hcju/isabelleps huggingface dataset:
base_subtype: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and C' :: "char list"   and M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "char list list \<times> expr"   and Cs :: "char list list"   and D :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr" assumes "wf_cdecl wf_md P (C, Bs, fs, ms)"   and "C' \<in> baseClasses Bs"   and "P \<turnstile> C' has M = (Ts', T', m') via Cs @\<^sub>p [D]"   and "(M, Ts, T, m) \<in> set ms" shows "Ts' = Ts \<and> P \<turnstile> T \<le> T'"
le: shows "count_const_lt_components (fst aux) \<le> fst data"
Smith_Impl_def: fixes Smith_1x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat"   and Smith_2x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat"   and div_op :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "Smith_Impl Smith_1x2 Smith_2x2 div_op \<equiv> (\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2 A)) \<and> (\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2 A)) \<and> is_div_op div_op"
formulaContainsItsClausesVariables: fixes clause :: "Literal list"   and formula :: "Literal list list" assumes "clause el formula" shows "vars clause \<subseteq> vars formula"
integral_combine: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and c :: "real" assumes "integrable (lebesgue_on {a..b}) f"   and "a \<le> c"   and "c \<le> b" shows "integral\<^sup>L (lebesgue_on {a..b}) f = integral\<^sup>L (lebesgue_on {a..c}) f + integral\<^sup>L (lebesgue_on {c..b}) f"
singular_chain_0: fixes p :: "nat"   and X :: "'c topology" shows "singular_chain p X 0"
rep_eq: fixes x :: "('b, unit) RBT.rbt" shows "r_max_opt x = rbt_max_opt (rbt.impl_of x)"
vector_derivative_within: fixes x :: "real"   and S :: "real set"   and f :: "real \<Rightarrow> 'b"   and y :: "'b" assumes "at x within S \<noteq> bot"   and "(f has_vector_derivative y) (at x within S)" shows "vector_derivative f (at x within S) = y"
ord_Zp_p: shows "ord_Zp ([p] \<cdot> \<one>) = 1"
valid_reqs2: fixes m :: "'b SecurityInvariant_configured"   and M :: "'b SecurityInvariant_configured list" assumes "valid_reqs (m # M)" shows "valid_reqs M"
acyclic_4a_4b: fixes x :: "'a" shows "acyclic_4a x = acyclic_4b x"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus" shows "-\<^sub>\<int> a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_uminus"
sb_simps1: fixes xs :: "'b"   and y :: "'a"   and x :: "'a" shows "x @xs[y / x]_xs = y"
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length xs"   and "xs ! i = x"   and "xs ! j = x" shows "thesis"
lookup_operator_wf_2: 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 name :: "char list"   and \<pi> :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "wf_ast_problem problem"   and "ast_problem.lookup_operator problem name = Some \<pi>" shows "fst \<pi> = name"
eq: fixes k :: "nat" shows "adjoint (tensor_P (testN k) (1\<^sub>m K)) * tensor_P post (1\<^sub>m K) * tensor_P (testN k) (1\<^sub>m K) = tensor_P (adjoint (testN k) * post * testN k) (1\<^sub>m K)"
suffix_tl: fixes xs :: "'b list" shows "tl xs \<le>s xs"
oSuc_less_omega: fixes x :: "ordinal" shows "(oSuc x < \<omega>) = (x < \<omega>)"
fact7: assumes "\<exists>v1 v2. fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') = 0 \<and> get_addr (snd v2) s'' AND 3 \<noteq> 0" shows "False"
abs_eq: fixes xa :: "('b \<times> 'b) list"   and x :: "('b \<times> 'b) list" shows "abs_prm xa \<diamondop> abs_prm x = abs_prm (preprm_compose xa x)"
cases: fixes r :: "'b graph" obtains nodes :: "'b set" and edges :: "('b \<times> 'b) set" where "r = \<lparr>nodes = nodes, edges = edges\<rparr>"
composites_are_identities_2: shows "ide ((\<nu> \<star> \<pi>) \<star> \<rho>)"
assms_4: shows "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some True"
assms_1: shows "r \<in> local.mk_linear_orders C A"
inv_safe_safe: fixes c :: "('loc, 't) configuration" assumes "\<And>loc. c_work c loc = {#}\<^sub>z"   and "inv_safe c" shows "safe c"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "C.arr f"
group_by'_def: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and s :: "'c set" shows "group_by' f s = (\<lambda>a. let (fa, uu_) = f a in (fa, (snd \<circ> f) ` Set.filter (\<lambda>a'. fa = fst (f a')) s)) ` s"
homeomorphic_to_disc: fixes S :: "complex set" assumes "SC_Chain S"   and "S \<noteq> {}"   and "S = UNIV \<or> (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z))" shows "S homeomorphic ball 0 1"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "R.in_hom (R.comp g f) a c" obtains b :: "'a" where "R.in_hom f a b" and "R.in_hom g b c"
sheaf_morphisms_sheaf_spec: fixes s :: "nat set \<Rightarrow> (nat \<times> nat) set"   and U :: "nat set set" assumes "s \<in> cring0.sheaf_spec U" shows "cring0.sheaf_spec_morphisms U U s = s"
ballE: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('u \<Rightarrow> 'u) \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme"   and v :: "'u" assumes "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
polyfun_mult_mat_vec: fixes v :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a vec"   and n :: "nat"   and N :: "nat set"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a mat"   and m :: "nat"   and j :: "nat" assumes "\<And>x. v x \<in> carrier_vec n"   and "\<And>j. j < n \<Longrightarrow> polyfun N (\<lambda>x. v x $ j)"   and "\<And>x. A x \<in> carrier_mat m n"   and "\<And>i j. \<lbrakk>i < m; j < n\<rbrakk> \<Longrightarrow> polyfun N (\<lambda>x. A x $$ (i, j))"   and "j < m" shows "polyfun N (\<lambda>x. (A x *\<^sub>v v x) $ j)"
ih2: fixes t :: "'a trm" shows "vars_of (t \<lhd> \<theta>2) \<subseteq> vars_of (N \<lhd> \<theta>1) \<union> vars_of (N' \<lhd> \<theta>1) \<union> vars_of t"
finfun_comp_update_code: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow>f 'e"   and a :: "'c"   and b :: "'e" shows "g \<circ>$ finfun_update_code f a b = finfun_update_code (g \<circ>$ f) a (g b)"
algebraic_mono: fixes K :: "'a set"   and K' :: "'a set"   and x :: "'a" assumes "K \<subseteq> K'"   and "(algebraic over K) x" shows "(algebraic over K') x"
short_vector_hybrid_def: shows "short_vector_hybrid = (if enable_external_lll_solver then short_vector_external else short_vector)"
root_def: fixes n :: "nat" shows "FFT.root n \<equiv> cis (2 * pi / real n)"
singleton_eq_union_iff: fixes x :: "'d"   and M :: "'d multiset"   and y :: "'d" shows "({#x#} = M + {#y#}) = (M = {#} \<and> x = y)"
fixp_preserves_mcont1: fixes F :: "('c \<Rightarrow> 'd llist) \<Rightarrow> 'c \<Rightarrow> 'd llist"   and f :: "'c \<Rightarrow> 'd llist"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. mono_llist (\<lambda>f. F f x)"   and "f \<equiv> llist.fixp_fun F"   and "\<And>f. mcont lubb ordb lSup lprefix f \<Longrightarrow> mcont lubb ordb lSup lprefix (F f)" shows "mcont lubb ordb lSup lprefix f"
is_ground_cls_def: fixes C :: "'a literal multiset" shows "is_ground_cls C = (\<forall>L. L \<in># C \<longrightarrow> is_ground_lit L)"
new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def: fixes h :: "(_) heap" shows "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (let new_character_data_ptr = character_data_ptr.Ref (Suc (fMax (character_data_ptr.the_ref |`| character_data_ptrs h))) in (new_character_data_ptr, put new_character_data_ptr (create_character_data_obj []) h))"
Y_G: shows "\<F>\<G> \<phi> w = \<G> \<phi> (suffix (SOME i. \<F>\<G> \<phi> w = \<G> \<phi> (suffix i w)) w)"
induct: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" assumes "orthogonal_current \<Gamma> f S" obtains "\<And>x. \<lbrakk>x \<in> S; x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> weight \<Gamma> x \<le> d_IN f x" and "\<And>x. \<lbrakk>x \<in> S; x \<in> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x" and "\<And>u v. \<lbrakk>v \<in> RF\<^bsub>\<Gamma>\<^esub> S; u \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S\<rbrakk> \<Longrightarrow> f (u, v) = 0"
map_poly_pcompose: fixes f :: "'a poly"   and g :: "'a poly" shows "map_poly hom (f \<circ>\<^sub>p g) = map_poly hom f \<circ>\<^sub>p map_poly hom g"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_singleton ext" shows "ext ext"
app_insf2_Seq: fixes s :: "hf"   and k :: "hf"   and k' :: "hf"   and y :: "hf" assumes "Seq s k"   and "k' \<noteq> k" shows "app (insf s k y) k' = app s k'"
th1: shows "inj_on (\<lambda>p. p \<circ> q) {p. p permutes S}"
f_def: shows "f = (\<lambda>x. (b - a) * x + a)"
assms_2: shows "ipresIGVar h' MOD' MOD''"
SUP_cong_simp: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "lub (C ` A) = lub (D ` B)"
tD_gchain_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "tD_gchain G n g = (if n = 0 then g 0 = carrier G \<and> g 0 = {\<one>\<^bsub>G\<^esub>} else D_gchain G n g \<and> g 0 = carrier G \<and> g n = {\<one>\<^bsub>G\<^esub>})"
nn_integral_return_pmf: fixes f :: "'a \<Rightarrow> ennreal"   and x :: "'a" assumes "0 \<le> f x" shows "integral\<^sup>N (measure_pmf (return_pmf x)) f = f x"
l_Sup: shows "l \<longlonglongrightarrow> Sup A"
braun_Node': fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "braun \<langle>l, x, r\<rangle> = (size r \<le> size l \<and> size l \<le> size r + 1 \<and> braun l \<and> braun r)"
dcharacter_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "dcharacter n \<chi> \<equiv> residues_nat n \<and> dcharacter_axioms n \<chi>"
that_1: fixes s :: "'a set" assumes "s \<in> S" shows "f differentiable_on s"
distinct_1: fixes x1 :: "'b"   and x21 :: "('a, 'b) com"   and x22 :: "('a, 'b) com" shows "Atm x1 \<noteq> x21 ;; x22"
group_by_def: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and s :: "'c set" shows "group_by f s = {(fst (f a), {snd (f a') |a'. a' \<in> s \<and> fst (f a) = fst (f a')}) |a. a \<in> s}"
eql_map_neg_lit_eql_atm: fixes \<eta> :: "'s"   and As' :: "'a list"   and As :: "'a list" assumes "map (\<lambda>L. L \<cdot>l \<eta>) (map Neg As') = map Neg As" shows "As' \<cdot>al \<eta> = As"
that: fixes as' :: "'edge list" assumes "(_Entry_) -as'\<rightarrow>\<^sub>\<surd>* n"   and "set (sourcenodes as') \<subseteq> set (sourcenodes as)"   and "\<forall>a'\<in>set as'. intra_kind (kind a') \<or> (\<exists>Q r p fs. kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs)" shows "thesis"
t: shows "eval r_result1 [t, the (s (e_append_zeros b k)), e_length b + k] \<down>= Suc 0"
simps_1: fixes x2 :: "'b"   and y2 :: "'b" shows "(extB.Val x2 = extB.Val y2) = (x2 = y2)"
assms_1: fixes i :: "nat" assumes "i \<le> x" shows "f1 ! i = f2 ! i"
simps: fixes as :: "atom set"   and x :: "'a" shows "supp_set ([as]set. x) = supp x - as"
cases: fixes x :: "('c \<times> ('d, 'e) fmap) list" obtains "x = []"   | a :: "'c \<times> ('d, 'e) fmap" and as :: "('c \<times> ('d, 'e) fmap) list" where "x = a # as"
llast_prepend: fixes ys :: "'b llist"   and xs :: "'b list" assumes "\<not> lnull ys" shows "llast (prepend xs ys) = llast ys"
numeral_plus_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus n) one = numeral.numeral one plus (n + num.One)"
inter_guards_Throw: fixes c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Throw \<inter>\<^sub>g c2) = Some c) = (c2 = Throw \<and> c = Throw)"
paths_tri: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and ab :: "'b set"   and a :: "'b"   and b :: "'b"   and bc :: "'b set"   and c :: "'b"   and ca :: "'b set" assumes "MinkowskiPrimitive \<E> \<P>"   and "ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b"   and "bc \<in> \<P> \<and> b \<in> bc \<and> c \<in> bc \<and> b \<noteq> c"   and "ca \<in> \<P> \<and> c \<in> ca \<and> a \<in> ca \<and> c \<noteq> a"   and "a \<notin> bc" shows "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c"
lemScaleScale: fixes a :: "'a"   and b :: "'a"   and u :: "'a Vector" shows "a ** (b ** u) = (a * b) ** u"
tricky_subst: fixes b :: "coname"   and c :: "coname"   and N :: "trm"   and z :: "name"   and x :: "name"   and P :: "trm"   and M :: "trm" assumes "b \<sharp> (c, N)"   and "z \<sharp> (x, P)"   and "M \<noteq> Ax z b" shows "Cut <c>.N z.M{b:=x.P} = Cut <c>.N z.M{b:=x.P}"
not_finals_ConsI: fixes e :: "expr"   and es :: "expr list" assumes "\<not> final e" shows "\<not> finals (e # es)"
elims: fixes x :: "('b, 'c) chain"   and y :: "nat" assumes "chainlen x = y" obtains uu_ :: "'b" where "x = \<lbrace> uu_ \<rbrace>" and "y = 0"   | uv_ :: "'b" and uw_ :: "'c" and \<Pi> :: "('b, 'c) chain" where "x = \<lbrace> uv_ \<rbrace> \<cdot> uw_ \<cdot> \<Pi>" and "y = 1 + chainlen \<Pi>"
single_def: shows "single \<equiv> map_fun id (map_fun id Abs_poly_mapping) (\<lambda>k v k'. v when k = k')"
tau_seq_prop2: fixes y :: "'a"   and x :: "'a" assumes "\<tau> y = y" shows "\<tau> (x \<cdot> y) = x \<cdot> y"
f: shows "f \<equiv> \<lambda>a. SOME B. B \<in> C \<and> a \<in> B"
seq_def: shows "seq = rec_nat {} (\<lambda>_. fun)"
select_convs_3: fixes dwa_cond :: "'a \<Rightarrow> bool"   and dwa_step :: "'a \<Rightarrow> 'a"   and dwa_initial :: "'a"   and dwa_invar :: "'a set"   and more :: "'b" shows "dwa_initial \<lparr>dwa_cond = dwa_cond, dwa_step = dwa_step, dwa_initial = dwa_initial, dwa_invar = dwa_invar, \<dots> = more\<rparr> = dwa_initial"
fiter_fun_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "fiter_fun f g = (\<sqinter>) f \<circ> (\<circ>) g"
getEnv_swapEnv1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and x :: "'var" assumes "rho xs (x @xs[z1 \<and> z2]_zs) = None" shows "(rho &[z1 \<and> z2]_zs) xs x = None"
permutations_of_multiset_Cons_iff: fixes x :: "'a"   and xs :: "'a list"   and A :: "'a multiset" shows "(x # xs \<in> permutations_of_multiset A) = (x \<in># A \<and> xs \<in> permutations_of_multiset (A - {#x#}))"
usubstr_union_undef: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and A :: "fml list"   and B :: "fml list"   and C :: "fml" shows "(usubstr \<sigma> (A @ B, C) \<noteq> None) = (usubstr \<sigma> (A, C) \<noteq> None \<and> usubstr \<sigma> (B, C) \<noteq> None)"
1: shows "wf (Inl ` Field r \<times> Inr ` Field r')"
nth_equalityI: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
init_syzygy_list_nth: fixes i :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "i < length bs" shows "init_syzygy_list bs ! i = lift_poly_syz (length bs) (bs ! i) i"
aux_lemma: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "h a \<otimes>\<^bsub>H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>\<^esub> h b = h (a \<otimes> b)"
consequence_graph_def: fixes Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and G :: "('e, 'g) labeled_graph" shows "consequence_graph Rs G \<equiv> graph G \<and> (\<forall>R\<in>Rs. subgraph (fst R) (snd R) \<and> maintained R G)"
factors_exist: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<not> a dvd (1::'a)" shows "\<exists>fs. set fs \<subseteq> UNIV - {0::'a} \<and> comm_semiring_1_class.factors fs a"
ec: shows "spr_simAbs ec = spr_jkbpCSt t"
simple_fw_iff_generalized_fw_accept: fixes fw :: "'c simple_rule list"   and p :: "('c, 'd) simple_packet_scheme" shows "(simple_fw fw p = Decision FinalAllow) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, Accept))"
FunctorCompM: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and f :: "'i"   and X :: "'g"   and Y :: "'g" assumes "FunctorM F"   and "f maps\<^bsub>CatDom F\<^esub> X to Y" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
a_type_wf_def: fixes h :: "(_) heap" shows "NodeClass.type_wf h = (ObjectClass.type_wf h \<and> (\<forall>node_ptr\<in>fset (node_ptr_kinds h). get node_ptr h \<noteq> None))"
AE_contr: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "prob_space M"   and "almost_everywhere M P"   and "AE \<omega> in M. \<not> P \<omega>" shows "False"
RG_proj_mult_leftdelta': fixes g :: "'g"   and r :: "'a"   and s :: "'a"   and g' :: "'g" assumes "g \<in> G" shows "RG_proj (r \<delta>\<delta> g * (s \<delta>\<delta> g')) = r \<delta>\<delta> g * RG_proj (s \<delta>\<delta> g')"
map_replicate_const: fixes k :: "'k"   and lst :: "'l list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
simps_14: fixes e :: "'a exp" shows "max_vars (throw e) = max_vars e"
frees_list_comb: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a"   and xs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "frees (term.list_comb app t xs) = frees t |\<union>| pre_term.freess frees xs"
Grp_iff: fixes B :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" shows "BNF_Def.Grp B g x y = (y = g x \<and> x \<in> B)"
partial_latin_square_def: fixes s :: "(latin_type \<Rightarrow> nat) set"   and n :: "nat" shows "partial_latin_square s n \<equiv> (\<forall>t. inj_on (without t) s) \<and> (\<forall>e\<in>s. \<forall>t. e t < n)"
fpairwiseI: fixes M :: "'c fset"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>m n. \<lbrakk>m |\<in>| M; n |\<in>| M\<rbrakk> \<Longrightarrow> P m n" shows "fpairwise P M"
ideal_of_antimono: fixes A :: "('c \<Rightarrow> 'd) set"   and B :: "('c \<Rightarrow> 'd) set" assumes "A \<subseteq> B" shows "\<I> B \<subseteq> \<I> A"
leg0_composite: shows "fg.leg0 = g.leg0 \<cdot> fg.prj\<^sub>0"
rk_B'C'PR: fixes P :: "Points"   and Q :: "Points"   and R :: "Points"   and B' :: "Points"   and C' :: "Points" assumes "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {B', C', P, Q} = 4" shows "rk {B', C', P, R} = 4"
tab_in_vhom': shows "\<guillemotleft>(r \<star> \<phi>) \<cdot> \<rho> : g \<Rightarrow> r \<star> f'\<guillemotright>"
qp_init_post: shows "is_quantum_predicate (tensor_P proj_psi (proj_k 0))"
iteratei_def: fixes S :: "'s3" shows "s3.iteratei S \<equiv> it_to_it (s3.list_it S)"
neighborhoods_def: fixes x :: "'a" shows "neighborhoods x \<equiv> {U. is_open U \<and> x \<in> U}"
set_cond_bind_pmf: fixes A :: "'h set"   and f :: "'i \<Rightarrow> 'h pmf"   and p :: "'i pmf" assumes "A \<inter> (\<Union>x\<in>set_pmf p. set_pmf (f x)) \<noteq> {}" shows "set_pmf (cond_bind_pmf p f A) = {x \<in> set_pmf p. set_pmf (f x) \<inter> A \<noteq> {}}"
k: shows "max 0 (- log y x) + 1 < real k"
arctic_delta_interpretation: fixes \<delta> :: "'c" assumes "(0::'c) < \<delta>" shows "SN_both_mono_ordered_semiring_1 1 (gt_arctic_delta \<delta>) pos_arctic_delta"
in_listsI: fixes xs :: "'g list"   and A :: "'g set" assumes "\<forall>x\<in>set xs. x \<in> A" shows "xs \<in> lists A"
solves_ode_cong: fixes T :: "real set"   and x :: "real \<Rightarrow> 'a"   and y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "real set"   and X :: "'a set"   and Y :: "'a set" assumes "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(x solves_ode f) T X = (y solves_ode g) S Y"
IH: fixes f :: "complex \<Rightarrow> complex"   and x :: "complex" assumes "f holomorphic_on A"   and "x \<in> A"   and "open A" shows "(deriv ^^ j) (\<lambda>x. c * f x) x = c * (deriv ^^ j) f x"
one_less_inverse_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less one (inverse x) = (less zero x \<and> less x one)"
Diff_Int2: fixes A :: "'d set"   and C :: "'d set"   and B :: "'d set" shows "A \<inter> C - B \<inter> C = A \<inter> C - B"
simps: fixes xs :: "(int \<times> int) list" shows "closest_pair_rec_code xs = (let n = length xs in if n \<le> 3 then (mergesort snd xs, closest_pair_bf_code xs) else let (xs\<^sub>L, xs\<^sub>R) = split_at (n div 2) xs; l = fst (hd xs\<^sub>R); (ys\<^sub>L, p\<^sub>L) = closest_pair_rec_code xs\<^sub>L; (ys\<^sub>R, p\<^sub>R) = closest_pair_rec_code xs\<^sub>R; ys = Common.merge snd ys\<^sub>L ys\<^sub>R in (ys, combine_code p\<^sub>L p\<^sub>R l ys))"
cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "\<omega>[F](f) = \<omega>[F](g)"
prems_1: shows "l V' = \<lfloor>v\<rfloor>"
generate_subgroup_incl: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "generate G K \<subseteq> H"