hcju/isabelleps huggingface dataset:
astG_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.astG problem \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> G"
io_target_implies_L: fixes q :: "'d"   and M :: "('e, 'f, 'd) FSM"   and io :: "('e \<times> 'f) list" assumes "q \<in> io_targets M (initial M) io" shows "io \<in> L M"
bind_fv2_unchanged: fixes tm :: "Core.term"   and lev :: "nat"   and v :: "variable \<times> typ" assumes "\<not> loose_bvar tm lev"   and "bind_fv2 v lev tm = tm" shows "v \<notin> fv tm"
R_def: fixes a :: "bool"   and b :: "bool" shows "R a b = b"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "c \<in> S" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
unitary_sound: fixes P :: "'c \<Rightarrow> real" assumes "unitary P" shows "sound P"
starset_finite: fixes A :: "'a set" assumes "finite A" shows "*s* A = star_of ` A"
le_refl: fixes L :: "('d, 'e) gorder_scheme"   and x :: "'d" assumes "weak_partial_order L"   and "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x"
dgrad_p_setD_lp: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "p \<in> punit.dgrad_p_set d m"   and "p \<noteq> 0" shows "d (punit.lp p) \<le> m"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>n insert x A = sup x (\<Squnion>\<^sub>f\<^sub>i\<^sub>n A)"
fmlookup_drop_fset: fixes A :: "'d fset"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmdrop_fset A m) x = (if x |\<notin>| A then fmlookup m x else None)"
is_empty_update: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "\<not> is_empty (update k v m)"
pow_generator_mod_int: fixes G :: "('d, 'e) cyclic_group_scheme"   and k :: "int" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> (k mod int (order G)) = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> k"
mask_int_def: fixes n :: "nat" shows "mask n = 2 ^ n - 1"
proj3_invalidate: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat" shows "\<pi>\<^sub>3 (the (invalidate rt dests dip)) = \<pi>\<^sub>3 (the (rt dip))"
f: shows "f \<in> L \<rightarrow>\<^sub>M prob_algebra N"
unl_lift4: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'b"   and x :: "'g \<Rightarrow> 'c"   and y :: "'g \<Rightarrow> 'd"   and z :: "'g \<Rightarrow> 'e"   and zz :: "'g \<Rightarrow> 'f"   and w :: "'g" shows "(f<x, y, z, zz>) w \<equiv> f (x w) (y w) (z w) (zz w)"
distinct_rotate: fixes n :: "nat"   and xs :: "'b list" shows "cycle (rotate n xs) = cycle xs"
restrict_shared_out': fixes a :: "nat"   and L :: "nat set"   and m :: "nat \<Rightarrow> bool option"   and writeable :: "bool"   and A :: "nat set" assumes "a \<notin> L"   and "m a = Some writeable" shows "(m \<ominus>\<^bsub>A\<^esub> L) a = Some (a \<in> A \<or> writeable)"
tpick_def: fixes \<sigma> :: "'tp"   and V :: "Preliminaries.var set" shows "tpick \<sigma> V \<equiv> pick (V \<inter> tpOfV -` {\<sigma>})"
normalize_rules_concat_map: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr list"   and rs :: "'b rule list" shows "normalize_rules f rs = concat (map (\<lambda>r. map (\<lambda>m. Rule m (get_action r)) (f (get_match r))) rs)"
prems_2: shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
simps: fixes \<Gamma> :: "'b formula set"   and a :: "'b formula" shows "(\<Gamma> \<turnstile>\<^sub>H a) = ((\<exists>F. a = F \<and> F \<in> \<Gamma>) \<or> (\<exists>F G. a = G \<and> (\<Gamma> \<turnstile>\<^sub>H F) \<and> (\<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G)))"
Units_one_side_I_2: fixes a :: "'a mat"   and p :: "'a mat" assumes "a \<in> carrier (ring_mat TYPE('a) n ())"   and "p \<in> Units (ring_mat TYPE('a) n ())"   and "a \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> p = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>" shows "a \<in> Units (ring_mat TYPE('a) n ())"
l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.to_tree_order Shadow_DOM.first_in_tree_order get_attribute get_attribute_locs Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_class_name Shadow_DOM.get_elements_by_tag_name ShadowRootClass.type_wf"
inv_can: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "local.inv (can u t) = can t u"
prj_in_hom_3: shows "\<guillemotleft>Prj\<^sub>0 : \<mu>\<nu>_\<pi>.chine \<rightarrow>\<^sub>C \<pi>.apex\<guillemotright>"
simps: fixes \<I> :: "('c, 'd) \<I>"   and a :: "('c, 'd) resource" shows "\<I> \<turnstile>res a \<surd> = (\<exists>res. a = res \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (run_resource res x) \<longrightarrow> xa \<in> responses_\<I> \<I> x \<and> \<I> \<turnstile>res xb \<surd>))"
Lp_1: fixes p :: "real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "1 \<le> p" shows "eNorm (\<LL> (ennreal p) M) f = (if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)) else \<infinity>)"
in_inter_fset: fixes S :: "'b fset"   and T :: "'b fset"   and x :: "'b" shows "(x |\<in>| S |\<inter>| T) = (x |\<in>| S \<and> x |\<in>| T)"
neg_diag_empty_spec: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "Beta_Regions' X v n not_in_X"   and "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
Q_init_Q_2: fixes G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and g :: "'a"   and G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and s :: "'b" assumes "Dijkstra_Impl G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>" shows "Q_lookup (Dijkstra_Impl_Defs.Q_initQ Q_empty Q_update s) = D.initQ s"
msb_conv_bin_sign: fixes x :: "int" shows "msb x = (bin_sign x = - 1)"
composition_invertible: fixes x :: "'a"   and y :: "'a" assumes "additive.invertible x"   and "additive.invertible y"   and "x \<in> R"   and "y \<in> R" shows "additive.invertible (x + y)"
PO_m1_inv4_rfresh_step5_lemmas: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat" shows "{m1_inv4_rfresh \<inter> m1_inv3r_init \<inter> m1_inv2r_serv \<inter> m1_inv1r_cache \<inter> m1_secrecy \<inter> m1_inv1_ifresh} m1_step5 Rb A B Kab Ts Ta {> m1_inv4_rfresh}"
Q_hit'_v0_v1: shows "v0 \<leadsto>Q_hit'\<leadsto> v1"
validn_elim: fixes x :: "name"   and B :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "validn ((x, B) # \<Gamma>)" shows "validn \<Gamma> \<and> x \<sharp> \<Gamma>"
Abrupt_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Abrupt s'" shows "t = Abrupt s'"
vrestr_eq: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T" shows "(a \<in> qmpt.vimage_restr M f A) = (a \<in> space M \<and> f a \<in> A \<inter> space M)"
self_in_downset: fixes x :: "'a" shows "x \<in> \<down>x"
equiv_up_to_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com" shows "(P \<Turnstile> c \<sim> c') = (\<forall>s s'. P s \<longrightarrow> (c, s) \<Rightarrow> s' = (c', s) \<Rightarrow> s')"
G_is_faithful: shows "faithful_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Pos x1 = Pos y1) = (x1 = y1)"
stc_unique: fixes y :: "complex star"   and x :: "complex star" assumes "y \<in> SComplex"   and "y \<approx> x" shows "stc x = y"
assms_5: shows "D.adjoint_pair (F f) (F g)"
join_shunt: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "(sup x y = top) = less_eq (uminus x) y"
env_restr_env_delete_other: fixes x :: "'c"   and S :: "'c set"   and m :: "'c \<Rightarrow> 'd" assumes "x \<notin> S" shows "env_delete x m f|` S = m f|` S"
weak_conv_m_subseq: fixes M_seq :: "nat \<Rightarrow> real measure"   and M :: "real measure"   and r :: "nat \<Rightarrow> nat" assumes "weak_conv_m M_seq M"   and "strict_mono r" shows "weak_conv_m (\<lambda>n. M_seq (r n)) M"
step_z_V: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"   and "Z \<subseteq> V" shows "Z' \<subseteq> V"
simps_4: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "thd (unfold_tllist p g1 g21 g22 a) = g21 a"
cond_hiding_entailsD: fixes t :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'c option"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'a"   and x :: "'b"   and y :: "'c" assumes "t hiding s under C entails R"   and "t k = Some x"   and "s k = Some y"   and "C x y" shows "R x y"
sub_K_implies_NRR: fixes NRO :: "msg"   and A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat"   and M :: "msg"   and NRR :: "msg"   and sub_K :: "msg"   and evs :: "event list" assumes "NRO = Crypt (priEK A) \<lbrace>Number f_nro, Agent B, Nonce L, Crypt K M\<rbrace>"   and "NRR = Crypt (priEK B) \<lbrace>Number f_nrr, Agent A, Nonce L, Crypt K M\<rbrace>"   and "sub_K \<in> parts (knows Spy evs)"   and "NRO \<in> parts (knows Spy evs)"   and "sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>"   and "A \<notin> bad"   and "evs \<in> zg" shows "Gets A \<lbrace>Number f_nrr, Agent A, Nonce L, NRR\<rbrace> \<in> set evs"
simps_7: fixes c :: "int"   and e :: "MIR.num" shows "\<beta> (Gt (CN 0 c e)) = [Neg e]"
ide_cod: fixes f :: "'s" assumes "arr f" shows "ide (cod f)"
nb\<^sub>1: shows "\<forall>C\<in>cnf ((\<Phi>\<^sub>G \<Pi>) t). clause_semantics \<A> C"
simps_27: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x21 :: "nat"   and x22 :: "'e"   and x23 :: "'e"   and x24 :: "'f"   and x25 :: "'f"   and x26 :: "'g" shows "(case Send x21 x22 x23 x24 x25 x26 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25 x26"
linear_scale: fixes f :: "('a \<Rightarrow> real) \<Rightarrow> real"   and b :: "'a \<Rightarrow> real"   and r :: "real" assumes "\<forall>x. x \<in> src.diff_fun_space \<longrightarrow> f x \<in> UNIV"   and "b \<in> src.diff_fun_space"   and "linear_on src.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
distinct_3: fixes x1 :: "'a"   and x3 :: "'a" shows "Terminates x1 \<noteq> Goes_wrong x3"
1: shows "\<exists>!h. \<guillemotleft>h : C.dom e' \<rightarrow> C.dom e\<guillemotright> \<and> D.cones_map h (D.mkCone e) = D.mkCone e'"
ntrp_split: fixes s :: "'n list"   and c1 :: "'n list multiset"   and c2 :: "'n list multiset"   and w :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c1 + c2), w, s', c') \<in> trcl (ntrp fg)"   and "valid fg ({#s#} + c1 + c2)" shows "\<exists>w1 w2 c1' c2'. w \<in> w1 \<otimes>\<^bsub>\<alpha>nl fg\<^esub> map ENV w2 \<and> c' = c1' + c2' \<and> ((s, c1), w1, s', c1') \<in> trcl (ntrp fg) \<and> (c2, w2, c2') \<in> trcl (ntr fg) \<and> mon_ww fg (map le_rem_s w1) \<inter> mon_c fg c2 = {} \<and> mon_ww fg w2 \<inter> mon_c fg ({#s#} + c1) = {}"
list_comb_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 abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f (xs @ ys) = term.list_comb app (term.list_comb app f xs) ys"
in_annotate_rlen: fixes a :: "nat"   and x :: "'b"   and l :: "'b list" assumes "(a, x) \<in> set (annotate_rlen l)" shows "x \<in> set l"
permute_fun_app_eq: fixes p :: "perm"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "p \<bullet> f x = (p \<bullet> f) (p \<bullet> x)"
induct: fixes x1 :: "(Vars.var \<times> exp) list"   and x2 :: "exp"   and x3 :: "Vars.var list"   and x4 :: "(Vars.var \<times> exp) list"   and x5 :: "exp"   and P :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Vars.var list \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 : x2 \<Down>\<^bsub>x3\<^esub> x4 : x5"   and "\<And>\<Gamma> x e L. P \<Gamma> (Lam [x]. e) L \<Gamma> (Lam [x]. e)"   and "\<And>y \<Gamma> e x L \<Delta> \<Theta> z e'. \<lbrakk>atom y \<sharp> (\<Gamma>, e, x, L, \<Delta>, \<Theta>, z); \<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'; P \<Gamma> e L \<Delta> (Lam [y]. e'); \<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z; P \<Delta> e'[y::=x] L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (App e x) L \<Theta> z"   and "\<And>\<Gamma> x e L \<Delta> z. \<lbrakk>map_of \<Gamma> x = Some e; delete x \<Gamma> : e \<Down>\<^bsub>x # L\<^esub> \<Delta> : z; P (delete x \<Gamma>) e (x # L) \<Delta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (Var x) L ((x, z) # \<Delta>) z"   and "\<And>\<Delta> \<Gamma> L body \<Theta> z. \<lbrakk>atom ` domA \<Delta> \<sharp>* (\<Gamma>, L); \<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z; P (\<Delta> @ \<Gamma>) body L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (Let \<Delta> body) L \<Theta> z"   and "\<And>\<Gamma> b L. P \<Gamma> (Bool b) L \<Gamma> (Bool b)"   and "\<And>\<Gamma> scrut L \<Delta> b e\<^sub>1 e\<^sub>2 \<Theta> z. \<lbrakk>\<Gamma> : scrut \<Down>\<^bsub>L\<^esub> \<Delta> : Bool b; P \<Gamma> scrut L \<Delta> (Bool b); \<Delta> : (if b then e\<^sub>1 else e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z; P \<Delta> (if b then e\<^sub>1 else e\<^sub>2) L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (scrut ? e\<^sub>1 : e\<^sub>2) L \<Theta> z" shows "P x1 x2 x3 x4 x5"
inject: fixes x1 :: "form multiset"   and x2 :: "form multiset"   and y1 :: "form multiset"   and y2 :: "form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
in_homE: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" obtains "\<phi>.B.arr f" and "\<phi>.B.dom f = a" and "\<phi>.B.cod f = b"
Min_ge_iff: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Min A) = (\<forall>a\<in>A. x \<le>s a)"
lincomb_replicate0_left: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and k :: "nat" assumes "RModule R smult M"   and "set ms \<subseteq> M" shows "scalar_mult.lincomb smult (replicate k (0::'a)) ms = (0::'b)"
swap: fixes a :: "atom"   and b :: "atom" assumes "sort_of a = sort_of b"   and "a \<noteq> b" shows "P (a \<rightleftharpoons> b)"
small_UN: fixes X :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "small X"   and "\<And>x. x \<in> X \<Longrightarrow> small (B x)" shows "small (\<Union> (B ` X))"
execute_preserves_valid: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "nat \<Rightarrow> nat option"   and name :: "char list" assumes "wf_ast_problem problem"   and "s \<in> ast_problem.valid_states problem"   and "ast_problem.enabled problem name s" shows "ast_problem.execute problem name s \<in> ast_problem.valid_states problem"
le_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
Suc: shows "fact n * fact n \<le> fact (2 * n)"
S_union: fixes i :: "'i" assumes "i \<in> I" shows "\<Union> (range (S i)) = space (M i)"
is_natural_2: fixes f :: "'c" assumes "C.arr f" shows "C (C.map (C.cod f)) (C.map f) = C.map f"
l_i_m: assumes "l \<le> i" shows "l < length fs"
elims_2: fixes x :: "'b list \<times> 'b list" assumes "is_empty x" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "fs = []"
erf_real_eq_iff: fixes a :: "real"   and b :: "real" shows "(erf a = erf b) = (a = b)"
assms: shows "\<forall>x\<in>OUT S. \<forall>Z\<in>AbstrLevel i. x \<notin> IN Z"
intros_8: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "\<langle>X, Y\<rangle> \<in> synth H"
iwlsFSw_iterAll_termFSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Iteration.model" assumes "iwlsFSw MOD" shows "termFSwImorph (iter MOD) (iterAbs MOD) MOD"
equivp_gen_cong: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "equivp (gen_cong R)"
coset_assoc: fixes x :: "'a"   and y :: "'a"   and H :: "'a set" assumes "x \<in> carrier H'"   and "y \<in> carrier H'"   and "H \<subseteq> carrier H'" shows "x <#\<^bsub>H'\<^esub> (H #>\<^bsub>H'\<^esub> y) = x <#\<^bsub>H'\<^esub> H #>\<^bsub>H'\<^esub> y"
intros: fixes Q :: "'a \<Rightarrow> 'b tree option"   and \<delta> :: "('a, 'b) ta_rule set" assumes "\<forall>q t. Q q = Some t \<longrightarrow> (\<exists>r\<in>\<delta>. rhsq r = [] \<and> q = lhs r \<and> t = NODE (rhsl r) [])"   and "\<forall>r\<in>\<delta>. rhsq r = [] \<longrightarrow> Q (lhs r) \<noteq> None" shows "Q \<in> brw_iq \<delta>"
n_inter_num_choose_design: fixes b1 :: "'a set"   and b2 :: "'a set"   and n :: "nat" assumes "b1 \<in># \<B>"   and "b2 \<in># \<B>" shows "(b1 |\<inter>|\<^sub>n b2) = int (card (b1 \<inter> b2) choose n)"
map\<^sub>0_simp: fixes a :: "'a" assumes "B.obj a" shows "map\<^sub>0 a = G.map\<^sub>0 (F.map\<^sub>0 a)"
group_axioms_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a" shows "group_axioms f z inverse \<equiv> (\<forall>a. f z a = a) \<and> (\<forall>a. f (inverse a) a = z)"
logical_instr_sub1_privilege: fixes s' :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (logical_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
frac_le: fixes t :: "real" assumes "t \<in> {a..b}" shows "g' t / g t \<le> K"
ck_unifier_def: fixes t :: "'c trm"   and s :: "'c trm"   and \<sigma> :: "('c \<times> 'c trm) list"   and type :: "inferences" shows "ck_unifier t s \<sigma> type = (if type = FirstOrder then MGU \<sigma> t s else Unifier \<sigma> t s)"
map_bset_eq_bempty_iff: fixes f :: "'e \<Rightarrow> 'c"   and X :: "'e set['d]" shows "(map_bset f X = bempty) = (X = bempty)"
left_commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ac_operator f" shows "f a (f b c) = f b (f a c)"
induct: fixes P :: "int \<Rightarrow> Cooper.num \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "Cooper.num" assumes "\<And>i j. P i (C j)"   and "\<And>i n c t. P i t \<Longrightarrow> P i (CN n c t)"   and "\<And>i v. P i (Bound v)"   and "\<And>i v. P i (Neg v)"   and "\<And>i v va. P i (Add v va)"   and "\<And>i v va. P i (Sub v va)"   and "\<And>i v va. P i (Mul v va)" shows "P a0 a1"
rw: shows "\<bar>1 / 4 - x_coord x * x_coord x\<bar> = 1 / 4 - x_coord x * x_coord x"
Gramian_determinant_2: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list"   and k :: "nat" assumes "LLL.LLL_invariant_weak n m fs_init fs"   and "k \<le> m" shows "0 < gram_schmidt.Gramian_determinant n fs k"
INF_of_enat_infty_iff1: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set" shows "((INF x\<in>A. enat (f x)) = \<infinity>) = (A = {})"
post_meas1_bell00_fst: shows "post_meas1 2 |\<beta>\<^sub>0\<^sub>0\<rangle> 0 = |unit_vec 4 3\<rangle>"
finite_UNIV: shows "finite_UNIV = Phantom('c) (finite UNIV)"
inject: fixes x1 :: "meth_sig"   and x2 :: "meth_body"   and y1 :: "meth_sig"   and y2 :: "meth_body" shows "(meth_def_def x1 x2 = meth_def_def y1 y2) = (x1 = y1 \<and> x2 = y2)"
unitary_gen_unitary: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary_gen M = (\<exists>k M'. 0 < k \<and> unitary M' \<and> M = cor k *\<^sub>s\<^sub>m eye *\<^sub>m\<^sub>m M')"
g_orbit_eq: fixes X :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'c set" shows "\<gamma> X G U = {X t |t. t \<in> U \<and> (\<forall>\<tau>\<in>down U t. G (X \<tau>))}"
simps: fixes xs :: "real_alg_2_list" shows "real_alg_2_list_convert xs = (if real_alg_2_list_nil xs then [] else real_alg_2_list_hd xs # real_alg_2_list_convert (real_alg_2_list_tl xs))"
combine_infinity_invariant_6: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_strong_release (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"