hcju/isabelleps huggingface dataset:
map_scale_assoc: fixes c :: "'b"   and d :: "'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "c \<cdot> d \<cdot> t = (c * d) \<cdot> t"
comp_def: fixes g :: "'a"   and f :: "'a" shows "L.comp g f = (if left a f \<and> left a g \<and> cod f = local.dom g then g \<cdot> f else null)"
af_G_nested_propos: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "nested_propos (af_G_letter \<phi> \<nu>) \<subseteq> nested_propos \<phi>"
Sup_insert: fixes a :: "'a"   and A :: "'a set" shows "Sup (insert a A) = sup a (Sup A)"
image_mset_eq_implies_permutes: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "finite A"   and "image_mset f (mset_set A) = image_mset f' (mset_set A)" obtains p :: "'a \<Rightarrow> 'a" where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
\<phi>_in_terms_of_\<Phi>': fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = (\<Phi>\<Psi>.\<psi>D (y, G x) \<circ> \<Phi>\<Psi>.\<Phi>.FUN (y, x) \<circ> inC) f"
gcd_dvd2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) b"
wUntilE: fixes \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "(\<gamma>' \<WW>\<^sub>c \<gamma>) t n" shows "(\<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')) \<or> (\<forall>n'\<ge>n. \<gamma>' t n')"
y: shows "spmf (trace_callee run_resource p xs x) y \<noteq> spmf (trace_callee run_resource q xs x) y"
\<C>_def: shows "\<C>\<^sub>C \<equiv> \<Union> (range \<C>_vars\<^sub>C)"
eventuallyI: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<And>x. P x" shows "eventually P F"
gext_new: fixes s :: "st"   and r :: "oref"   and G :: "prog"   and oi :: "obj_tag" assumes "globs s r = None" shows "s\<le>|init_obj G oi r s"
s_def: shows "s = lcs (lp p) (lp q) - lp p"
isl_map_sum: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and s :: "'f + 'g" shows "isl (map_sum f g s) = isl s"
evals_length: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "length es = length es'"
swap_cs_Send_RecvMarker: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
extract_weights_Conv: fixes shared :: "bool"   and A :: "real mat"   and m :: "real mat convnet" shows "extract_weights shared (Conv A m) = (\<lambda>x. if x < dim_row A * dim_col A then flatten_matrix A x else extract_weights shared m (x - dim_row A * dim_col A))"
gfp_dual: shows "\<partial> \<circ> gfp = lfp \<circ> \<partial>\<^sub>F"
hash_oracle''_def: shows "hash_oracle'' = (\<lambda>(s, \<sigma>) x. (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>)) \<bind> (\<lambda>(h, \<sigma>'). return_spmf (h, insert x s, \<sigma>')))"
1: shows "\<exists>f. f c * c = x * c + y * c + z * c"
spec_ops_def: fixes ops :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" shows "spec_ops ops deps \<equiv> sorted (map fst ops) \<and> distinct (map fst ops) \<and> (\<forall>oid oper ref. (oid, oper) \<in> set ops \<and> ref \<in> deps oper \<longrightarrow> ref < oid)"
l_delete_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<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_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
b: shows "aux = f k (lookup_pair ko (sort_oalist_aux ko xs) k) (lookup_pair ko (sort_oalist_aux ko ys) k)"
card_pderivs_lang_UNIV1_le_awidth: fixes r :: "'a rexp" shows "card (pderivs_lang UNIV1 r) \<le> awidth r"
succ_in_omega: fixes n :: "V" assumes "n \<in> elts \<omega>" shows "ZFC_in_HOL.succ n \<in> elts \<omega>"
bij_betw_finite: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "finite A = finite B"
tape_of_list_def: shows "tape_of \<equiv> tape_of_nat_list"
reduced_S_list_min_gallery: fixes ss :: "'a permutation list" assumes "ss \<noteq> []"   and "reduced_word (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) ss" shows "min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))"
hyps_2: shows "\<forall>e f g. e,f,g,G \<Turnstile> Exists a"
simps_4: fixes uu :: "'b list"   and zs :: "'b list" shows "l_sort_inv_1 (uu, [], zs) = l_sorted zs"
representation_ne_zero: fixes basis :: "'b set"   and v :: "'b"   and b :: "'b" assumes "vs1.representation basis v b \<noteq> (0::'a)" shows "b \<in> basis"
M'_def: shows "M' \<equiv> \<lambda>i' j'. if i' = i \<and> j' = 0 then Le d else if i' = 0 \<and> j' = i then Le (- d) else M i' j'"
dim_subset_UNIV: fixes S :: "'a set" shows "dim S \<le> DIM('a)"
cring_simprules_22: shows "- (0::'a) = (0::'a)"
a_group_hom: shows "group_hom (add_monoid G) (add_monoid H) h"
run_one_step_basic_relop_i_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x291 :: "t"   and x292 :: "relop_i"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Relop_i x291 x292) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
cong: fixes scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "module.dependent scale \<equiv> module.dependent scale"
A_authenticates_B_r: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and A :: "agent"   and Ta :: "nat"   and authTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
bij_betw_conv_Ex1: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" shows "bij_betw f A B = ((\<forall>y\<in>B. \<exists>!x. x \<in> A \<and> f x = y) \<and> B = f ` A)"
tendsto_0_smallo_1: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> o(\<lambda>x. 1)" shows "(f \<longlongrightarrow> 0) at_top"
nlpaux: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators"   and a :: "((int \<times> int) set set, int) Combinators" assumes "x \<notin> dom (C b)" shows "C (a \<oplus> b) x = C a x"
to_nat_plus_one_less_card': fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "mod_type.to_nat Rep a + 1 < CARD('d)"
multiset_of_list: fixes i :: "'a list" shows "multiset (of_list i) = mset i"
induct: fixes P :: "'a list list \<Rightarrow> bool"   and a0 :: "'a list list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>v vb vc. P (local.merge_pairs (v # vb # vc)) \<Longrightarrow> P (v # vb # vc)" shows "P a0"
fact_mem_2: fixes fi :: "int poly"   and i :: "nat" assumes "(fi, i) \<in> set qis" shows "degree fi \<noteq> 0"
map_poly_degree_leq: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "degree (map_poly f p) \<le> degree p"
intro: fixes R :: "'b set" assumes "Ring1 R" shows "R_scalar_mult R"
splitset_consl: fixes x :: "'c"   and xs :: "'c list" shows "splitset (x # xs) = insert ([], x # xs) (apfst ((#) x) ` splitset xs)"
IH: shows "pderiv a (flatten PLUS X) = pderiv_set a X"
vars_snd_rcv_strand_subset2_4: fixes S :: "('e, 'd) strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>t S \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
col_cong2_bet4: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Col A B C"   and "Bet A B D"   and "Cong A B C D"   and "Cong A D B C" shows "Bet B D C"
prems_3: fixes i :: "nat" assumes "i < length (x # xs)" shows "length (yys ! i) = length ((x # xs) ! i)"
equivalence_map_def: fixes f :: "'a" shows "C.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. C.internal_equivalence f g \<eta> \<epsilon>"
semIntAbs_prWlsAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prWlsAbs (semIntAbs SEM) SEM"
eq3_1: shows "y2 \<bullet> y2 = x \<bullet> y2"
"fun": fixes rs :: "(name \<times> pterm) fset"   and t :: "pterm"   and t' :: "pterm"   and u :: "pterm" assumes "rs \<turnstile>\<^sub>p t \<longrightarrow> t'" shows "rs \<turnstile>\<^sub>p t $\<^sub>p u \<longrightarrow> t' $\<^sub>p u"
monoid_add_hom_0_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_add_hom_0 hom \<equiv> monoid_add_hom hom \<and> zero_hom_0 hom"
fds_integral_0: fixes a :: "'b" shows "fds_integral a 0 = fds_const a"
1: fixes x :: "real" assumes "x \<in> {0<..<1}" shows "(F has_real_derivative f x) (at x)"
bit_iff_odd: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a n = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)))"
af_simps_2: fixes w :: "'a set list" shows "af false\<^sub>n w = false\<^sub>n"
sup_right_isotone: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x" shows "z \<sqinter> y \<le> z \<sqinter> x"
graeffe: shows "graeffe_poly c as m = Polynomial.smult (c ^ 2 ^ m) (\<Prod>a\<leftarrow>map (\<lambda>a. a ^ 2 ^ m) as. [:- a, 1:])"
eq_size_1: fixes kdt :: "'b kdt" shows "(KD_Tree.size_kdt kdt = 1) = (\<exists>p. kdt = Leaf p)"
term_subst_eq_conv: fixes t :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and \<tau> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "(t \<cdot> \<sigma> = t \<cdot> \<tau>) = (\<forall>x\<in>fv t. \<sigma> x = \<tau> x)"
fpxs_compose_power_mult: fixes f :: "'a fpxs"   and g :: "'a fpxs"   and r :: "rat" shows "fpxs_compose_power (f * g) r = fpxs_compose_power f r * fpxs_compose_power g r"
pr_symk_used: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "Key (symK n) \<in> U"
consistent_eq1: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s = (\<forall>sv\<in>Store.symvars s. \<sigma>\<^sub>s\<^sub>y\<^sub>m sv = \<sigma> (fst sv))"
span_swap: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and E :: "'g set"   and u :: "'g"   and v :: "'g" assumes "vectorspace K V"   and "finite E"   and "E \<subseteq> carrier V"   and "u \<in> carrier V"   and "u \<notin> module.span K V E"   and "v \<in> carrier V"   and "u \<in> module.span K V (insert v E)" shows "module.span K V (insert u E) \<subseteq> module.span K V (insert v E)"
sliding: fixes r :: "'a language"   and s :: "'a language" shows "Times (Star (Times r s)) r = Times r (Star (Times s r))"
eSubst_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and s :: "'sort"   and X :: "'c"   and y :: "'var" assumes "igWls MOD (asSort ys) Y"   and "igWls MOD s X" shows "eSubst MOD ys (OK Y) y (OK X) = OK (igSubst MOD ys Y y X)"
induct: fixes P :: "exp \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "exp"   and a1 :: "nat" assumes "\<And>x k. P (EVar x) k"   and "\<And>n k. P (ENat n) k"   and "\<And>x e k. P e k \<Longrightarrow> P (ELam x e) k"   and "\<And>e1 e2. \<lbrakk>P e1 0; P e2 0\<rbrakk> \<Longrightarrow> P (EApp e1 e2) 0"   and "\<And>e1 e2 k. \<lbrakk>P e1 (Suc k); \<And>x. x = opt e1 (Suc k) \<Longrightarrow> P e2 (Suc k); \<And>x xa x31 x32. \<lbrakk>x = opt e1 (Suc k); xa = opt e2 (Suc k); x = ELam x31 x32; is_value xa\<rbrakk> \<Longrightarrow> P (subst x31 xa x32) k\<rbrakk> \<Longrightarrow> P (EApp e1 e2) (Suc k)"   and "\<And>f e1 e2 k. \<lbrakk>P e1 k; \<And>x. x = opt e1 k \<Longrightarrow> P e2 k\<rbrakk> \<Longrightarrow> P (EPrim f e1 e2) k"   and "\<And>e1 e2 e3 k. \<lbrakk>P e1 k; \<And>x. x = opt e1 k \<Longrightarrow> P e2 k; \<And>x xa. \<lbrakk>x = opt e1 k; xa = opt e2 k\<rbrakk> \<Longrightarrow> P e3 k\<rbrakk> \<Longrightarrow> P (EIf e1 e2 e3) k" shows "P a0 a1"
Sup_lessThan: fixes y :: "'b" shows "Sup {..<y} = y"
is_arg_min_on_def: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a" shows "is_arg_min_on f S x = (x \<in> S \<and> (\<forall>y\<in>S. f x \<le> f y))"
assms_3: shows "Poly_Mapping.keys b \<subseteq> S"
AndR: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "\<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta>"
IH_1: assumes "M \<subseteq> M'" shows "\<langle>M';P\<rangle> \<turnstile>\<^sub>r t"
Nml_VcompNml_3: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t"   and "bicategorical_language.Nml V src trg u"   and "bicategorical_language.Dom V src trg t = bicategorical_language.Cod V src trg u" shows "bicategorical_language.Cod V src trg (bicategorical_language.VcompNml V t u) = bicategorical_language.Cod V src trg t"
IntE: fixes c :: "'f"   and A :: "'f set"   and B :: "'f set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
integrable_divide: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "integrable M f" shows "integrable M (\<lambda>x. f x / y)"
unadjoin_def: fixes x :: "'a" shows "unadjoin x = (THE z. z \<in> H \<times> {..<subgroup_indicator G H a} \<and> x = fst z \<otimes> a [^] snd z)"
monoI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "mono f"
zero_transforms: fixes FH :: "('f, 'g) aezfun set" shows "0 \<in> smultfunset G FH"
select_convs_6: fixes Obj :: "'b set"   and Mor :: "'a set"   and Dom :: "'a \<Rightarrow> 'b"   and Cod :: "'a \<Rightarrow> 'b"   and Id :: "'b \<Rightarrow> 'a"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'c" shows "(;;\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub>) = Comp"
Infinite_Soundness_axioms: shows "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>"
\<F>_2: shows "\<F> \<subseteq> \<V>"
zeta_cnj: fixes z :: "complex" shows "zeta (cnj z) = cnj (zeta z)"
continuous_at_closest_point: fixes S :: "'b set"   and x :: "'b" assumes "convex S"   and "closed S"   and "S \<noteq> {}" shows "isCont (closest_point S) x"
enum_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(enum i \<le> enum j) = (i \<le> j)"
right_neutral: fixes a :: "'a list set" shows "p_prod a p_one = a"
homeomorphic_compact_space: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homeomorphic_space Y" shows "compact_space X = compact_space Y"
assms_3: shows "- b1 / a1 \<le> - b2 / a2"
Field_ofilter: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "ofilter r (Field r)"
Ints_cases: fixes 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 q :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "q \<in> ring_1.Ints one plus zero minus" obtains z :: "int" where "q = ring_1.of_int one plus zero minus z"
nonbossy_def: fixes ds :: "'d set"   and \<phi> :: "('d \<Rightarrow> ('x \<times> 'x) set) \<Rightarrow> ('h \<Rightarrow> 'x set \<Rightarrow> 'x set) \<Rightarrow> 'd set \<Rightarrow> 'x set" shows "nonbossy ds \<phi> = ((\<forall>Pd Pd' Ch. \<forall>d\<in>ds. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<and> ContractsWithBilateralSubstitutesAndIRC Xd Xh (Pd(d := Pd')) Ch \<longrightarrow> dX (\<phi> Pd Ch ds) d = dX (\<phi> (Pd(d := Pd')) Ch ds) d \<longrightarrow> \<phi> Pd Ch ds = \<phi> (Pd(d := Pd')) Ch ds) \<and> (\<forall>Pd Ch Ch' h. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<and> ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd (Ch(h := Ch')) \<longrightarrow> hX (\<phi> Pd Ch ds) h = hX (\<phi> Pd (Ch(h := Ch')) ds) h \<longrightarrow> \<phi> Pd Ch ds = \<phi> Pd (Ch(h := Ch')) ds))"
vwalk_to_vpath_induct3: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x xs. \<lbrakk>x \<in> set xs; P (dropWhile (\<lambda>y. y \<noteq> x) xs)\<rbrakk> \<Longrightarrow> P (x # xs)"   and "\<And>x xs. \<lbrakk>x \<notin> set xs; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
1: shows "Suc (card X - 2) = card X - 1"
bezout_ring_imp_diagonal_admits_SNF_mod_ring_admits: assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus" shows "\<forall>A. admits_SNF_HA A"
total_def_var_2: fixes x :: "'a" shows "is_total x = (- x \<le> x ; - 1')"
outside_reduces_domain: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "Domain (Outside' X P) = Domain P - X"
small_arr_Set: fixes \<alpha> :: "V" shows "small {T. arr_Set \<alpha> T}"
surjI: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. g (f x) = x" shows "top_pres ((`) g)"
res_def: shows "CIPHER.AUTH.res \<equiv> RES (CIPHER.AUTH.auth_oracle \<oplus>\<^sub>O CIPHER.AUTH.send_oracle \<oplus>\<^sub>O CIPHER.AUTH.recv_oracle) Void"
frequently_at_top_realE: fixes P :: "nat \<Rightarrow> real \<Rightarrow> bool" assumes "\<forall>n. frequently (P n) at_top" obtains s :: "nat \<Rightarrow> real" where "\<And>i. P i (s i)" and "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
star_n_def: fixes X :: "nat \<Rightarrow> 'b" shows "star_n X = Abs_star (starrel `` {X})"
clop_def: fixes f :: "'c \<Rightarrow> 'c" shows "clop f = (id \<le> f \<and> mono f \<and> f \<circ> f \<le> f)"
sort_leq_ex: fixes cs :: "(String.literal \<times> String.literal) set"   and s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_leq cs s1 s2" shows "sort_ex cs s2"
zero: shows "\<zero>\<^bsub>\<Z>\<^esub> = 0"
results'_gpv_Cont: fixes generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat"   and gpv :: "('d, 'e, 'f) gpv"   and c :: "('d, 'e, 'f) rpv"   and x :: "'d"   and input :: "'f" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "c \<in> generat_conts generat"   and "x \<in> results'_gpv (c input)" shows "x \<in> results'_gpv gpv"
lift_spmf_parametric': fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_gpv'' A C R) lift_spmf lift_spmf"
add_minus_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and t :: "'a" assumes "class.comm_powerprod plus minus zero" shows "minus (plus s t) s = t"
negate_sat: fixes \<sigma> :: "nat \<Rightarrow> bool"   and l :: "lit" shows "sat_lit \<sigma> (negate l) = (\<not> sat_lit \<sigma> l)"
hash_ineq_card: fixes W :: "'\<beta> set" assumes "finite W" shows "spmf (local.hash.game_hash_set W) True \<le> local.hash.\<epsilon>_uh * real (card W) * real (card W)"
lift_RealIntVal_Real: fixes x :: "val"   and f :: "int \<Rightarrow> int"   and g :: "real \<Rightarrow> real" assumes "x \<in> space (stock_measure REAL)" shows "lift_RealIntVal f g x = lift_RealVal g x"
\<tau>move2_IfThenD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>move2 P h stk (if (e) e1 else e2) (Suc (length (compE2 e) + pc')) xcp'"   and "pc' < length (compE2 e1)" shows "\<tau>move2 P h stk e1 pc' xcp'"