hcju/isabelleps huggingface dataset:
induct: fixes P1 :: "'a FOL_Fitting.term \<Rightarrow> bool"   and P2 :: "'a FOL_Fitting.term list \<Rightarrow> bool"   and term_list :: "'a FOL_Fitting.term list" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 term_list"
sep_conj_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P ** Q \<equiv> \<lambda>s. \<exists>p q. p \<oplus> q = Some s \<and> P p \<and> Q q"
g_g': fixes z :: "complex" shows "(g has_field_derivative h z * g z) (at z)"
\<tau>instr_stk_drop_exec_move: 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 ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and fr' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and vs :: "'addr val list" assumes "exec_move ci P t e h (stk, loc, pc, None) ta h' fr'" shows "\<tau>instr (compP2 P) h (stk @ vs) (compE2 e ! pc) = \<tau>instr (compP2 P) h stk (compE2 e ! pc)"
co_out12c: shows "connected (outside (path_image c1 \<union> path_image c2))"
suminf_minus: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "(\<Sum>n. - f n) = - (\<Sum>n. f n)"
ex: shows "\<forall>x\<in>X. x \<in> {y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r}"
bound_intersect_2d_lower: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and m :: "real"   and M :: "real" assumes "bound_intersect_2d prec Xs x = Some (m, M)" obtains X :: "(real \<times> real) \<times> real \<times> real" and M' :: "real" where "X \<in> set Xs" and "intersect_segment_xline prec X x = Some (m, M')" and "\<And>X m' M'. \<lbrakk>X \<in> set Xs; intersect_segment_xline prec X x = Some (m', M')\<rbrakk> \<Longrightarrow> m \<le> m'"
circ_plus_one: fixes x :: "'a" shows "x\<^sup>\<circ> = (1::'a) \<squnion> x\<^sup>\<circ>"
igSwapClsSTR_imp_igSwapCls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igSwapClsSTR MOD" shows "igSwapCls MOD"
tab_simps_5: shows "cod f.tab = f \<star> tab\<^sub>0 f"
le_less: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x < y \<or> x \<approx> y)"
stable_pair_on_CD_on_CH: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set" assumes "stable_pair_on ds XD_XH" shows "CD_on ds (fst XD_XH) = CH (snd XD_XH)"
norm_stack_append: fixes xs :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and ys :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "norm_stack (xs @ ys) = norm_stack xs @ norm_stack ys"
eq: shows "matrix_change_of_basis Z X = matrix_change_of_basis Y X"
stab_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "stab s i j = (case inst.tab (inst s ! i) of None \<Rightarrow> None | Some k \<Rightarrow> stab_s s k j)"
collect_pseudominimals_below_in_poset: fixes P :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b set"   and A :: "'b set"   and a :: "'a set" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a" shows "the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a)) \<in> P"
scaleRat_left_distrib: fixes a :: "rat"   and b :: "rat"   and x :: "'a" shows "(a + b) *R x = a *R x + b *R x"
simps_3: fixes p :: "real poly"   and q :: "real poly"   and r :: "real poly"   and ps :: "real poly list"   and x :: "real" shows "split_sign_changes (p # q # r # ps) x = (if poly p x \<noteq> 0 \<and> poly q x = 0 then [p, q, r] # split_sign_changes (r # ps) x else [p, q] # split_sign_changes (q # r # ps) x)"
evn_def: shows "evn = IF (\<lambda>s. s<Arg> = 0) THEN Loc Res:==(\<lambda>s. 0) ELSE (Loc Res:=CALL Odd(\<lambda>s. s<Arg> - 1);; Loc Arg:=CALL Odd(\<lambda>s. s<Arg> - 1);; Loc Res:==(\<lambda>s. s<Res> * s<Arg>))"
trans: fixes b :: "'k multiset"   and a :: "'k multiset"   and c :: "'k multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
in_M_3: shows "\<Union> (A ` F) - A x \<in> M"
append_eq_append_conv: fixes xs :: "'g list"   and ys :: "'g list"   and us :: "'g list"   and vs :: "'g list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
that: fixes i :: "nat" assumes "i < num_params p" shows "xs i = ys ! i"
simps_1: fixes n :: "nat" shows "V (TNat n) = VNat n"
insert_noteq_member: fixes b :: "'k"   and B :: "'k multiset"   and c :: "'k"   and C :: "'k multiset" assumes "add_mset b B = add_mset c C"   and "b \<noteq> c" shows "c \<in># B"
marked_insertions_def: fixes m :: "'a" shows "mut_m.marked_insertions m = (\<lambda>s. \<forall>x. x \<in> set (sys_mem_store_buffers (mutator m) s) \<longrightarrow> marked_insertion x s)"
NotInQFrozenQReachability: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and Q :: "'a set"   and cfg2 :: "('a, 'c, 'b) configuration"   and p :: "'a" assumes "asynchronousSystem.qReachable trans sends cfg1 Q cfg2"   and "p \<notin> Q" shows "states cfg1 p = states cfg2 p"
assms_1: shows "\<nexists>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> \<I>\<^sub>\<tau> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>"
shift_var_set_def: fixes V :: "nat set" shows "shift_var_set V = insert 0 (Suc ` V)"
intro: fixes G :: "('a, 'b) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x" shows "Group.monoid G"
tgeq0: shows "0 \<le> trace (A - f 0)"
simps_38: fixes P :: "'c prog"   and uu :: "char list"   and v :: "char list"   and va :: "char list list"   and vb :: "bool"   and vc :: "'d exp" shows "icheck P uu (INIT v (va,vb) \<leftarrow> vc) = False"
disj_assoc: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> bool" shows "((P \<or> Q) \<or> S) = (P \<or> Q \<or> S)"
halfspace_eq_empty_lt: fixes a :: "'b"   and b :: "real" shows "({x. a \<bullet> x < b} = {}) = (a = (0::'b) \<and> b \<le> 0)"
int_encode_eq: fixes x :: "int"   and y :: "int" shows "(int_encode x = int_encode y) = (x = y)"
"case": fixes f :: "real \<Rightarrow> real \<Rightarrow> 'a"   and x1 :: "real"   and x2 :: "real" shows "(case Complex x1 x2 of Complex x xa \<Rightarrow> f x xa) = f x1 x2"
intros_14: fixes \<chi> :: "'a ltln"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<chi> \<turnstile>\<^sub>s \<phi>"   and "\<chi> \<turnstile>\<^sub>s \<psi>" shows "\<chi> \<turnstile>\<^sub>s \<phi> R\<^sub>n \<psi>"
mult_right: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)" shows "(\<lambda>x. f x * h x) \<in> \<omega>[F](\<lambda>x. g x * h x)"
swapAbs_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs set" shows "A $[x \<and> y]_xs = asAbs (pick A $[[x \<and> y]]_xs)"
cBii: fixes i :: "nat" assumes "i < n" shows "csqrt (B $$ (i, i)) * cnj (csqrt (B $$ (i, i))) = B $$ (i, i)"
not_arr_null: shows "\<not> A_BxA.arr A_BxA.null"
vbrelationE1: fixes r :: "V"   and x :: "V" assumes "vbrelation r"   and "x \<in>\<^sub>\<circ> r" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>"
atMost_eq_iff: fixes x :: "'c"   and y :: "'c" shows "({..x} = {..y}) = (x = y)"
homeomorphism_imp_closed_map: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and U :: "'a set" assumes "homeomorphism S T f g"   and "closedin (top_of_set S) U" shows "closedin (top_of_set T) (f ` U)"
content_Pair: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b" shows "content (cbox (a, c) (b, d)) = content (cbox a b) * content (cbox c d)"
linear_split_req2: fixes xs :: "('b \<times> 'c) list"   and p :: "'c"   and ls :: "('b \<times> 'c) list"   and sub :: "'b"   and sep :: "'c"   and rs :: "('b \<times> 'c) list" assumes "linear_split xs p = (ls @ [(sub, sep)], rs)"   and "sorted_less (separators xs)" shows "sep < p"
prems_1: fixes l :: "'b list"   and l' :: "'b list" assumes "ls = l @ l'"   and "map f l = x # xs"   and "map f l' = fl'" shows "thesis"
RECT_eq_REC: fixes body :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'d" assumes "REC\<^sub>T body x \<noteq> top" shows "REC\<^sub>T body x = REC body x"
b_assn_subtyping_match_tL: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'a \<Rightarrow> bool" assumes "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"   and "vassn_tag (hn_ctxt A x y) \<Longrightarrow> P' x" shows "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn A' P') x y"
independent_span_bound: fixes T :: "('a, 'b) vec set"   and S :: "('a, 'b) vec set" assumes "finite T"   and "vec.independent S"   and "S \<subseteq> vec.span T" shows "finite S \<and> card S \<le> card T"
sets_T: fixes s :: "'b cfg" shows "MC.T.events s = sets (stream_space (count_space UNIV))"
select_convs_4: fixes Obj :: "'h set"   and Mor :: "'g set"   and Dom :: "'g \<Rightarrow> 'h"   and Cod :: "'g \<Rightarrow> 'h"   and Id :: "'h \<Rightarrow> 'g"   and Comp :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and more :: "'i" shows "Cod \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Cod"
simps_75: fixes x51 :: "'c"   and x52 :: "('b, 'a, 'c) expr" shows "LID\<^sub>V (Lambda x51 x52) = LID\<^sub>E x52"
Theap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "Theap \<Gamma> e = (\<Lambda> a. if nonrec \<Gamma> then ccTTree (edom (Aheap \<Gamma> e\<cdot>a)) (\<G>\<^bsub>a\<^esub> e) else ttree_restr (edom (Aheap \<Gamma> e\<cdot>a)) anything)"
frontier_disjoint_eq: fixes S :: "'b set" shows "(frontier S \<inter> S = {}) = open S"
DIAndCut12Intro_def: shows "DIAndCut12Intro = (([[$\<alpha> vid1]]Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2)) \<rightarrow> ([[$\<alpha> vid1]]Pc pid2) \<rightarrow> [[$\<alpha> vid1]]Pc pid1 && Pc pid2)"
mult_mono': fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d"   and "(0::'a) \<le> a"   and "(0::'a) \<le> c" shows "a * c \<le> b * d"
sub_complement: fixes x :: "'b"   and y :: "'b" shows "! (d x) = ! (d (! (d (! (d x))) \<squnion> ! (d y))) \<squnion> ! (d (! (d (! (d x))) \<squnion> ! (d (! (d y)))))"
rep_eq: fixes x :: "'c" shows "(*\<^sub>V) (cblinfun_cinner_right x) = (\<bullet>\<^sub>C) x"
eqvt_at_def: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "eqvt_at f x \<equiv> \<forall>p. p \<bullet> f x = f (p \<bullet> x)"
diff_fun_spaceD: fixes f :: "'b \<Rightarrow> real" assumes "f \<in> dest.diff_fun_space" shows "diff_fun k charts2 f"
T_eval: fixes x :: "dB"   and f :: "dB" shows "\<T> \<degree> x \<degree> f \<rightarrow>\<^sub>\<beta>\<^sup>* f \<degree> x"
A: shows "g_max m P = iterate_sel_no_map (reverse_iterateoi m) P"
is_nth_power_prime_power_nat_iff: fixes p :: "nat"   and n :: "nat"   and k :: "nat" assumes "prime p" shows "is_nth_power n (p ^ k) = (n dvd k)"
Skip_comp_absorb: shows "(action.relation_of Skip ;; action.relation_of Skip) = action.relation_of Skip"
cut_ge_not_empty_iff: fixes I :: "'b set"   and t :: "'b" shows "(I \<down>\<ge> t \<noteq> {}) = (\<exists>x\<in>I. t \<le> x)"
wsumSS_2: shows "(\<Sum>t\<in>S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
IH: shows "{l \<in> lists A. length l = n} \<lesssim> A"
primitiveD: fixes f :: "'a poly"   and x :: "'a" assumes "primitive f"   and "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y" shows "x dvd (1::'a)"
coalesce_eq_OK1_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [\<squnion>\<^bsub>f\<^esub>] ys) = OK zs" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
ediff_def: shows "ediff \<equiv> \<lambda>D j i. enum (D j) (Suc i) - enum (D j) i"
s: fixes g' :: "'a"   and g :: "'a" assumes "g' \<in> set (succs g)"   and "P g" shows "P g'"
l_new_element_get_shadow_root_axioms_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_shadow_root_axioms get_shadow_root get_shadow_root_locs \<equiv> (\<forall>ptr' new_element_ptr h h' r. ptr' \<noteq> new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_element_ptr h'. h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_shadow_root new_element_ptr \<rightarrow>\<^sub>r None)"
ccSup_least: fixes A :: "'b set"   and z :: "'b" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "Sup A \<le> z"
continuous_on_blinfunI1: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'd" assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on X (\<lambda>x. blinfun_apply (f x) i)" shows "continuous_on X f"
invertible_left_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "additive.invertible x"   and "x \<in> R"   and "y \<in> R"   and "z \<in> R" shows "(x + y = x + z) = (y = z)"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym a1 a2 = ((\<exists>t s. a1 = t \<and> a2 = s \<and> tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s) \<and> \<not> ground_heads.comp_hd gt_sym ground_heads_var (head t) (head s) \<and> num_args t = 1 \<and> (\<exists>f\<in>ground_heads.ground_heads ground_heads_var (head t). arity_sym f = 1 \<and> wt_sym f = 0) \<and> (tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym (arg t) s \<or> arg t = s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s) \<and> ground_heads.gt_hd gt_sym ground_heads_var (head t) (head s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s) \<and> head t = head s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym) (args t) (args s))))"
zero_less_hnorm_iff: fixes x :: "'a star" shows "(0 < hnorm x) = (x \<noteq> 0)"
openin_delete: fixes u :: "'a set"   and s :: "'a set"   and a :: "'a" assumes "openin (top_of_set u) s" shows "openin (top_of_set u) (s - {a})"
borel_measurable_euclidean_space: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
naturality: fixes f :: "'a" assumes "A.arr f" shows "B (\<sigma> (A.cod f)) (F f) = B (G f) (\<sigma> (A.dom f))"
d2_def: fixes v :: "'a"   and v' :: "'a" shows "d2 v v' = (\<lambda>x. frechet_derivative (\<lambda>x. frechet_derivative f (at x) v') (at x) v)"
trans_MapOps_lookup_def: shows "trans_MapOps_lookup \<equiv> \<lambda>m ((cec, aec), obs). Option.bind (lookup_trie m (map fst (toList cec))) (\<lambda>m. Option.bind (lookup_trie m (map snd (toList cec))) (\<lambda>m. Option.bind (lookup_trie m (map fst (toList aec))) (\<lambda>m. Option.bind (lookup_trie m (map snd (toList aec))) (\<lambda>m. Mapping.lookup m obs))))"
Inter_insert: fixes a :: "'a set"   and B :: "'a set set" shows "\<Inter> (insert a B) = a \<inter> \<Inter> B"
zero_eq_power2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a 2 = zero) = (a = zero)"
encode_interfering_operator_pair_exclusion_def: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and op\<^sub>1 :: "'b strips_operator"   and op\<^sub>2 :: "'b strips_operator" shows "encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2 \<equiv> let ops = \<Pi>\<^sub>\<O> in \<^bold>\<not> (Atom (Operator k (index ops op\<^sub>1))) \<^bold>\<or> \<^bold>\<not> (Atom (Operator k (index ops op\<^sub>2)))"
MN_eq_app_right: fixes L :: "'b list set" assumes "finite (UNIV // eq_app_right L)" shows "MyhillNerode L (eq_app_right L)"
converse_converse: fixes r :: "('c \<times> 'd) set" shows "\<smile> (\<smile> r) = r"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs" shows "l_new_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
gkernTr1: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> gker\<^bsub>F,G\<^esub> f" shows "x \<in> carrier F"
p_in_prime_factorsE: fixes p :: "nat"   and n :: "nat" assumes "p \<in># prime_factorization n"   and "0 < n" obtains "2 \<le> p" and "p \<le> n" and "p dvd n" and "prime p"
sig_red_rtrancl_subset: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q"   and "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)" shows "(sig_red sing_reg top_tail {f \<in> F. sing_reg (lt f) (lt p)})\<^sup>*\<^sup>* p q"
under_mono: fixes r :: "('a \<times> 'a) set"   and i :: "'a"   and j :: "'a" assumes "Well_order r"   and "(i, j) \<in> r" shows "under r i \<subseteq> under r j"
set_ys: shows "set (sorted_list_of_set J) = J"
take_eq_Nil: fixes n :: "nat"   and xs :: "'d list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
fact_bounds_2: fixes n :: "nat" assumes "0 < n" shows "fact n \<le> sqrt (2 * pi * real n) * (real n / exp 1) ^ n * exp (1 / (12 * real n))"
cases: fixes a :: "'b list"   and C :: "'b list set"   and b :: "'b list" assumes "a \<in> no_head_gen C b" obtains u :: "'b list" where "a = ua__" and "ua__ \<in> C" and "ua__ \<noteq> b"   | u :: "'b list" where "a = u \<cdot> b" and "u \<in> no_head_gen C b"
term_ok_mk_eqI: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> s"   and "term_ok \<Theta> t"   and "typ_of s = typ_of t" shows "term_ok \<Theta> (mk_eq s t)"
termMOD_igWlsAllDisj: shows "igWlsAllDisj termMOD"
C\<^sub>0_def: fixes c :: "'c" shows "C\<^sub>0 c \<equiv> monomial c 0"
atLeastAtMost_eq_cball: fixes a :: "real"   and b :: "real" shows "{a..b} = cball ((a + b) / 2) ((b - a) / 2)"
Pareto_strictI': fixes x :: "'alt pmf"   and y :: "'alt pmf"   and i :: "'agent" assumes "\<And>i. i \<in> agents \<Longrightarrow> (SD \<circ> R) i x y"   and "i \<in> agents"   and "\<not> (SD \<circ> R) i y x" shows "x \<prec>[Pareto (SD \<circ> R)] y"
step_t_z: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l, (Z \<inter> {u. u \<turnstile> inv_of A l})\<^sup>\<up> \<inter> {u. u \<turnstile> inv_of A l}\<rangle>"
sublist_prefix: fixes h :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" assumes "subseq (h # l1) l2" shows "\<exists>l2a l2b. l2 = l2a @ [h] @ l2b \<and> \<not> ListMem h l2a"
circuit_def: fixes X :: "'a set" shows "\<E>.circuit X = (X \<subseteq> \<E> \<and> \<not> indep_in \<E> X \<and> (\<forall>x\<in>X. indep_in \<E> (X - {x})))"
assms_3: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call p (Modif \<sigma>),{}"
\<delta>_3: shows "subst_domain \<delta> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
GSMP_disjoint_def: fixes P1 :: "('fun, 'var) Term.term set"   and P2 :: "('fun, 'var) Term.term set"   and Secrets :: "('fun, 'var) Term.term set" shows "GSMP_disjoint P1 P2 Secrets \<equiv> GSMP P1 \<inter> GSMP P2 \<subseteq> Secrets \<union> {m. {} \<turnstile>\<^sub>c m}"