs))"
assms_3: shows "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"
r_u: shows "(tab(r \<mapsto> w, u \<mapsto> v, xs [\<mapsto>] ys)) vn = Some z"
Process_spec: fixes P :: "'a process" shows "Abs_process (F P, D P) = P"
l2product_radd: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S"   and "h square_integrable S" shows "l2product S f (\<lambda>x. g x + h x) = l2product S f g + l2product S f h"
uniform_limit_minus: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'd" assumes "uniform_limit X f l F"   and "uniform_limit X g m F" shows "uniform_limit X (\<lambda>a b. f a b - g a b) (\<lambda>a. l a - m a) F"
Domain_iff: fixes a :: "'e"   and r :: "('e \<times> 'f) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
suffixE: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" obtains zs :: "'a list" where "ys = zs @ xs"
strategyproofness_2: shows "(pmf (sds R31) c + pmf (sds R31) d < pmf (sds R45) b + pmf (sds R45) a \<or> pmf (sds R31) c + (pmf (sds R31) d + pmf (sds R31) b) < pmf (sds R45) b + (pmf (sds R45) a + pmf (sds R45) c)) \<or> pmf (sds R31) c + pmf (sds R31) d = pmf (sds R45) b + pmf (sds R45) a \<and> pmf (sds R31) c + (pmf (sds R31) d + pmf (sds R31) b) = pmf (sds R45) b + (pmf (sds R45) a + pmf (sds R45) c)"
poly_rel_def: fixes x :: "'i list"   and x' :: "'a poly" shows "poly_rel x x' = list_all2 R x (coeffs x')"
c_f_list_nth: fixes z :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "z \<le> y \<longrightarrow> c_nth (c_f_list f y x) (y - z) = f z x"
while_true_symeval: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and b :: "'a upred"   and P :: "'a hrel" assumes "\<Gamma> \<dagger> b = true" shows "\<Gamma> \<Turnstile> while\<^sup>\<top> b do P od = \<Gamma> \<Turnstile> P ;; while\<^sup>\<top> b do P od"
semAssignSC: fixes x :: "'id"   and e :: "'exp" assumes "AssignSideCondition x e" shows "e \<equiv>\<^bsub>DA x\<^esub> e"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl' newWl state in \<forall>c. c \<notin> set Wl' \<longrightarrow> getWatch1 state' c = getWatch1 state c \<and> getWatch2 state' c = getWatch2 state c"
equality_axioms_imp_Pf: fixes \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "\<alpha> \<in> equality_axioms" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
transfer: shows "(\<lambda>x y. Abs_atoms (Rep_atoms x \<union> Rep_atoms y)) = (\<squnion>)"
set_entails_clause_member: fixes C :: "'b literal set"   and S :: "'b literal set set" assumes "C \<in> S" shows "set_entails_clause S C"
finite1: shows "finite ((\<lambda>(y, d). Fun.swap x y d) ` {(y, d). y \<in> S \<and> y \<noteq> x \<and> d \<in> derangements (S - {x, y})})"
declclass_fieldm_simp: fixes n :: "vname"   and f :: "qtname \<times> Decl.field" shows "declclass (fieldm n f) = declclass f"
sum_weights_append_singleton: fixes xs :: "'lbl list"   and x :: "'lbl" shows "sum_weights (xs @ [x]) = sum_weights xs + x"
case_2: fixes f1 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "'b" shows "(case Geq x21 x22 of Leq x xa \<Rightarrow> f1 x xa | Geq x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
S_transition_Pred_iff: fixes P :: "'state"   and \<phi> :: "unit"   and P' :: "'state" shows "S_transform.S_transition P \<langle>S_action.Pred \<phi>,P'\<rangle> = (P' = P \<and> P \<turnstile>\<^sub>S \<phi>)"
meet_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "i1 \<sqinter> i2 = (if is_empty i1 \<or> is_empty i2 then Abs_Int2_ivl.empty else case (i1, i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (max_option False l1 l2) (min_option True h1 h2))"
that: shows "(A ===> A ===> (=)) (\<bullet>) (\<bullet>)"
domain_eq: shows "Domainp (pcr_set_dlist (=)) = (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None)"
A_3: shows "\<forall>x\<in>X. \<forall>y\<in>X. (isGreater (I' x) \<longrightarrow> J x y = J' x y) \<and> (isGreater (I' y) \<longrightarrow> J x y = J' x y)"
zero_order_4: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
id_ring_hom: fixes R :: "('e, 'f) ring_scheme" shows "id \<in> ring_hom R R"
fps_cos_def: fixes c :: "'a" shows "fps_cos c = Abs_fps (\<lambda>n. if even n then (- (1::'a)) ^ (n div 2) * c ^ n / of_nat (fact n) else (0::'a))"
intros_8: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c\<^sub>2 :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and c\<^sub>1 :: "com" assumes "\<not> bval b s"   and "\<pi>: (c\<^sub>2, s) \<Rightarrow> t" shows "\<pi>: (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
get_proc_return: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "get_proc (sourcenode a) = p"
c_def: shows "c = f \<circ> geodesic_segment_param G x"
gR_galois_imp21: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y"   and "z \<preceq>\<^sub>R rquot y x" shows "x \<oplus> z \<preceq>\<^sub>R y"
integral_cts_step_conv_imp_weak_conv: assumes "\<And>x y. x < y \<Longrightarrow> (\<lambda>n. integral\<^sup>L (M_seq n) (cts_step x y)) \<longlonglongrightarrow> integral\<^sup>L M (cts_step x y)" shows "weak_conv_m M_seq M"
valid_history_nth_update: fixes i :: "nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and sb :: "'a memref list"   and "is" :: "instr list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "history_consistent \<theta> (hd_prog p sb) sb" shows "valid_history program_step (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
evnodd_iff: fixes i :: "nat"   and j :: "nat"   and A :: "(nat \<times> nat) set"   and b :: "nat" shows "((i, j) \<in> evnodd A b) = ((i, j) \<in> A \<and> (i + j) mod 2 = b)"
knows2a: fixes A :: "specID"   and e1 :: "Expression list"   and e :: "Expression list" shows "knows A (e1 @ e) \<longrightarrow> knows A e"
MN_eq_nextl: fixes M :: "'b dfa" assumes "dfa M" shows "MyhillNerode (dfa.language M) (dfa.eq_nextl M)"
P1: shows "Bet A B A' \<and> D E F CongA C B A'"
c_2: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>sum_upto (\<lambda>x. real x powr - s) x - x powr (1 - s) / (1 - s)\<bar> \<le> c"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Convert of Convert \<Rightarrow> f1 | Reinterpret \<Rightarrow> f2) = f1"
induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>x y. P x y 0"   and "\<And>x y n. P (x div y) y n \<Longrightarrow> P x y (Suc n)" shows "P a0 a1 a2"
isometry_preserves_infdist: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b" assumes "isometry_on X f"   and "A \<subseteq> X"   and "x \<in> X" shows "infdist (f x) (f ` A) = infdist x A"
ceiling_eq_iff: fixes x :: "'a"   and a :: "int" shows "(\<lceil>x\<rceil> = a) = (of_int a - (1::'a) < x \<and> x \<le> of_int a)"
measure_closure: fixes S :: "'b set" assumes "bounded S"   and "negligible (frontier S)" shows "Sigma_Algebra.measure lebesgue (closure S) = Sigma_Algebra.measure lebesgue S"
diff_times_axiom_def: shows "diff_times_axiom \<equiv> Equals (Differential (Times (state_fun fid1) (state_fun fid2))) (Plus (Times (Differential (state_fun fid1)) (state_fun fid2)) (Times (state_fun fid1) (Differential (state_fun fid2))))"
rtranclp_reflclp_absorb: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R\<^sup>*\<^sup>*\<^sup>=\<^sup>= = R\<^sup>*\<^sup>*"
ide_char: fixes a :: "'a" shows "Left_a.ide a = (Left_a.arr a \<and> ide a)"
dual_Sup: fixes X :: "'a set" shows "\<Squnion>X ^ o = \<Sqinter>(dual ` X)"
union_mset_add_mset_left: fixes a :: "'d"   and A :: "'d multiset"   and B :: "'d multiset" shows "add_mset a A + B = add_mset a (A + B)"
min_max_for_sol: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "rat_LP A b c m n"   and "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 = b \<bullet> y"
tok_id_pubk_prik_def: fixes n :: "nat"   and m :: "nat" shows "tok_id_pubk_prik n m \<equiv> insert (Spy, \<langle>n, ID n (Sec_PubKey m)\<rangle>) s\<^sub>0"
path_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "path (reversepath g) = path g"
Compl_insert: fixes x :: "'d"   and A :: "'d set" shows "- insert x A = - A - {x}"
right_edge_explicit: shows "(\<lambda>y. twoC (1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
rA': shows "local.sub1 (A $$ (l, l)) (dim_row A - Suc l) l (r, A) = (r', A')"
inv_def: fixes f :: "'a" shows "A.inv f = (SOME g. A.inverse_arrows f g)"
prop_prop_nec_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[(\<^bold>\<forall>p. F \<^bold>\<noteq> (\<^bold>\<lambda>x. p)) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>p. F \<^bold>\<noteq> (\<^bold>\<lambda>x. p)) in v]"
case_2: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case flat_complete_lattice.Bot of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f2"
weak_hb_def: fixes m1 :: "'a \<times> ('a, 'b) operation"   and m2 :: "'a \<times> ('a, 'b) operation" shows "weak_hb m1 m2 \<equiv> hb m1 m2 \<or> m1 = m2"
ord_p_lt: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.ord_p ord_term_strict p q" shows "ord_term (ordered_term.lt term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term q)"
oalist_inv_raw_ConsD3: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and k :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list"   and k' :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp ((k, v) # xs)"   and "k' \<in> fst ` set xs" shows "lt_of_comp comp k k'"
sat: fixes I :: "'b set"   and v' :: "(nat, QDelta) mapping"   and ics :: "('b \<times> constraint) list" assumes "(I, \<langle>v'\<rangle>) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set (to_ns ics)" shows "(I, \<langle>from_ns v' (map snd (to_ns ics))\<rangle>) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics"
norm_pths_1: fixes x :: "'d"   and y :: "'d" shows "(x = y) = (norm (x - y) \<le> 0)"
assms_2: shows "list_all2 (\<approx>) z z'"
SD_pref_profileD_1: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents"   and "SD (R i) q p" shows "p \<in> lotteries_on alts"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray"   and i :: "nat"   and fi :: "int vec"   and mu_i :: "int iarray" shows "upd_fi_mu_state (f, mu, d) i fi mu_i = (update_i f fi, iarray_update mu i mu_i, d)"
trail_def: fixes u :: "'c"   and p :: "'d list"   and v :: "'c" shows "H.trail u p v \<equiv> H.awalk u p v \<and> distinct p"
space_measure_of_eq: fixes M :: "'c measure"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "space (measure_of (space M) (sets M) \<mu>) = space M"
distinct_2: fixes x21 :: "'a Trace"   and x22 :: "'a"   and x1 :: "'a" shows "x21 \<leadsto> x22 \<noteq> tInit x1"
test_sup_complement_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y"   and "\<forall>u v. test u \<and> test v \<longrightarrow> ! u \<sqinter> ! v \<sqinter> (u \<squnion> v) = bot" shows "test (x \<squnion> y)"
weak_partial_order_bottom_axioms: shows "weak_partial_order_bottom L"
has_field_derivative_def: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and F :: "'c filter" shows "(f has_field_derivative D) F = (f has_derivative (*) D) F"
Abs_def: fixes x2 :: "vrs"   and x3 :: "trm"   and x1 :: "ty" shows "\<lambda>x2:x1. x3 \<equiv> Abs_trm (Abs_Rep ([x2].Rep_trm x3) x1)"
Rep_Msg: fixes x :: "msg" shows "Rep_Msg x \<in> Msg"
lexp_head: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "order x y"   and "length xs = length ys" shows "lexp order (x # xs) (y # ys)"
resActRight: fixes x :: "name"   and \<alpha> :: "act"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "x \<sharp> \<alpha>"   and "Id \<subseteq> Rel" shows "\<alpha>.(\<lparr>\<nu>x\<rparr>P) \<leadsto>[Rel] \<lparr>\<nu>x\<rparr>\<alpha>.P"
rel_spmf_return_spmf2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b spmf"   and x :: "'c" shows "rel_spmf R p (return_spmf x) = (lossless_spmf p \<and> (\<forall>a\<in>set_spmf p. R a x))"
Sublists_def: fixes A :: "'b list set" shows "Sublists A = {w. \<exists>q\<in>A. sublist w q}"
zero_eq_add_iff_both_eq_0: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(zero = plus x y) = (x = zero \<and> y = zero)"
source_stability: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "wb_lens x" shows "\<exists>v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
image_insert: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and B :: "'c set" shows "f ` (a \<triangleright> B) = f a \<triangleright> f ` B"
circ_star: fixes x :: "'a" shows "x\<^sup>\<circ>\<^sup>\<star> = x\<^sup>\<circ>"
grid_dim_remove: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and d :: "nat"   and ds :: "nat set" assumes "p \<in> grid b ({d} \<union> ds)"   and "d < length b \<Longrightarrow> p ! d = b ! d" shows "p \<in> grid b ds"
Bell_eq: fixes n :: "nat"   and m :: "nat" shows "Bell (n + m) = (\<Sum>k\<le>n. \<Sum>j\<le>m. j ^ (n - k) * Stirling m j * (n choose k) * Bell k)"
min_r_def: shows "min_r \<equiv> Min (pre_ranks (\<lambda>q. state_rank q n) (w n) q)"
true_cls_mset_def: fixes I :: "'c set"   and CC :: "'c literal multiset multiset" shows "(I \<Turnstile>m CC) = (\<forall>C\<in>#CC. I \<Turnstile> C)"
nat_pow_eq': shows "([^]\<^bsub>G\<^esub>) = (\<lambda>x k. x ^ k mod n)"
cases: fixes a1 :: "comb"   and a2 :: "comb" assumes "a1 \<Rrightarrow>\<^sup>1 a2" obtains x :: "comb" where "a1 = x__" and "a2 = x__"   | x :: "comb" and y :: "comb" where "a1 = K \<bullet> xa__ \<bullet> y__" and "a2 = xa__"   | x :: "comb" and y :: "comb" and z :: "comb" where "a1 = S \<bullet> xb__ \<bullet> ya__ \<bullet> z__" and "a2 = xb__ \<bullet> z__ \<bullet> (ya__ \<bullet> z__)"   | x :: "comb" and y :: "comb" and z :: "comb" and w :: "comb" where "a1 = x \<bullet> z" and "a2 = y \<bullet> w" and "x \<Rrightarrow>\<^sup>1 y" and "z \<Rrightarrow>\<^sup>1 w"
leg1_in_hom_1: shows "\<guillemotleft>tab\<^sub>1 (r \<star> s) : src (tab\<^sub>0 (r \<star> s)) \<rightarrow> trg (r \<star> s)\<guillemotright>"
lprefix_lnull: fixes ys :: "'c llist"   and xs :: "'c llist" assumes "lnull ys" shows "lprefix xs ys = lnull xs"
sqn_update_same: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "sqn (rt(ip \<mapsto> v)) ip = \<pi>\<^sub>2 v"
wfP_accp_iff: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wfP r = (\<forall>x. Wellfounded.accp r x)"
assoc_comp: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set"   and W :: "'c set"   and x :: "'d" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "is_open U"   and "is_open V"   and "is_open W"   and "V \<subseteq> U"   and "W \<subseteq> V"   and "x \<in> \<FF> U" shows "\<rho> U W x = (\<rho> V W \<circ> \<rho> U V) x"
matrix_inv_1: fixes M :: "(('a, 'b) vec, 'c) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'a)"
compser_nsg: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "0 < n"   and "compseries G n f"   and "i \<le> n - 1" shows "\<natural>f i \<triangleright> f (Suc i)"