hcju/isabelleps huggingface dataset:
bij_betw_apply: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set"   and B :: "'ah set"   and a :: "'ag" assumes "bij_betw f A B"   and "a \<in> A" shows "f a \<in> B"
Phase1or2Read_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "Phase1or2Read s s' p d q = (Phase1or2ReadThen s s' p d q \<or> Phase1or2ReadElse s s' p d q)"
injectivity_solver_CollectE: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool"   and a' :: "'d"   and P' :: "'d \<Rightarrow> bool" assumes "a \<in> {x. P x} \<and> a' \<in> {x. P' x}" obtains "P a \<and> P' a'"
sublens_implies_sublens': fixes Y :: "'e \<Longrightarrow> 'f"   and X :: "'g \<Longrightarrow> 'f" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "X \<subseteq>\<^sub>L' Y"
in_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "prefix xs ys"   and "x \<in> set xs" shows "x \<in> set ys"
T2: shows "[\<lambda>w. \<forall>x. (G x m\<rightarrow> G ess x) w]"
asympartpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "asympartp r x y" obtains "r x y" and "\<not> r y x"
healthy_wp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "healthy (wp a)" shows "healthy (wp (repeat n a))"
CartProd: fixes x :: "ZF"   and A :: "ZF"   and B :: "ZF" shows "x |\<in>| A |\<times>| B = (\<exists>a b. a |\<in>| A \<and> b |\<in>| B \<and> x = Opair a b)"
\<delta>_states_complete: fixes q :: "'Q option" assumes "q \<in> \<delta>_states \<delta>complete" shows "q \<in> Qcomplete"
basis_row_space_iarrays_def: fixes A :: "'b iarray iarray" shows "basis_row_space_iarrays A = (let GJ = Gauss_Jordan_iarrays A; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of GJ)) in set (map (\<lambda>i. row_iarray i GJ) [0..<rank_A]))"
connected_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" shows "Graph.connected c u v \<equiv> \<exists>p. Graph.isPath c u p v"
tendsto_iff: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" assumes "class.metric_space dist uniformity open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
merge_guards_simps_9: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "merge_guards (Guard f g c) = (let c' = merge_guards c in if is_Guard c' then let (f', g', c'') = dest_Guard c' in if f = f' then Guard f (g \<inter> g') c'' else Guard f g (Guard f' g' c'') else Guard f g c')"
bet_suma__per: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C A B C D E F" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C"
axioms_5: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element" shows "l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
elims_2: fixes x :: "'d"   and xa :: "'e"   and xb :: "'d set list"   and xc :: "('d, 'e) ta_rule" assumes "r_matchc x xa xb xc" obtains q :: "'d" and l :: "'e" and Qs :: "'d set list" and qr :: "'d" and lr :: "'e" and qsr :: "'d list" where "x = q" and "xa = l" and "xb = Qs" and "xc = qr \<rightarrow> lr qsr" and "q = qr \<and> l = lr \<and> r_match Qs qsr"
FWdelay_bisimulation_obs_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_obs final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) (flip bisim_wait) \<tau>move2 \<tau>move1"
r_coset_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})"
ldropn_all: fixes xs :: "'b llist"   and m :: "nat" assumes "llength xs \<le> enat m" shows "ldropn m xs = LNil"
cpc: shows "cp TYPE('y \<Rightarrow> 'a) TYPE('x) TYPE('y)"
simps_1: fixes \<Gamma> :: "char list set"   and v :: "char list"   and e :: "expr" shows "wfs \<Gamma> (Assign v e) = (v \<in> \<Gamma> \<and> wfe \<Gamma> e)"
select_convs_5: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_proto \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_proto"
uminus_vec_def: fixes v :: "'b vec" shows "- v \<equiv> vec (dim_vec v) (\<lambda>i. - v $ i)"
oInv_inverse: fixes x :: "ordinal" shows "oInv F (F x) = x"
simps_1: fixes x :: "'b" shows "is_a_variable (Var x) = True"
mset_set_eq_iff: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(mset_set A = mset_set B) = (A = B)"
FaultProp: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and f :: "'c" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> Fault f"
simps: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and id1 :: "nat"   and id2 :: "nat"   and s1 :: "('i, 'j, 'k, 'l) SLin_state_scheme"   and s2 :: "('m, 'n, 'o, 'p) SLin_state_scheme" assumes "Idempotence \<delta> \<gamma> bot id1 id2" shows "P13 (s1, s2) = (finite (initVals s2) \<and> finite (abortVals s1) \<and> finite (abortVals s2))"
set_cases_1: fixes e :: "'a"   and a :: "('a, 'b) message" assumes "e \<in> set1_message a" obtains z2 :: "bool" where "a = <e, inM z2__>"   | z2 :: "'b" where "a = <e, z2>"
eq: fixes t :: "'a \<times> nat" assumes "t \<in> set (dense_repr p)" shows "g [fst t] \<otimes>\<^bsub>A\<^esub> g X [^]\<^bsub>A\<^esub> snd t = h [fst t] \<otimes>\<^bsub>A\<^esub> h X [^]\<^bsub>A\<^esub> snd t"
is_natural_1: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi> (C.VV.dom f) = \<Phi> f"
in_homI: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
iIN_iFROM_iTILL_conv: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] = [n\<dots>] \<inter> [\<dots>n + d]"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "Bossy''.dX X d = {x \<in> X. Xd x = d}"
fgw2: fixes w :: "'b" shows "g w \<le> f w"
vangle_inverse: fixes v\<^sub>1 :: "'b"   and v\<^sub>2 :: "'b" shows "vangle (- v\<^sub>1) v\<^sub>2 = pi - vangle v\<^sub>1 v\<^sub>2"
ras_shrinkI: fixes s :: "'c array \<times> nat"   and a :: "'d list"   and R :: "('c \<times> 'd) set" assumes "(s, a) \<in> \<langle>R\<rangle>ras_rel" shows "(ras_shrink s, a) \<in> \<langle>R\<rangle>ras_rel"
ringI: fixes R :: "('e, 'f) ring_scheme" assumes "abelian_group R"   and "Group.monoid R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y" shows "ring R"
Edges_rev: fixes zs :: "nat list" shows "Edges (rev zs) = {(b, a). (a, b) \<in> Edges zs}"
splitter_pslice: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and k :: "'b" shows "splitter \<pi> k = pmap_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> {v. \<exists>v'. map the v' = v \<and> wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v' \<and> k \<in> strategy v'}) \<pi>"
induct: fixes P :: "'b list \<Rightarrow> 'b trie \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b trie" assumes "\<And>b m. P [] (Nd b m)"   and "\<And>x xs b m. (\<And>xa. xa = (case m x of None \<Rightarrow> Trie_Fun.empty | Some t \<Rightarrow> t) \<Longrightarrow> P xs xa) \<Longrightarrow> P (x # xs) (Nd b m)" shows "P a0 a1"
even_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_parity plus minus zero times one divide modulo uminus" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (minus a b) = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b)"
simps_1: fixes p :: "perm" shows "p \<bullet> None = None"
interval_lebesgue_integral_of_real: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> real" shows "complex_interval_lebesgue_integral M a b (\<lambda>x. complex_of_real (f x)) = complex_of_real (interval_lebesgue_integral M a b f)"
right_comp_S: fixes x :: "'b"   and y :: "'b"   and S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" assumes "(x, y) \<in> S O (S O S\<^sup>* O NS\<^sup>* \<union> NS\<^sup>*)" shows "(x, y) \<in> S O S\<^sup>* O NS\<^sup>*"
del_inorder: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t"   and "bal t"   and "sorted_less (inorder t)" shows "inorder (del k x t) = del_list x (inorder t)"
rell_F_eq_onp: fixes P1 :: "'n \<Rightarrow> bool"   and P2 :: "'o \<Rightarrow> bool"   and P3 :: "'p \<Rightarrow> bool" shows "rell_F (eq_onp P1) (eq_onp P2) (eq_onp P3) = eq_onp (\<lambda>x. (\<forall>z\<in>set1_F x. P1 z) \<and> (\<forall>z\<in>set2_F x. P2 z) \<and> (\<forall>z\<in>set3_F x. P3 z))"
c_tr_append: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'g"   and xs :: "'e list"   and ys :: "'e list" shows "c_tr step out s (xs @ ys) = c_tr step out s xs @ c_tr step out (foldl step s xs) ys"
degree_pderiv: fixes p :: "'a poly" shows "degree (pderiv p) = degree p - 1"
assms_5: fixes i :: "nat" assumes "i < nr"   and "f i < nc" shows "A $$ (i, f i) = (1::'a)"
rel_eq_transfer: shows "rel_fun pcr_real (rel_fun pcr_real (=)) realrel (=)"
suffix_def: fixes s :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "s |\<^sub>s i \<equiv> \<lambda>n. s (n + i)"
mat_abelian_group: fixes n :: "nat"   and b :: "'d" shows "abelian_group (mat_ring n b)"
prefixes_monotone: fixes t :: "nat"   and j :: "nat"   and b :: "nat"   and d :: "nat"   and b' :: "nat" assumes "local.prefixes t j \<down>= b"   and "local.prefixes (t + d) j \<down>= b'" shows "e_length b \<le> e_length b'"
K_\<Lambda>_GDD_axioms: shows "K_\<Lambda>_GDD \<V> \<B> {\<k>} \<G> 1"
rsp_foldE: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and u :: "'b"   and v :: "'b" assumes "rsp_fold f" obtains "f u \<circ> f v = f v \<circ> f u"
atomic_LtK: fixes X :: "ltkey" shows "atomic (LtK X)"
pair_completeD: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "pair_complete A r"   and "x \<in> A"   and "y \<in> A" shows "Ex (extreme_bound A r {x, y})"
UPSpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "UPSpec ch mm rs p \<equiv> Init PInit rs p \<and> \<box>[UNext ch mm rs p]_(rtrner ch!p, rs!p) \<and> WF(RNext ch mm rs p)_(rtrner ch!p, rs!p) \<and> WF(MemReturn ch rs p)_(rtrner ch!p, rs!p)"
AA_I: fixes X :: "'a set" assumes "X \<subseteq> A"   and "f ` X \<subseteq> X"   and "\<And>Y s. \<lbrakk>Y \<subseteq> X; extreme_bound A (\<sqsubseteq>) Y s\<rbrakk> \<Longrightarrow> s \<in> X" shows "X \<in> AA"
env_delete_env_restr_swap: fixes x :: "'d"   and S :: "'d set"   and e :: "'d \<Rightarrow> 'e" shows "env_delete x (e f|` S) = env_delete x e f|` S"
Units_r_inv: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "x \<otimes> inv x = \<one>"
wf_rexp_of_alt: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of_alt n \<phi>)"
cis_arg_power: fixes x :: "complex"   and n :: "nat" assumes "x \<noteq> 0" shows "cis (arg (x ^ n)) = cis (arg x * real n)"
fps_const_to_fls: fixes c :: "'a" shows "fps_to_fls (fps_const c) = fls_const c"
proves_strip_all_single: 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 \<Gamma> :: "Core.term set"   and B :: "Core.term"   and \<tau> :: "typ"   and t :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> B"   and "strip_all_single_var B = Some \<tau>"   and "typ_of t = Some \<tau>"   and "term_ok \<Theta> t" shows "\<Theta>,\<Gamma> \<turnstile> subst_bv t (strip_all_single_body B)"
circ_transitive_equal: fixes x :: "'a" shows "idempotent (x\<^sup>\<star>)"
freshTauTransition: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
cInf_eq_minimum: fixes z :: "'b multiset"   and X :: "'b multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "\<Sqinter> X = z"
cos_eq_arccos_Ex: fixes x :: "real"   and y :: "real" shows "(cos x = y) = (- 1 \<le> y \<and> y \<le> 1 \<and> (\<exists>xa. x = arccos y + 2 * real_of_int xa * pi \<or> x = - arccos y + 2 * real_of_int xa * pi))"
co01: shows "compact (sphere (0::'a) 1 \<inter> affine hull S)"
rows_to_keep_hyp: shows "\<forall>y. y \<in> set (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))) \<longrightarrow> y < length subsets"
divide_poly_list: fixes f :: "'b poly"   and g :: "'b poly" shows "f div g = divide_poly_list f g"
simps_1: fixes G :: "'b graph" shows "generate_valid_topology_SOME [] G = G"
is_methodI: fixes C :: "qtname"   and m :: "mdecl" shows "is_method (C, mdecl m)"
gf: fixes x :: "'a" assumes "x \<in> box a b" shows "g x = f x"
fac_pref_suf: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>f v" shows "\<exists>p. p \<le>p v \<and> u \<le>s p"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and x :: "'b" shows "ipurge_tr_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then ipurge_tr_aux I D U xs else ipurge_tr_aux I D U xs @ [x])"
scount_infinite: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "alw (ev P) \<omega>" shows "scount P \<omega> = \<infinity>"
WTrt_binop_simps_14: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>BinXor\<guillemotright>T2 : T) = (T1 = Boolean \<and> T2 = Boolean \<and> T = Boolean \<or> T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
strips_equivalent_to_sas_plus_i_a_V: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "\<not> (are_all_operators_applicable_in s ops \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent ops)" shows "\<not> (are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) (map (sasp_op_to_strips \<Psi>) ops) \<and> STRIPS_Semantics.are_all_operator_effects_consistent (map (sasp_op_to_strips \<Psi>) ops))"
simps_1: fixes uu :: "('e \<times> 'e) set"   and uv :: "'f \<Rightarrow> 'e"   and U :: "'e set" shows "sinks_aux uu uv U [] = U"
represents_irr_non_0: fixes p :: "int poly"   and x :: "'b" assumes "irreducible p"   and "p represents x"   and "x \<noteq> (0::'b)" shows "poly p 0 \<noteq> 0"
Pd_range: shows "\<forall>d. Field (Pd d) \<subseteq> {x. Xd x = d}"
H1: shows "geodesic_segment_between H (geodesic_segment_param G p D) x"
adds_funI: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" assumes "s \<le> t" shows "s adds t"
lessalive_antisym: fixes s :: "Store"   and t :: "Store" assumes "s \<lless> t"   and "t \<lless> s" shows "s = t"
gcd_mute_fun: shows "mute_fun gcd_mute_fun"
memberid_fdecl_simp1: fixes f :: "fdecl" shows "memberid (fdecl f) = fid (fst f)"
Euclidean_dist_upper: fixes i :: "'a"   and x :: "'a"   and y :: "'a" assumes "i \<in> Basis" shows "dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
pmf_prob_code_alt_finite: fixes A :: "'b set"   and p :: "'b pmf" assumes "finite A" shows "pmf_prob p A = sum (pmf p) A"
directed_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "directed A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. \<exists>z\<in>A. r x z \<and> r y z"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int" assumes "\<And>d i. (\<not> d < i \<Longrightarrow> P d (i + 1)) \<Longrightarrow> P d i" shows "P a0 a1"
ideal_subgroup_generated: fixes \<aa> :: "'a set"   and \<bb> :: "'a set" assumes "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>" shows "ideal (ideal_gen_by_prod \<aa> \<bb>) R (+) (\<cdot>) \<zero> \<one>"
funct_OT_12_def: fixes input\<^sub>1 :: "'d \<times> 'd"   and \<sigma> :: "bool" shows "funct_OT_12 input\<^sub>1 \<sigma> = return_spmf ((), if \<sigma> then snd input\<^sub>1 else fst input\<^sub>1)"
kmp_impl_correct': shows "(uncurry kmp_impl, uncurry kmp_SPEC) \<in> (arl_assn id_assn)\<^sup>k *\<^sub>a (arl_assn id_assn)\<^sup>k \<rightarrow>\<^sub>a option_assn nat_assn"
3: fixes x :: "'a" assumes "x \<in> A" shows "ennreal e \<le> f x"
nonneg_dirichlet_series_power: fixes f :: "'a fds"   and n :: "nat" assumes "nonneg_dirichlet_series f" shows "nonneg_dirichlet_series (f ^ n)"
le_term_order_rel': shows "(\<le>) = (\<lambda>x y. x = y \<or> term_order_rel' x y)"
nonzero_minus_divide_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "uminus (divide a b) = divide a (uminus b)"
hrelation_def: fixes r :: "hf" shows "hrelation r = (\<forall>z. z \<^bold>\<in> r \<longrightarrow> is_hpair z)"
wf_trm_Var: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and x :: "'c" assumes "intruder_model arity Ana" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (Var x)"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_CHAOS_refine_F: fixes A :: "'a set" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F CHAOS A"
has_mode\<^sub>A: fixes x :: "'Var\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and m :: "Mode" assumes "var\<^sub>C_of x \<in> mds\<^sub>C m" shows "x \<in> mds\<^sub>A_of mds\<^sub>C m"
sets_lebesgue_on_refl: fixes S :: "'b set" shows "S \<in> sets (lebesgue_on S)"
a: shows "R[x\<turnstile>n>y] = ImpL <c>.M w.N u"
singleton_ordering: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "\<exists>f. TernaryOrdering.ordering f ord {a}"
group_left_neutral: fixes a :: "'b" shows "(0::'b) + a = a"
pcr_cr_eq: shows "pcr_CoCalls = cr_CoCalls"
mult_not_zero: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
strict_implies_order: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
eq_\<I>_generat_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and D :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and D' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and \<I>' :: "('e, 'f) \<I>" assumes "A \<le> A'"   and "D \<le> D'"   and "\<I> \<le> \<I>'" shows "eq_\<I>_generat A \<I> D \<le> eq_\<I>_generat A' \<I>' D'"
neg: shows "\<nexists>K. openin (top_of_set S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
chg_map_def: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'b"   and m :: "'b \<Rightarrow> 'c option" shows "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a \<mapsto> f b))"
intensional_rews_1: fixes c :: "'b"   and w :: "'c" shows "(#c) w \<equiv> c"
irc_on_consistency_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "irc_on A f"   and "finite A" shows "consistency_on A f"
Bij_def: fixes S :: "'d set" shows "Bij S = extensional S \<inter> {f. bij_betw f S S}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>'"