hcju/isabelleps huggingface dataset:
interchange_rows_preserves: fixes i :: "'b"   and a :: "'b"   and j :: "'b"   and A :: "(('c, 'd) vec, 'b) vec" assumes "i \<noteq> a"   and "j \<noteq> a" shows "interchange_rows A i j $ a = A $ a"
nonpos_upto_def: fixes a :: "('loc \<times> 't) zmultiset"   and t :: "'loc \<times> 't" shows "cri.nonpos_upto a t = (\<forall>s. s\<le>\<^sub>pt \<longrightarrow> zcount a s \<le> 0)"
ide_char: fixes a :: "'a \<times> 'a \<times> 'a" shows "H.VVV.ide a = (H.VVV.arr a \<and> H.VxVxV.ide a)"
prems_3: shows "fst ` set (tx # txs) \<subseteq> trm"
\<chi>_def: shows "\<chi> \<equiv> exi yy (cnj (subst \<phi> (Var yy) xx) (SS (Var xx) (Var yy)))"
NewC: fixes G :: "prog"   and s0 :: "st"   and C :: "qtname"   and s1 :: "state"   and a :: "loc"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>halloc CInst C\<succ>a\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<rightarrow> s2"
mult_mono_nonneg_nonpos: fixes a :: "'a"   and c :: "'a"   and d :: "'a"   and b :: "'a" assumes "a \<le> c"   and "c \<le> (0::'a)"   and "(0::'a) \<le> d"   and "d \<le> b" shows "b * a \<le> d * c"
M_def: shows "M = (\<lambda>xa. ln (real xa) * (C / real xa powr (x + 1) + 1 / (real xa powr x * (real xa powr x - 1))))"
interleave_rev_append2: fixes w :: "'b list"   and a :: "'b list"   and b :: "'b list"   and w' :: "'b list" assumes "w \<in> a \<otimes> b" shows "w @ w' \<in> a \<otimes> b @ w'"
CastThrow: fixes C :: "char list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>Cast C (throw e),s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
intros_22: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "P \<oplus> R \<equiv>\<^sub>e Q \<oplus> R"
times_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f * g = (\<lambda>x. f x * g x)"
arg_complex_of_real_times_i_negative: fixes k :: "real" assumes "k < 0" shows "arg (cor k * \<i>) = - pi / 2"
types_conf: shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> S # Ss"
map_cong: fixes xs :: "'f list"   and ys :: "'f list"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
subset_UNIV: fixes A :: "'a set" shows "A \<subseteq> UNIV"
similar_mat_witD_2: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "Q * P = 1\<^sub>m n"
step: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a" shows "f x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"
simps_1: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "wpw b c 0 Q l s = (\<not> bval b s \<and> Q l s)"
JF2col_0: fixes x :: "'c JF2" shows "JF2col 0 x = {}"
rel_coinduct: fixes P :: "('b, 'c) dds \<Rightarrow> ('b, 'd) dds \<Rightarrow> bool"   and x :: "('b, 'c) dds"   and y :: "('b, 'd) dds"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "P x y"   and "\<And>dds dds'. P dds dds' \<Longrightarrow> ((=) ===> rel_prod R2 P) (run dds) (run dds')" shows "rel_dds' R2 x y"
ThrowRedsNull: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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 E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s'\<rangle>"
callee_invariant_on_def: fixes callee :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h) spmf"   and I :: "'h \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<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>)"
Close2: fixes P :: "pi"   and a :: "name"   and y :: "name"   and P' :: "pi"   and Q :: "pi"   and Q'' :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>y> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>ly in Q''\<rightarrow>a<x> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> <\<nu>y>(P' \<parallel> Q')"
size_multiset_union: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset"   and N :: "'k multiset" shows "size_multiset f (M + N) = size_multiset f M + size_multiset f N"
aux2: fixes P :: "ccs" shows "bangPred P (P \<parallel> !P)"
proots_withinI: fixes p :: "'b poly"   and x :: "'b"   and s :: "'b set" assumes "poly p x = (0::'b)"   and "x \<in> s" shows "x \<in> proots_within p s"
lt1: shows "trace (c \<cdot>\<^sub>m (A - f 0)) < 1"
coeff_of_sum_diff_degree0: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < n" shows "(p \<oplus>\<^bsub>P\<^esub> q) n = p n"
frag_induction: fixes c :: "'c \<Rightarrow>\<^sub>0 int"   and S :: "'c set"   and P :: "('c \<Rightarrow>\<^sub>0 int) \<Rightarrow> bool" assumes "Poly_Mapping.keys c \<subseteq> S"   and "P 0"   and "\<And>x. x \<in> S \<Longrightarrow> P (frag_of x)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (a - b)" shows "P c"
append_interleave: fixes xs :: "'b list"   and ys :: "'b list" shows "xs @ ys \<in> xs \<otimes> ys"
Un_absorb2: fixes B :: "'a set"   and A :: "'a set" assumes "order.greater_eq A B" shows "A \<union> B = A"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (Ge (CN 0 c e)) = T"
bounded_matrix_for_marginals_finite: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and g :: "nat \<Rightarrow> real"   and R :: "(nat \<times> nat) set" assumes "sum f {..n} = sum g {..n}"   and "\<And>X. X \<subseteq> {..n} \<Longrightarrow> sum f X \<le> sum g (R `` X)"   and "\<And>x. 0 \<le> f x"   and "\<And>y. 0 \<le> g y"   and "R \<subseteq> {..n} \<times> {..n}" obtains h :: "nat \<Rightarrow> nat \<Rightarrow> real" where "\<And>x y. \<lbrakk>x \<le> n; y \<le> n\<rbrakk> \<Longrightarrow> 0 \<le> h x y" and "\<And>x y. \<lbrakk>0 < h x y; x \<le> n; y \<le> n\<rbrakk> \<Longrightarrow> (x, y) \<in> R" and "\<And>x. x \<le> n \<Longrightarrow> f x = sum (h x) {..n}" and "\<And>y. y \<le> n \<Longrightarrow> g y = (\<Sum>x\<le>n. h x y)"
eval_simps_2: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "local.dom (eval b c) = HF'.prod (exp b c) b"
trans_sym: fixes A :: "'a set" shows "trans (A \<times> A)"
c: shows "\<Sqinter> ((;) (c::'a::refinement_lattice) ` (d::'b::type \<Rightarrow> 'a::refinement_lattice) ` (X::'b::type set)) = (\<Sqinter>x::'b::type\<in>X. c ; d x)"
simps_1: fixes uu :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" shows "merge_wrt uu xs [] = xs"
inv_P: shows "invertible (mult_row (mat (1::'a)) i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))"
lt_def: fixes p :: "tm" shows "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F | CP _ \<Rightarrow> Lt p | _ \<Rightarrow> Lt p)"
splitex: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "Ifm bs (E p) = (\<exists>i. Ifm (real_of_int i # bs) (E (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))"
\<delta>ss_subset: fixes f :: "'L"   and ss :: "'Q set list" shows "\<delta>ss_lhs f ss \<subseteq> Q"
functional_relation: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "right_unique R"   and "left_total R" shows "\<forall>x. \<exists>!y. R x y"
pnop_n: shows "2\<^sub>\<nat> \<in>\<^sub>\<circ> \<omega>"
lens_plus_mono_left: fixes Y :: "'d \<Longrightarrow> 'e"   and Z :: "'f \<Longrightarrow> 'e"   and X :: "'g \<Longrightarrow> 'e" assumes "Y \<bowtie> Z"   and "X \<subseteq>\<^sub>L Y" shows "X +\<^sub>L Z \<subseteq>\<^sub>L Y +\<^sub>L Z"
wf_graph_remove_edges_union: fixes V :: "'c set"   and E :: "('c \<times> 'c) set"   and E' :: "('c \<times> 'c) set" assumes "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
set_induct: fixes x :: "'d"   and a :: "'d stream"   and P :: "'d \<Rightarrow> 'd stream \<Rightarrow> bool" assumes "x \<in> sset a"   and "\<And>z1 z2. P z1 (z1 ## z2)"   and "\<And>z1 z2 xa. \<lbrakk>xa \<in> sset z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (z1 ## z2)" shows "P x a"
le_onorm: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "bounded_linear f" shows "norm (f x) / norm x \<le> onorm f"
psubst_Ax_aux: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and a :: "coname"   and y :: "name"   and N :: "trm"   and x :: "name"   and c :: "coname"   and P :: "trm" assumes "\<theta>_c cmaps a to Some (y, N)" shows "lookupb x a \<theta>_c c P = Cut <c>.P y.N"
pa_subset_pm_kde: shows "path_image (linepath (a + complex_of_real kde) (a + complex_of_real e) +++ p +++ linepath (a - complex_of_real d) (a - complex_of_real kde)) \<inter> closed_segment (a - complex_of_real kde) (a + complex_of_real kde) \<subseteq> {a - complex_of_real kde, a + complex_of_real kde}"
baEANow: fixes c :: "'bid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<phi> :: "'BB \<Rightarrow> bool" assumes "pb.eval c t t' n [\<phi>]\<^sub>b"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"
rewrite_modal_sound: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n rewrite_modal \<phi> = w \<Turnstile>\<^sub>n \<phi>"
accept_eq_lang: shows "(!!) ` jsys.language = Collect lv.sa.accept"
prime_nat_int_transfer: fixes n :: "nat" shows "prime (int n) = prime n"
comp_exp_nat: fixes i :: "'c"   and I :: "'c set"   and G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme"   and x :: "'c \<Rightarrow> 'd"   and k :: "nat" assumes "i \<in> I" shows "(x [^]\<^bsub>DirProds G I\<^esub> k) i = x i [^]\<^bsub>G i\<^esub> k"
lsided_fres: fixes x :: "'a" shows "lsd x = (\<top> \<le> x \<leftarrow> x)"
Cauchy_Schwarz_ineq: fixes x :: "'a"   and y :: "'a" shows "x \<bullet>\<^sub>C y * (y \<bullet>\<^sub>C x) \<le> x \<bullet>\<^sub>C x * (y \<bullet>\<^sub>C y)"
E: shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ y # ys)), inv p ` Y) \<in> failures P"
cosh_dist_perp: fixes l :: "proj2_line"   and m :: "proj2_line"   and a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "M_perp l m"   and "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "proj2_incident a l"   and "proj2_incident b l"   and "proj2_incident b m"   and "proj2_incident c m" shows "cosh_dist a c = cosh_dist b a * cosh_dist b c"
fiter2: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> ; c\<^sup>\<star> = c\<^sup>\<star>"
Fvars_Ls: shows "Fvars Ls = {xx, yy}"
gen_akiyama_tanigawa_row_eq_Nil_iff: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and l :: "nat"   and u :: "nat" shows "(gen_akiyama_tanigawa_row f n l u = []) = (u \<le> l)"
coset_mult_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier H'"   and "g \<in> carrier H'"   and "h \<in> carrier H'" shows "M #>\<^bsub>H'\<^esub> g #>\<^bsub>H'\<^esub> h = M #>\<^bsub>H'\<^esub> g \<otimes>\<^bsub>H'\<^esub> h"
lg_power: fixes x :: "nat"   and rs :: "nat" assumes "Suc 0 < x" shows "lg (x ^ rs) x = rs"
degenerate_path_is_valid_path: fixes c :: "'a" shows "valid_path (\<lambda>x. c)"
lpfp\<^sub>c_def: fixes f :: "'b option acom \<Rightarrow> 'b option acom"   and c :: "com" shows "lpfp\<^sub>c f c = pfp f (\<bottom>\<^sub>c c)"
span_base: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and S :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "a \<in> S" shows "a \<in> local.punit.pmdl S"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>) (\<star>) \<a> \<i> src trg UP cmp\<^sub>U\<^sub>P"
filter_inter: fixes Fa :: "'b set set"   and Ta :: "'b set set"   and F :: "'c set set"   and T :: "'c set set"   and A :: "'c set"   and B :: "'c set" assumes "Topology.filter Fa Ta"   and "F \<in> Filters\<^bsub>T\<^esub>"   and "A \<in> F"   and "B \<in> F" shows "A \<inter> B \<in> F"
case_eq_if: fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and stateT :: "('e, 'f) stateT" shows "(case stateT of StateT x \<Rightarrow> f x) = f (run_state stateT)"
bprv_\<phi>T_eqv: shows "bprv (eqv \<phi>T (neg (subst T \<langle>\<phi>T\<rangle> xx)))"
fps_of_poly_add: fixes p :: "'a poly"   and q :: "'a poly" shows "fps_of_poly (p + q) = fps_of_poly p + fps_of_poly q"
unitary_gen_scale: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex" assumes "unitary_gen M"   and "k \<noteq> 0" shows "unitary_gen (k *\<^sub>s\<^sub>m M)"
lens_plus_def: fixes X :: "'e \<Longrightarrow> 'g"   and Y :: "'f \<Longrightarrow> 'g" shows "X +\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. (get\<^bsub>X\<^esub> \<sigma>, get\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> (u, v). put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u\<rparr>"
homogeneous_monom_mult: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p" shows "homogeneous (MPoly_Type_Class.punit.monom_mult c t p)"
items_subset: fixes H :: "msg set" shows "H \<subseteq> items H"
atLeastAtMost_eq_UNIV_iff: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_lattice inf less_eq less sup bot top" shows "(ord.atLeastAtMost less_eq x y = UNIV) = (x = bot \<and> y = top)"
borderD_nemp: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x \<noteq> \<epsilon>"
im_sheaf_axioms: shows "Comm_Ring.im_sheaf Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec Spec is_zariski_open (identity Spec)"
division_unit_def: fixes a :: "'a" shows "division_unit a \<equiv> \<forall>x. x = inv_resr a x \<and> x = inv_resl x a"
grid_not_child: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir"   and ds :: "nat set" assumes "d < length p" shows "p \<notin> grid (child p dir d) ds"
weak_unit_self_left: fixes a :: "'a" assumes "weak_unit a" shows "left a a"
filter_code_3: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('a, 'b) mapping_rbt" shows "Mapping.filter P (RBT_Mapping t) = (case ID ccompare of None \<Rightarrow> Code.abort STR ''filter RBT_Mapping: ccompare = None'' (\<lambda>_. Mapping.filter P (RBT_Mapping t)) | Some x \<Rightarrow> RBT_Mapping (RBT_Mapping2.filter (\<lambda>(k, v). P k v) t))"
"0_1": shows "list_all stateful_strand_step.is_Receive (unlabel (transaction_receive T))"
assms_3: shows "(g has_derivative g') (at (f x) within f ` S)"
width_spec_def: fixes X :: "'c" shows "width_spec X = SPEC top"
lens_comp_quotient: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'd" assumes "weak_lens Y" shows "(X ;\<^sub>L Y) /\<^sub>L Y = X"
assocs_eqD: fixes b :: "'a"   and a :: "'a" assumes "assocs G b = assocs G a"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<sim> b"
isunit: fixes ba :: "'a" assumes "ba \<in> carrier G \<and> properfactor G ba (a \<otimes> b)" shows "ba \<in> Units G"
mset_pos_plus: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" shows "mset_pos (A + B) = mset_pos A - mset_neg B + (mset_pos B - mset_neg A)"
max_suml: shows "b \<bullet> ys \<le> maxne0 ys b * sum_list ys"
FactGroup_nonempty: fixes X :: "'a set" assumes "X \<in> carrier (G Mod kernel G H h)" shows "X \<noteq> {}"
IH_readable_2: shows "cnf G \<union> \<Union> (cnf ` \<Gamma>) \<turnstile> \<box>"
REMOVE_INTERNAL_def: fixes x :: "'b" shows "REMOVE_INTERNAL x \<equiv> x"
IFC_def: fixes v :: "'b"   and t :: "'b ifex"   and e :: "'b ifex" shows "IFC v t e \<equiv> if t = e then t else IF v t e"
distinct_18: fixes ccs' :: "ccs"   and act :: "act"   and ccs :: "ccs" shows "!ccs' \<noteq> act.ccs"
log_eq_div_ln_mult_log: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "log a x = real_ln b / real_ln a * log b x"
update_convs_2: fixes ps' :: "('a \<Rightarrow> 'e) \<Rightarrow> 'a \<Rightarrow> 'e"   and es :: "'c"   and ps :: "'a \<Rightarrow> 'e"   and pubActs :: "'b \<times> ('a \<Rightarrow> 'd)"   and more :: "'f" shows "ps_update ps' \<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr> = \<lparr>es = es, ps = ps' ps, pubActs = pubActs, \<dots> = more\<rparr>"
simps_1: fixes ks :: "bool list" shows "isinP LfP ks = False"
JF1col_Suc: fixes n :: "nat"   and x :: "'d JF1" shows "JF1col (Suc n) x = F1set1 (dtor1 x) \<union> (\<Union> (JF1col n ` F1set2 (dtor1 x)) \<union> \<Union> (JF2col n ` F1set3 (dtor1 x)))"
is_span: shows "C.span H\<nu>H\<pi>\<rho>.leg0 H\<nu>H\<pi>\<rho>.leg1"
beforex_appendD1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt\<^sub>1 :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and xt\<^sub>2 :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and f :: "nat"   and t :: "nat"   and D :: "char list"   and h :: "nat"   and d :: "nat"   and I :: "nat set"   and J :: "nat set" assumes "P,C,M \<rhd> xt\<^sub>1 @ xt\<^sub>2 @ [(f, t, D, h, d)] / I,d"   and "pcs xt\<^sub>1 \<subseteq> J"   and "J \<subseteq> I"   and "J \<inter> pcs xt\<^sub>2 = {}" shows "P,C,M \<rhd> xt\<^sub>1 / J,d"
simps_1: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b" shows "Partial_Cost_Model.config' A s [] = s"
insert_ops_def: fixes list :: "('c \<times> 'c option) list" shows "insert_ops list \<equiv> spec_ops list set_option"
ranking_stuck_odd: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and f :: "nat \<times> 'b \<Rightarrow> nat"   and v :: "nat \<times> 'b"   and r :: "(nat \<times> 'b) stream" assumes "ranking A w f"   and "v \<in> gunodes A w"   and "gurun A w r v" obtains n :: "nat" where "\<forall>a\<in>sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))). odd a"
inducts: fixes P :: "freeRes \<Rightarrow> bool"   and freeRes :: "freeRes" assumes "\<And>name1 name2. P (name1[name2])"   and "P (\<tau>)" shows "P freeRes"
\<pi>_at_top: shows "filterlim primes_pi at_top at_top"
sup_compl_top: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (uminus x) = top"
b: shows "(\<not> X ! Suc y) = (\<not> zs ! y)"
simps_15: fixes s :: "nat \<Rightarrow> floatarith"   and a :: "floatarith"   and i :: "nat" shows "subst_floatarith s (a ^\<^sub>e i) = subst_floatarith s a ^\<^sub>e i"
length_map_dup: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "length (map_dup f g xs) = length xs"
gm_fin: shows "AE x in M. ennreal (f x) * nn_transfer_operator gm x \<noteq> \<infinity>"
Abs_triangle_inverse: fixes y :: "real list" assumes "y \<in> triangle" shows "Rep_triangle (Abs_triangle y) = y"
rb_aux_inv_invariant: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_inv (fst args)" shows "rb_aux_inv (fst (rb_aux args))"
smult_distrib_right: fixes r :: "('f, 'g) aezfun"   and s :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "s \<in> FG"   and "m \<in> V" shows "(r + s) \<cdot> m = r \<cdot> m + s \<cdot> m"
Sup_Set_fold: fixes t :: "('b, unit) RBT.rbt" shows "Sup (Set t) = (if RBT.is_empty t then bot else r_max_opt t)"
assms_1: shows "x * f $$ fls_subdegree f = (1::'a)"
in_new_spairsI: fixes b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b \<in> set bs"   and "is_regular_spair p b" shows "Inl (p, b) \<in> set (new_spairs bs p)"
arrI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "D.arr f"
intro: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c" assumes "Deduct2_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv"   and "Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S"   and "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P" shows "Goedel_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P"
i_2: shows "(n - 1) div 2 + 1 \<le> 2 ^ (i + 1)"
prems_2: shows "fmdom (xm(i \<mapsto>\<^sub>f x)) = fmdom ym"
S_in_E: fixes i :: "'i" assumes "i \<in> I" shows "range (S i) \<subseteq> E i"