hcju/isabelleps huggingface dataset:
HPT_source_target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "t \<in> TS"   and "source t \<in> States A"   and "A \<in> SAs (HA ST)" shows "target t \<in> States A"
commutes_with_unit: shows "\<eta> \<zero> = \<zero>'"
implements_oreach: fixes pi :: "'a \<Rightarrow> 'b"   and Sa :: "('c, 'b) spec"   and Sc :: "('d, 'a) spec" assumes "implements pi Sa Sc" shows "pi ` oreach Sc \<subseteq> oreach Sa"
not_fin_subst2: fixes M :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "\<not> fin M x" shows "\<not> fin (M{y:=<c>.P}) x"
finite_extension_is_subring: fixes K :: "'a set"   and xs :: "'a list" assumes "subring K R"   and "set xs \<subseteq> carrier R" shows "subring (finite_extension K xs) R"
locally_finite_onI: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set" assumes "\<And>p. p \<in> X \<Longrightarrow> \<exists>N. p \<in> N \<and> open N \<and> finite {i \<in> I. U i \<inter> N \<noteq> {}}" shows "locally_finite_on X I U"
step_1: shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"
stream_dec_Inr: fixes i :: "nat"   and FO :: "nat set"   and n :: "nat"   and s :: "('a \<times> bool list) stream" assumes "i \<notin> FO"   and "i < n" shows "\<exists>P. stream_dec n FO s ! i = Inr P"
surj: fixes f :: "'a \<Rightarrow> 'b" assumes "surjective f" shows "surj f"
effective_scalar_product_times: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and w1 :: "'b list"   and w2 :: "'b list"   and x :: "'b"   and y :: "'b" assumes "plus_mult id f zer g inver"   and "length w1 = length w2" shows "f (f x y) (plus_mult.scalar_product f zer g w1 w2) = plus_mult.scalar_product f zer g (mult.times f x w1) (mult.times f y w2)"
split_vec_dims_1: fixes i :: "nat"   and j :: "nat"   and x :: "(nat, 'a) mapping"   and a :: "'a vec"   and b :: "'a vec" assumes "split_i_j_x i j x = (a, b)" shows "dim_vec a = i"
iFROM_subset_conv: fixes n' :: "nat"   and n :: "nat" shows "([n'\<dots>] \<subseteq> [n\<dots>]) = (n \<le> n')"
negE: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "sb.eval cid t t' n (\<not>\<^sup>b \<gamma>)" shows "\<not> sb.eval cid t t' n \<gamma>"
effs_eq_abs_effs: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" shows "effect_of (abs_ast_operator (name, pres, effs, layer)) = map (\<lambda>(uu_, x, uu_, v). (x, v)) effs"
plus_current_def: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" shows "plus_current f g = (\<lambda>e. f e + g e)"
g'_def: shows "g' = monom_mult (lc a / lc g) (punit.lt (rep_list a) - punit.lt (rep_list g)) g"
lazy_channel_send'_def: shows "lazy_channel_send' = \<dagger>lazy_channel_send"
B4a: shows "\<^bold>\<forall>x xa. dom x \<cdot> xa \<cong> dom (dom x) \<cdot> xa"
semiring_simprules_4: shows "\<one> \<in> carrier R"
axioms_1: fixes X :: "'c set"   and is_open\<^sub>X :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>X :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and Y :: "'e set"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'f set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and d :: "'f"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'f \<Rightarrow> 'd" assumes "morphism_locally_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f" shows "morphism_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
norm_less_1_non_empty: shows "{\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> < 1} \<noteq> {}"
p0: shows "of_int (lead_coeff p) \<noteq> (0::'a)"
dist_fs_triangle_infinite1: fixes b :: "complex"   and c :: "complex" shows "1 / sqrt (1 + (cmod b)\<^sup>2) \<le> 1 / sqrt (1 + (cmod c)\<^sup>2) + cmod (b - c) / (sqrt (1 + (cmod b)\<^sup>2) * sqrt (1 + (cmod c)\<^sup>2))"
\<sigma>\<tau>z': shows "subst \<sigma>\<tau> (Var z') out = exi z (cnj (subst \<tau> (Var z) out) (subst (subst \<sigma> (Var z) inp) (Var z') out))"
gso_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "gs.gso i \<in> carrier_vec n"
pi_foldli: fixes l :: "'b list" shows "proper_it (foldli l) (foldli l)"
soldAllocationsEquivalence: fixes \<Omega> :: "'d set"   and N :: "integer set" shows "soldAllocations N \<Omega> = soldAllocations' N \<Omega> \<and> soldAllocations' N \<Omega> = soldAllocations'' N \<Omega>"
Lt_lt_LeI: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "Lt x < Le y"
squareE: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state"   and B :: "state \<times> state \<Rightarrow> bool" assumes "(s, t) \<Turnstile> [A]_v"   and "A (s, t) \<Longrightarrow> B (s, t)"   and "v t = v s \<Longrightarrow> B (s, t)" shows "B (s, t)"
quot_fm_def: fixes A :: "fm" shows "\<guillemotleft>A\<guillemotright> = quot_dbfm (trans_fm [] A)"
OFSC_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" shows "A B C D OFSC A' B' C' D' \<equiv> Bet A B C \<and> Bet A' B' C' \<and> Cong A B A' B' \<and> Cong B C B' C' \<and> Cong A D A' D' \<and> Cong B D B' D'"
enum_0_bot: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> t" shows "(a = t.enum 0) = (\<forall>a'\<in>t. a \<le> a')"
inverse_le_imp_le_neg: fixes a :: "'a"   and b :: "'a" assumes "inverse a \<le> inverse b"   and "b < (0::'a)" shows "b \<le> a"
max2_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "wo_rel r" shows "max2 r a b \<equiv> if (a, b) \<in> r then b else a"
hd_prog_append_Ghost\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and A :: "nat set"   and R :: "nat set"   and L :: "nat set"   and W :: "nat set" shows "hd_prog p (sb @ [Ghost\<^sub>s\<^sub>b A R L W]) = hd_prog p sb"
bigo_sum_main: fixes A :: "'a \<Rightarrow> 'b set"   and h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>x. \<forall>y\<in>A x. (0::'c) \<le> h x y"   and "\<exists>c. \<forall>x. \<forall>y\<in>A x. \<bar>f x y\<bar> \<le> c * h x y" shows "(\<lambda>x. sum (f x) (A x)) \<in> O(\<lambda>x. sum (h x) (A x))"
lnth_lappend_llist_of: fixes xs :: "'c list"   and ys :: "'c llist"   and n :: "nat" shows "lnth (lappend (llist_of xs) ys) n = (if n < length xs then xs ! n else lnth ys (n - length xs))"
A: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(\<lambda>_. self \<tau>.boss@pre \<doteq> null) \<tau> = (self.boss@pre \<doteq> null) \<tau>"
fin: fixes A :: "rat vec set" assumes "gs.fin_dim"   and "gs.lin_indpt A"   and "A \<subseteq> Rn" shows "finite A"
RestrictedP_LstSeqP_Eats: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and y :: "tm"   and z :: "tm" shows "{RestrictedP s (SUCC k) s', LstSeqP s k y} \<turnstile> LstSeqP (Eats s' (HPair (SUCC k) z)) (SUCC k) z"
k_steps_to_rtrancl: fixes k :: "nat"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "nat \<Rightarrow> ('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<forall>i<k. \<Gamma>\<turnstile> p i \<rightarrow> p (Suc i)" shows "\<Gamma>\<turnstile> p 0 \<rightarrow>\<^sup>* p k"
ltake_eq_ltake_antimono: fixes n :: "enat"   and xs :: "'a llist"   and ys :: "'a llist"   and m :: "enat" assumes "ltake n xs = ltake n ys"   and "m \<le> n" shows "ltake m xs = ltake m ys"
add_less_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
assms_5: fixes \<rho> :: "nat \<Rightarrow> val"   and x :: "val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "x \<in> space (stock_measure t)" shows "f \<rho> x = f' \<rho> x"
finite_dimensional_real_vector_space_pair_on_axioms: shows "finite_dimensional_real_vector_space_pair_on S1 S2 Basis1 Basis2"
not_in_complement_partition: fixes A :: "'a set" assumes "infinite UNIV" shows "(A \<notin> infinite_complement_partition) = (- A \<in> infinite_complement_partition)"
Cons_3: shows "\<S>\<bullet>\<C> \<turnstile> $$* vs : [] _> ts"
lift_constrains: fixes i :: "nat"   and F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and B :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "(lift i F \<in> lift_set i A co lift_set i B) = (F \<in> A co B)"
i_Exec_Stream_drop: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun input c \<Up> n = i_Exec_Comp_Stream trans_fun (input \<Up> n) (f_Exec_Comp trans_fun (input \<Down> n) c)"
assms_4: shows "\<forall>x. a < x \<and> x \<le> a' \<or> b' \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0"
nonneg_1: fixes i :: "'a" assumes "i \<in> I" shows "0 \<le> f i"
take_fin: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<infinity>" shows "t \<down> i \<in> A\<^sup>\<star>"
assms_2: shows "\<Gamma> (Var x) = \<Gamma> t"
monad_prob_writerT': assumes "monad_prob Monad_Overloading.return Monad_Overloading.bind sample" shows "monad_prob Monad_Overloading.return Monad_Overloading.bind sample"
eq: shows "(\<lambda>x. restrict x J) -` A \<inter> Pi\<^sub>E I S = (\<lambda>x. restrict x J) -` B \<inter> Pi\<^sub>E I S"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso (\<I>\<^sub>1 \<otimes>\<^sub>D a)"
in_degree_def: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" shows "in_degree G v \<equiv> card (in_arcs G v)"
Suc_innermost_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fw m n (Suc k) i j \<le> fw m n k i j"
rref_condition3_equiv: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'f" assumes "reduced_row_echelon_form A"   and "i < j"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row j A" shows "(LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))"
frees_app: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "frees (app u\<^sub>1 u\<^sub>2) = frees u\<^sub>1 |\<union>| frees u\<^sub>2"
deg_nzero_nzero: fixes p :: "nat \<Rightarrow> 'a" assumes "deg R p \<noteq> 0" shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
mset_set_set_mset_msubset: fixes A :: "'a multiset" shows "mset_set (set_mset A) \<subseteq># A"
circ_sup_9: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = (x\<^sup>\<circ> * y)\<^sup>\<circ> * x\<^sup>\<circ>"
run_one_step_label_return_imp_return: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and n :: "nat"   and les :: "e list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and res :: "v list" assumes "run_one_step d i (s, vs, ves, Label n les es) = (s', vs', RSReturn res)" shows "run_step d i (s, vs, es) = (s', vs', RSReturn res)"
distinct_3: fixes x1 :: "name"   and x3 :: "(Term_Class.term \<times> sterm) list" shows "Sconst x1 \<noteq> Sabs x3"
shadow_root_delete_get_3: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and object_ptr :: "(_) object_ptr" assumes "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h = Some h'"   and "object_ptr \<noteq> cast (cast shadow_root_ptr)" shows "get object_ptr h' = get object_ptr h"
measure_pmf_zero_iff: fixes p :: "'h pmf"   and s :: "'h set" shows "(measure_pmf.prob p s = 0) = (set_pmf p \<inter> s = {})"
is_parser_many: fixes P :: "char \<Rightarrow> bool" shows "is_parser (many P)"
less_aexp_trans: fixes x :: "'a aexp"   and y :: "'a aexp"   and z :: "'a aexp" assumes "x < y"   and "y < z" shows "x < z"
assms_4: fixes a' :: "'a"   and b' :: "'a" assumes "a' \<in> A"   and "b' \<in> B"   and "b .= a'"   and "a .= b'" shows "P"
11: shows "{} \<turnstile> local.P IMP Q\<^sub>1"
fd_map_split: fixes list :: "(cl \<times> char list) list" shows "map (case_fd (\<lambda>cl f. f)) (map (\<lambda>(x, y). fd_def x y) list) = map (\<lambda>(cl, f). f) list"
Mp_product_modulus: fixes m' :: "int"   and k :: "int"   and f :: "int poly" assumes "m' = p * k"   and "0 < k" shows "poly_mod.Mp m' f =m f"
dg_10_components_2: fixes \<aa> :: "V" shows "dg_10 \<aa>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
EH\<^sub>N_sum_2: shows "integrable (measure_pmf (H\<^sub>N n)) real"
fix_const: fixes c :: "'c" shows "(\<mu> x. c) = c"
R: shows "\<R>\<^sub>G \<alpha> \<beta> s\<^sub>1 = s\<^sub>1'"
partition_filter1: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "fst (partition P xs) = filter P xs"
Order: fixes G :: "('a, 'b) form list"   and G' :: "('a, 'b) form list" assumes "\<turnstile> G"   and "set G = set G'" shows "\<turnstile> G'"
as_needed_asses_submap_exec_iii: fixes a :: "('a, 'b) fmap \<times> 'c"   and s :: "('a, 'b) fmap" shows "fmdom' (action_needed_asses a s) = {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
category_with_binary_products_axioms: shows "category_with_binary_products (\<cdot>)"
branch_prob_altdef: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val" assumes "density_context V V' \<Gamma> \<delta>"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "branch_prob (V, V', \<Gamma>, \<delta>) \<rho> = \<integral>\<^sup>+ x. \<delta> (merge V V' (x, \<rho>)) \<partial>state_measure V \<Gamma>"
choice_iff': fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) = (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = a \<squnion> b" shows "b \<le> a"
cases: fixes a1 :: "'s1"   and a2 :: "'s1" assumes "a1 -\<tau>1\<rightarrow> a2" obtains s :: "'s1" and s' :: "'s1" and tl :: "'tl1" where "a1 = s" and "a2 = s'" and "s -1-tl\<rightarrow> s'" and "\<tau>move1 s tl s'"
swapidseq_inverse: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b" assumes "swapidseq n p" shows "swapidseq n (inv p)"
diagonal_to_Smith_i_PQ_JNF_dim: fixes P' :: "'b mat"   and m :: "nat"   and A' :: "'b mat"   and n :: "nat"   and Q' :: "'b mat"   and xs :: "nat list"   and i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" assumes "P' \<in> carrier_mat m m \<and> A' \<in> carrier_mat m n \<and> Q' \<in> carrier_mat n n"   and "diagonal_to_Smith_i_PQ_JNF xs i bezout (P', A', Q') = (P, A, Q)" shows "P \<in> carrier_mat m m \<and> A \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
del_list_sorted3: fixes xs :: "'b list"   and a :: "'b"   and ys :: "'b list"   and b :: "'b"   and zs :: "'b list"   and x :: "'b" assumes "sorted (xs @ a # ys @ b # zs)"   and "x < b" shows "del_list x (xs @ a # ys @ b # zs) = del_list x (xs @ a # ys) @ b # zs"
case_prodD: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f"   and b :: "'g" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
node_order_is_total: fixes e1 :: "'a event"   and i :: "nat"   and e2 :: "'a event" assumes "e1 \<in> set (history i)"   and "e2 \<in> set (history i)"   and "e1 \<noteq> e2" shows "e1 \<sqsubset>\<^sup>i e2 \<or> e2 \<sqsubset>\<^sup>i e1"
order_sym_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubseteq> x"   and "y \<sim> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
case_2: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
finprod_one_eqI: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "Group.comm_monoid G"   and "\<And>x. x \<in> A \<Longrightarrow> f x = \<one>\<^bsub>G\<^esub>" shows "finprod G f A = \<one>\<^bsub>G\<^esub>"
lookup_list_to_poly: fixes ts :: "'d list"   and cs :: "'e list" shows "lookup (list_to_poly ts cs) = list_to_fun ts cs"
aligned_mod_eq_0: fixes p :: "'b word"   and sz :: "nat" assumes "is_aligned p sz" shows "p mod 2 ^ sz = 0"
add_right_inj: fixes n :: "'a" shows "inj (\<lambda>x. x + n)"
default_def: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) mapping" shows "Mapping.default k v m = (if k \<in> Mapping.keys m then m else Mapping.update k v m)"
is_fixed_selector_def: fixes H :: "nat set \<times> nat set set"   and f :: "nat set \<Rightarrow> nat set \<times> nat set set" shows "is_fixed_selector H f = (\<forall>V. finite V \<and> card (uverts H) = card V \<longrightarrow> H \<simeq> f V \<and> uverts (f V) = V)"
irreflexive_axioms: shows "irreflexive (Collect derivable) (\<sqsubset>)"
is_Inf_uniq: fixes A :: "'b set"   and inf :: "'b"   and inf' :: "'b" assumes "is_Inf A inf"   and "is_Inf A inf'" shows "inf = inf'"
of_hypreal_diff: fixes x :: "real star"   and y :: "real star" shows "of_hypreal (x - y) = of_hypreal x - of_hypreal y"
simulation_silentst1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and \<mu>1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and \<mu>2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and s1' :: "'g" assumes "delay_bisimulation_measure trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 \<mu>1 \<mu>2"   and "bisim s1 s2"   and "(\<tau>trsys.silent_move trsys1 \<tau>move1)\<^sup>+\<^sup>+ s1 s1'" shows "bisim s1' s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>+\<^sup>+ s2 s2' \<and> bisim s1' s2')"
lookup_map_of: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "sorted1 (inorder t)" shows "lookup t x = AList_Upd_Del.map_of (inorder t) x"
power_swap: fixes \<alpha>0 :: "nat"   and r :: "nat" shows "(\<^bold>g [^] \<alpha>0) [^] r = (\<^bold>g [^] r) [^] \<alpha>0"
c_def: shows "c = (\<lambda>n. if N \<le> n then get_c a b B N (n - N) else undefined)"
1: shows "prv (all zz (imp (LLq (Var zz) (encPf prf)) (neg (PPf (Var zz) \<langle>neg \<phi>R\<rangle>))))"
l2product_rsum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i square_integrable S"   and "g square_integrable S" shows "l2product S g (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. l2product S g (f i))"
D: shows "set xs' \<subseteq> range p \<union> range q"
P_null_bot: fixes b :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "b = null \<or> b = \<bottom>" shows "\<not> (\<exists>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = b \<tau>)"
inverse_arrows_def: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "CC.inverse_arrows f g \<equiv> CC.ide (CC.comp g f) \<and> CC.ide (CC.comp f g)"
assms: shows "x \<in> rng_to_rng_of_frac ` S"
distinct_1: fixes x1 :: "nat"   and x21 :: "dB"   and x22 :: "dB" shows "Var x1 \<noteq> x21 \<degree> x22"
Expr_low_def: fixes e :: "Expr" shows "Expr_low e = (\<forall>s t \<beta>. s  \<approx>\<^sub>\<beta> t \<longrightarrow> (\<beta>, evalE e s, evalE e t) \<in> twiddleVal)"
P_0: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "P $ 0 = v0"
non_dom_eval: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom m" shows "m x = None"
eventually_eq_min: fixes f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "\<forall>\<^sub>F x in at_top. f x = g x" shows "\<forall>\<^sub>F x in at_top. min (f x) (g x) = f x"
mset_set_image_inj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "mset_set (f ` A) = image_mset f (mset_set A)"
sublist_EQNS_2: fixes h :: "'a"   and t :: "'a list" shows "subseq (h # t) [] = False"
bet2__out: fixes A :: "'p"   and B :: "'p"   and B' :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A \<noteq> B'"   and "Bet A B C"   and "Bet A B' C" shows "A Out B B'"
strip3_move3: fixes r :: "'b mrexp3"   and m :: "'b"   and c :: "bool" assumes "ok3 r" shows "strip3 (move3 m r c) = move m (strip3 r) c"
Bfun_prod_Zfun: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "Bfun f F"   and "Zfun g F" shows "Zfun (\<lambda>x. prod (f x) (g x)) F"