hcju/isabelleps huggingface dataset:
cd: fixes x :: "complex" assumes "dist \<xi> x < r" shows "(\<lambda>z. deriv g z / g z) field_differentiable at x"
fg: fixes x :: "'a" assumes "x \<in> topspace X"   and "p x = q x" shows "f x = g x"
an: fixes x :: "'a vec" assumes "x \<in> set lst" shows "x \<in> carrier_vec n"
InvariantReasonClausesAfterLearn: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and C :: "Literal list"   and F' :: "Literal list list" assumes "InvariantReasonClauses F M"   and "formulaEntailsClause F C"   and "F' = F @ [C]" shows "InvariantReasonClauses F' M"
assms_6: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "equalish A B"   and "reflexive_rewrite B C" shows "reflexive_rewrite A C"
well_com_hadamard_k: fixes k :: "nat" assumes "k \<le> n" shows "well_com (hadamard_n k)"
Smith_1x2: shows "\<forall>A\<in>carrier_mat 1 2. is_SNF A (Smith_1x2 A)"
coeff_pderiv: fixes p :: "'a poly"   and n :: "nat" shows "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
name_abs_alpha: fixes b :: "name"   and P :: "pi"   and a :: "name" assumes "b \<sharp> P" shows "[a].P = [b].([(a, b)] \<bullet> P)"
nc_meet_closed: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> nc \<sqinter> (y \<sqinter> nc) = x \<sqinter> y \<sqinter> nc"
formulaEntailsLiteralRemoveAllAppend: fixes clause :: "Literal list"   and formula1 :: "Literal list list"   and formula2 :: "Literal list list"   and literal :: "Literal" assumes "formulaEntailsLiteral (removeAll clause formula1 @ formula2) literal" shows "formulaEntailsLiteral (formula1 @ formula2) literal"
lens_comp_def: fixes Y :: "'a \<Longrightarrow> 'c"   and X :: "'c \<Longrightarrow> 'b" shows "Y ;\<^sub>L X = \<lparr>lens_get = get\<^bsub>Y\<^esub> \<circ> get\<^bsub>X\<^esub>, lens_put = \<lambda>\<sigma> v. put\<^bsub>X\<^esub> \<sigma> (put\<^bsub>Y\<^esub> (get\<^bsub>X\<^esub> \<sigma>) v)\<rparr>"
atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {..Suc n} = g 0 + (\<Sum>i\<le>n. g (Suc i))"
dim_left_null_space: fixes A :: "(('b, 'd) vec, 'c) vec" shows "vec.dim (left_null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A"
sifum_types_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d" shows "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False \<equiv> sifum_lang_no_dma ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars \<and> sifum_security dma \<C>_vars (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) \<and> sifum_types_axioms ev\<^sub>A ev\<^sub>B bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"
sdiv_poly_smult: fixes c :: "'a"   and f :: "'a poly" assumes "c \<noteq> (0::'a)" shows "sdiv_poly (smult c f) c = f"
INFM_const: fixes P :: "bool" shows "(\<exists>\<^sub>\<infinity>x. P) = (P \<and> infinite UNIV)"
bij_betwI: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'e" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> B \<Longrightarrow> f (g y) = y" shows "bij_betw f A B"
fine_Int: fixes d1 :: "'b \<Rightarrow> 'c set"   and d2 :: "'b \<Rightarrow> 'c set"   and p :: "('b \<times> 'c set) set" shows "((\<lambda>x. d1 x \<inter> d2 x) fine p) = (d1 fine p \<and> d2 fine p)"
zero_less_dist_iff: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "(0 < dist x y) = (x \<noteq> y)"
polyfun_mult: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"   and g :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "polyfun N f"   and "polyfun N g" shows "polyfun N (\<lambda>x. f x * g x)"
measurable_extract_real_pair: shows "extract_real_pair \<in> stock_measure (PRODUCT REAL REAL) \<rightarrow>\<^sub>M borel \<Otimes>\<^sub>M borel"
induct: fixes x1 :: "'a com"   and x2 :: "'a"   and x3 :: "'a"   and P :: "'a com \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Sem x1 x2 x3"   and "\<And>f s. P (Basic f) s (f s)"   and "\<And>c1 s s'' c2 s'. \<lbrakk>Sem c1 s s''; P c1 s s''; Sem c2 s'' s'; P c2 s'' s'\<rbrakk> \<Longrightarrow> P (Seq c1 c2) s s'"   and "\<And>s b c1 s' c2. \<lbrakk>s \<in> b; Sem c1 s s'; P c1 s s'\<rbrakk> \<Longrightarrow> P (Cond b c1 c2) s s'"   and "\<And>s b c2 s' c1. \<lbrakk>s \<notin> b; Sem c2 s s'; P c2 s s'\<rbrakk> \<Longrightarrow> P (Cond b c1 c2) s s'"   and "\<And>s b x y c. s \<notin> b \<Longrightarrow> P (While b x y c) s s"   and "\<And>s b c s'' x y s'. \<lbrakk>s \<in> b; Sem c s s''; P c s s''; Sem (While b x y c) s'' s'; P (While b x y c) s'' s'\<rbrakk> \<Longrightarrow> P (While b x y c) s s'" shows "P x1 x2 x3"
arrE: fixes f :: "'d \<times> 'c" assumes "DopxC.arr f" obtains "Dop.arr (fst f) \<and> C.arr (snd f)"
af_letter_congruent: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "af_letter \<phi> \<nu> \<sim>\<^sub>P af_letter \<psi> \<nu>"
runit_simps_4: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.dom (S.runit f) = f \<star>\<^sub>S S.src f"
game_def: shows "game = (\<lambda>\<A> recv_oracle. exec_gpv (lazy_channel_insec' \<oplus>\<^sub>O lazy_channel_send' \<oplus>\<^sub>O recv_oracle) \<A> (False, Void, None, Map.empty) \<bind> (\<lambda>(b, flg, ams, es, as). return_spmf (b, flg)))"
implementsI: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "\<And>t. t \<in> jkbpC \<Longrightarrow> set \<circ> actJP jp t = set \<circ> jAction MC t" shows "implements jp"
square_reflcl_confluent: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "square R R R\<^sup>=\<^sup>= R\<^sup>=\<^sup>=" shows "confluent R"
oalist_ntm_eq_iff: fixes xs :: "('a, 'b) oalist_ntm"   and ys :: "('a, 'b) oalist_ntm" shows "(xs = ys) = (list_of_oalist_ntm xs = list_of_oalist_ntm ys)"
right_neutral: fixes a :: "'a multiset" shows "a \<union># {#} = a"
elims_1: fixes x :: "nat \<Rightarrow> bool"   and xa :: "lit uprod"   and y :: "bool" assumes "sat_clause x xa = y" obtains \<sigma> :: "nat \<Rightarrow> bool" and l1 :: "lit" and l2 :: "lit" where "x = \<sigma>" and "xa = Upair l1 l2" and "y = (sat_lit \<sigma> l1 \<or> sat_lit \<sigma> l2)"
\<gamma>_ivl_def: shows "\<gamma>_ivl \<equiv> map_fun rep_ivl id \<gamma>_rep"
assms_2: fixes x :: "'c"   and a\<^sub>1 :: "'a"   and a\<^sub>2 :: "'a" assumes "x \<in> S"   and "f x a\<^sub>1 a\<^sub>2" shows "g a\<^sub>1 \<le> g a\<^sub>2"
ISpec_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "ISpec ic q oc \<equiv> BInit ic q oc \<and> \<box>[Nxt ic q oc]_(ic, q, oc) \<and> WF(Deq ic q oc)_(ic, q, oc)"
lexord_same_pref_if_irrefl: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "irrefl r" shows "((xs @ ys, xs @ zs) \<in> lexord r) = ((ys, zs) \<in> lexord r)"
equalsffemptyI: fixes A :: "'a fset" assumes "\<And>y. y |\<in>| A \<Longrightarrow> False" shows "A = {||}"
rep_eq: fixes x :: "'c word" shows "unsigned x = of_nat (nat (take_bit LENGTH('c) (Word.rep x)))"
norm_diff_temp_dyn: fixes a :: "real"   and s\<^sub>1 :: "(real, 4) vec"   and L :: "real"   and s\<^sub>2 :: "(real, 4) vec" assumes "0 < a" shows "\<parallel>temp_vec_field a L s\<^sub>1 - temp_vec_field a L s\<^sub>2\<parallel> = \<bar>a\<bar> \<cdot> \<bar>s\<^sub>1 $ 1 - s\<^sub>2 $ 1\<bar>"
indspace_sum_list_fddh: fixes fhs :: "('f \<times> 'g) list"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "fhs \<noteq> []"   and "set (map snd fhs) \<subseteq> H"   and "f \<in> indV" shows "f (\<Sum>(a, h)\<leftarrow>fhs. a \<delta>\<delta> h) = (\<Sum>(a, h)\<leftarrow>fhs. f (a \<delta>\<delta> h))"
cases: fixes x :: "('a, 'b) insert_message" obtains p :: "'a extended" and i :: "'a" and s :: "'a extended" and \<sigma> :: "'b" where "x = InsertMessage p i s \<sigma>"
bound_l: shows "2 ^ N\<^sup>2 * B2_LLL F ^ (2 * N) \<le> (p ^ l)\<^sup>2"
inR1: fixes \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> G, H, \<Delta>" shows "\<Gamma> \<Rightarrow> H, G, \<Delta>"
callee_invariant_on_def: fixes callee :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and I :: "'c \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>" shows "callee_invariant_on callee I \<I> \<equiv> (\<forall>s x y s'. (y, s') \<in> set_spmf (callee s x) \<longrightarrow> I s \<longrightarrow> x \<in> outs_\<I> \<I> \<longrightarrow> I s') \<and> (\<forall>s. I s \<longrightarrow> \<I> \<turnstile>c callee s \<surd>)"
prems: shows "[] \<in> Star r \<rightarrow> v2"
real_Icc_closed_segment: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "{a..b} = {a--b}"
reachable_ss_V: fixes s :: "nat" assumes "s \<in> V" shows "reachableNodes s \<subseteq> V"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_altc_axioms bind altc" shows "monad_altc return bind altc"
SeqStTermP_Var_same: fixes s :: "name"   and k :: "name"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, v, i)"   and "atom k \<sharp> (v, i)" shows "{VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i v i (Var s) (Var k)))"
finite_psubset_induct: fixes A :: "'d set"   and P :: "'d set \<Rightarrow> bool" assumes "finite A"   and "\<And>A. \<lbrakk>finite A; \<And>B. B \<subset> A \<Longrightarrow> P B\<rbrakk> \<Longrightarrow> P A" shows "P A"
case_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case \<bottom> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
assms_2: shows "\<forall>x\<in>S. I x"
one_le_ceiling: fixes x :: "'b" shows "(1 \<le> \<lceil>x\<rceil>) = ((0::'b) < x)"
lred_imp_sred: fixes s :: "lam"   and t :: "lam" assumes "s \<rightarrow>\<^sub>l t" shows "s \<rightarrow>\<^sub>s t"
inf_commute: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<inter># y = y \<inter># x"
pi_pred_it: fixes S :: "'G"   and v :: "'V" shows "proper_it (pred_it S v) (pred_it S v)"
assms: shows "k < length (trace_parallel_plan_strips I \<pi>)"
gba_impl_rel_eext_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "\<langle>Rm, Rv, Rl\<rangle>gba_impl_rel_eext \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_gba_impl_rel_eext"
cases_scheme: fixes r :: "('b, 'c) state_scheme" obtains nu :: "nat" and m :: "nat" and l :: "nat" and hare :: "'b" and tortoise :: "'b" and more :: "'c" where "r = \<lparr>nu = nu, m = m, l = l, hare = hare, tortoise = tortoise, \<dots> = more\<rparr>"
OAlist_tc_update_by_fun_def: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_update_by_fun k f xs = OAlist_tc (update_by_fun_pair_tc k f (list_of_oalist_tc xs))"
block_size_gt_t: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and index :: "int"   and sizes :: "int set"   and k :: "int" assumes "\<Lambda>_PBD point_set block_collection index sizes"   and "k \<in> sizes" shows "2 \<le> k"
adv: fixes \<A> :: "nat \<Rightarrow> ('a + 'b, 'c + 'd) distinguisher" assumes "\<And>\<eta>. \<I>_real1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta> \<turnstile>g \<A> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound1 \<eta>)"   and "\<And>\<eta>. lossless1 \<Longrightarrow> plossless_gpv (\<I>_real1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta>) (\<A> \<eta>)" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (sim1 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal1 \<eta>) (real1 \<eta>))"
inf_sup_aci_6: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
keys_filter: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.filter P m) \<subseteq> Mapping.keys m"
R: shows "(\<prec>) \<in> {(<), \<lambda>x y. y < x}"
inv: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "S.inverse_arrows (\<Phi> a) (\<Psi> a)"
p_ph_enabled_eq_imp: shows "eq_imp (\<lambda>_ s. s p) p_ph_enabled"
IH2: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p2) TT"   and "(s, connect(i, i'), s') \<in> automaton.trans (pnet np p2)" shows "netmap s' = netmap s"
arg_def: fixes tm :: "('c, 'd) tm" shows "arg tm = (case tm of Hd \<zeta> \<Rightarrow> Hd \<zeta> | App x21 x22 \<Rightarrow> x22)"
set_child_nodes_locs_def: shows "set_child_nodes_locs \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes_locs"
empty_graph_is_tree: fixes g :: "('d, 'e) rgraph_scheme" assumes "edges g = {}" shows "is_tree g"
trivial_hom: fixes H :: "('c, 'd) monoid_scheme"   and G :: "('e, 'f) monoid_scheme" assumes "Group.group H" shows "(\<lambda>x. \<one>\<^bsub>H\<^esub>) \<in> hom G H"
bounded_linear_zero: shows "bounded_linear (\<lambda>x. 0::'b)"
d1: fixes x :: "'a vec" assumes "x \<in> set ws" shows "x \<in> carrier_vec n"
card'_code_1: shows "local.card' (set xs) = length (remdups xs)"
results'_gpv_Cont: fixes generat :: "('a, 'b, ('a, 'b, 'c) rpv) generat"   and gpv :: "('a, 'b, 'c) gpv"   and c :: "('a, 'b, 'c) rpv"   and x :: "'a"   and input :: "'c" 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"
max_fun_diff_le_eq: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "(s.max_fun_diff f g, x) \<in> s"   and "f \<noteq> g"   and "f \<in> rs.FINFUNC"   and "g \<in> rs.FINFUNC"   and "x \<noteq> s.max_fun_diff f g" shows "f x = g x"
assms: shows "K\<^bsub>5\<^esub> G"
A: shows "c_fst n mod 7 \<noteq> 0"
Inr_in_Plus: fixes x :: "'d"   and A :: "'c set"   and B :: "'d set" shows "(Inr x \<in> A <+> B) = (x \<in> B)"
power_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "num"   and l :: "num" assumes "class.semiring_numeral one times plus" shows "power.power one times (numeral.numeral one plus k) (numeral l) = numeral.numeral one plus (Num.pow k l)"
simps: fixes v :: "val" shows "the_val (Lit v) = v"
to_RED_aux: fixes p :: "trm"   and x :: "name"   and k :: "stack"   and n :: "trm" assumes "SN p"   and "x \<sharp> p"   and "x \<sharp> k"   and "SN (n[x::=p] \<star> k)" shows "SN (([p] to x in n) \<star> k)"
simps: fixes a1 :: "char list \<Rightarrow> com option"   and a2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and a3 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" shows "small_steps a1 a2 a3 = ((\<exists>\<pi> cs. a1 = \<pi> \<and> a2 = cs \<and> a3 = Some cs) \<or> (\<exists>\<pi> cs. a1 = \<pi> \<and> a2 = cs \<and> a3 = None \<and> small_step \<pi> cs = None) \<or> (\<exists>\<pi> cs cs1 cs2. a1 = \<pi> \<and> a2 = cs \<and> a3 = cs2 \<and> small_step \<pi> cs = Some cs1 \<and> small_steps \<pi> cs1 cs2))"
offs_pred_cons: fixes ns :: "nat list"   and ub :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" assumes "offs_pred ns ub (x # xs) index key mi ma"   and "index key x (length ns) mi ma = i"   and "i < length ns" shows "offs_pred (ns[i := Suc (ns ! i)]) ub xs index key mi ma"
igSubstAllIPresIGWlsAllSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstAllIPresIGWlsAllSTR MOD \<equiv> igSubstIPresIGWlsSTR MOD \<and> igSubstAbsIPresIGWlsAbsSTR MOD"
skl1_def: shows "skl1 \<equiv> \<lparr>init = skl1_init, trans = skl1_trans, obs = id\<rparr>"
oconf_def: fixes P :: "'a prog"   and h :: "heap"   and obj :: "obj" shows "P,h \<turnstile> obj \<surd> \<equiv> let (C, fs) = obj in \<forall>F D T. P \<turnstile> C has F,NonStatic:T in D \<longrightarrow> (\<exists>v. fs (F, D) = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
vec_lambda_beta: fixes g :: "'b \<Rightarrow> 'a"   and i :: "'b" shows "vec_lambda g $ i = g i"
scaleRat_less2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less a b"   and "k < 0" shows "less (scaleRat k b) (scaleRat k a)"
\<psi>': shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g' f \<phi> \<psi>'"
\<delta>_states_union: fixes \<delta> :: "('a, 'b) ta_rule set"   and \<delta>' :: "('a, 'b) ta_rule set" shows "\<delta>_states (\<delta> \<union> \<delta>') = \<delta>_states \<delta> \<union> \<delta>_states \<delta>'"
obj_def: fixes a :: "'a" shows "B.obj a \<equiv> B.arr a \<and> src\<^sub>B a = a"
Rep_ge_0: fixes x :: "'a" shows "0 \<le> Rep x"
snd_iso_thme_recip: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "subgroup N G"   and "subgroup H (G\<lparr>carrier := normalizer G N\<rparr>)" shows "G\<lparr>carrier := H <#> N\<rparr> Mod N \<cong> G\<lparr>carrier := H\<rparr> Mod H \<inter> N"
add_ac_2: fixes a :: "'k"   and b :: "'k" shows "a + b = b + a"
cases: fixes a :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> SINK f" obtains x :: "'a" where "a = x" and "d_OUT f x = 0"
simps_2: fixes block :: "DiskBlock"   and proc :: "Proc"   and more :: "'a" shows "proc \<lparr>block = block, proc = proc, \<dots> = more\<rparr> = proc"
streamsE: fixes s :: "'a stream"   and A :: "'a set" assumes "s \<in> streams A" obtains "shd s \<in> A" and "stl s \<in> streams A"
new_tv_subst_te: fixes n :: "nat"   and S :: "nat \<Rightarrow> typ"   and t :: "typ" assumes "new_tv n S"   and "new_tv n t" shows "new_tv n ($ S t)"
Lcm_coprime: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> coprime a b" shows "Lcm A = normalize (\<Prod>A)"
nextState_def: fixes alg :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'a"   and st :: "'b"   and msgs :: "'a \<Rightarrow> 'c option"   and st' :: "'b" shows "nextState alg r p st msgs st' \<equiv> CnextState alg r p st msgs undefined st'"
Step1_4_2: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkFwd memCh crCh cst p \<and> $S2 rmhist p \<and> S3 rmhist p$ \<and> unchanged (e p, r p, m p, rmhist!p) \<longrightarrow> unchanged (rtrner memCh!p, resbar rmhist!p)"
path_single_Arr: fixes x :: "'arr" assumes "x \<in> Arr" shows "G.path (D x) (C x) [x]"
outSetOfComponents_def: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "outSetOfComponents i chSet \<equiv> {Y. OUT Y \<inter> chSet \<noteq> {} \<and> Y \<in> AbstrLevel i}"
esing_below_iff: fixes x :: "'a"   and a :: "'b"   and ae :: "'a \<Rightarrow> 'b" shows "(esing x\<cdot>a \<sqsubseteq> ae) = (a \<sqsubseteq> ae x)"
accepts_or: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "NAe.accepts (or L R) w = (NAe.accepts L w \<or> NAe.accepts R w)"
roots_with_def: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a poly" shows "roots_with P p = {x. P x \<and> poly p x = (0::'a)}"
conj: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> B)"
simps_5: fixes a :: "tm" shows "bound0 (Eq a) = tmbound0 a"
Poly_snoc_zero: fixes as :: "'a list" shows "Poly (as @ [0::'a]) = Poly as"
root_polyfun: fixes n :: "nat"   and z :: "'a"   and a :: "'a" assumes "1 \<le> n" shows "(z ^ n = a) = ((\<Sum>i\<le>n. (if i = 0 then - a else if i = n then 1::'a else (0::'a)) * z ^ i) = (0::'a))"
rel_with_target_impl_transC_TRel_is_strong_reduction_bisimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "strong_reduction_simulation (TRel\<^sup>+) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>+ \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>+))"
iwlsFSbSw_termFSwSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSbSw MOD" shows "termFSwSbImorph h hA MOD = ipresCons h hA MOD"
sound_def: fixes P :: "'b \<Rightarrow> real" shows "sound P \<equiv> Expectations.bounded P \<and> nneg P"
assms_1: shows "ncc ((\<lambda>x. x ! i) ` env)"
enumerate_mono_iff: fixes S :: "'a set"   and m :: "nat"   and n :: "nat" assumes "infinite S" shows "(enum S m < enum S n) = (m < n)"
l2_inv7D_aux: fixes x :: "l2_state"   and c :: "msg" assumes "x \<in> l2_inv7"   and "c \<in> synth (analz (ik x))" shows "c \<in> synth (analz generators)"
squarefree_prime: fixes p :: "'a" assumes "prime p" shows "squarefree p"
IH_2: assumes "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Trueif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Falseif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e a. \<lbrakk>\<And>val. P (restrict_top i a val) (restrict_top t a val) (restrict_top e a val); lowest_tops [i, t, e] = Some a\<rbrakk> \<Longrightarrow> P i t e" shows "P (restrict_top i a False) (restrict_top t a False) (restrict_top e a False)"
l_disjoint: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length l" shows "l ! i \<inter> l ! j = {}"
Rep_G_inverse: fixes x :: "'b G" shows "Abs_G (Rep_G x) = x"
getFr_FvarsT_Fvars: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "getFr xs ts \<phi>s \<in> var \<and> getFr xs ts \<phi>s \<notin> set xs \<and> (t \<in> set ts \<longrightarrow> getFr xs ts \<phi>s \<notin> FvarsT t) \<and> (\<phi> \<in> set \<phi>s \<longrightarrow> getFr xs ts \<phi>s \<notin> Fvars \<phi>)"
simps_2: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"