hcju/isabelleps huggingface dataset:
small_image_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "small (f ` A) = small A"
Cons_eq_appendI: fixes x :: "'b"   and xs1 :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "x # xs1 = ys"   and "xs = xs1 \<cdot> zs" shows "x # xs = ys \<cdot> zs"
swap_pair_fun': fixes C :: "'a \<times> 'b \<Rightarrow> real \<times> real" shows "(\<lambda>y. prod.swap (C (y, 1::'b))) = (\<lambda>x. (C (x, 1::'b) \<bullet> j, C (x, 1::'b) \<bullet> i))"
a_d_closed: fixes x :: "'a" shows "d (ad x) = ad x"
POR_base_lf_rep_def: shows "POR_base_lf_rep \<equiv> \<lambda>(mR, pR). {\<lambda>i. ValTT} \<union> {\<lambda>i. ValFF} \<union> (\<Union>n. {\<lambda>i. ValN\<cdot>n}) \<union> {f. f One = \<bottom>} \<union> {f. f Two = \<bottom>}"
F1_3: shows "\<forall>t\<in>fst ` set (linearize data_in). t \<le> ot \<and> left I \<le> ot - t"
alg_path_equiv_implies_valid_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s \<leftrightarrow> t : T" shows "valid \<Gamma>"
gmft_correct: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.is_measured_ftree t" shows "FingerTreeStruc.gmft t = sum_list (map snd (FingerTreeStruc.toList t))"
"1_1": shows "v = v\<^sub>1 @ [a] @ v\<^sub>2"
dgrad_set_le_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d" shows "dgrad_set_le d (pp_of_term ` keys (monom_mult c t p)) (insert t (pp_of_term ` keys p))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltlr \<Rightarrow> 'b ltlr \<Rightarrow> 'a"   and f6 :: "'b ltlr \<Rightarrow> 'b ltlr \<Rightarrow> 'a"   and f7 :: "'b ltlr \<Rightarrow> 'a"   and f8 :: "'b ltlr \<Rightarrow> 'b ltlr \<Rightarrow> 'a"   and f9 :: "'b ltlr \<Rightarrow> 'b ltlr \<Rightarrow> 'a"   and ltlr :: "'b ltlr" shows "P (case ltlr of true\<^sub>r \<Rightarrow> f1 | false\<^sub>r \<Rightarrow> f2 | prop\<^sub>r(x) \<Rightarrow> f3 x | nprop\<^sub>r(x) \<Rightarrow> f4 x | x and\<^sub>r xa \<Rightarrow> f5 x xa | x or\<^sub>r xa \<Rightarrow> f6 x xa | X\<^sub>r x \<Rightarrow> f7 x | x U\<^sub>r xa \<Rightarrow> f8 x xa | x R\<^sub>r xa \<Rightarrow> f9 x xa) = ((ltlr = true\<^sub>r \<longrightarrow> P f1) \<and> (ltlr = false\<^sub>r \<longrightarrow> P f2) \<and> (\<forall>x3. ltlr = prop\<^sub>r(x3) \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. ltlr = nprop\<^sub>r(x4) \<longrightarrow> P (f4 x4)) \<and> (\<forall>x51 x52. ltlr = x51 and\<^sub>r x52 \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x61 x62. ltlr = x61 or\<^sub>r x62 \<longrightarrow> P (f6 x61 x62)) \<and> (\<forall>x7. ltlr = X\<^sub>r x7 \<longrightarrow> P (f7 x7)) \<and> (\<forall>x81 x82. ltlr = x81 U\<^sub>r x82 \<longrightarrow> P (f8 x81 x82)) \<and> (\<forall>x91 x92. ltlr = x91 R\<^sub>r x92 \<longrightarrow> P (f9 x91 x92)))"
lcs: shows "lcs = (\<Prod>(f, i)\<leftarrow>fs'. lead_coeff f ^ Suc i)"
is_asig_def: fixes triple :: "'b signature" shows "is_asig triple \<equiv> inputs triple \<inter> outputs triple = {} \<and> outputs triple \<inter> internals triple = {} \<and> inputs triple \<inter> internals triple = {}"
finsertE: fixes a :: "'b"   and b :: "'b"   and A :: "'b fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
delete_id: fixes k :: "'c"   and al :: "('c \<times> 'd) list" assumes "k \<notin> fst ` set al" shows "AList.delete k al = al"
isAbr_simps_1: fixes s :: "'d" shows "isAbr (Normal s) = False"
ftimes_def: fixes a :: "V"   and b :: "V" shows "a \<times>\<^sub>\<bullet> b \<equiv> \<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> a | otherwise \<Rightarrow> b"
gbm_invar_def: fixes m1 :: "'m1" shows "gbm_invar m1 \<equiv> m1.invar m1 \<and> (\<forall>m2\<in>ran (m1.\<alpha> m1). m2.invar m2 \<and> Ball (ran (m2.\<alpha> m2)) s3.invar) \<and> valid_graph (gbm_\<alpha> m1)"
lfp_lowerbound: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'c \<Rightarrow> 'c"   and A :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq (f A) A" shows "less_eq (complete_lattice.lfp Inf less_eq f) A"
exec_gpv_invariant: fixes x :: "'c"   and s' :: "'b \<Rightarrow> bool list option"   and gpv :: "('c, 'b, bool list) gpv"   and s :: "'b \<Rightarrow> bool list option" assumes "(x, s') \<in> set_spmf (exec_gpv oracle gpv s)"   and "dom \<sigma> \<subseteq> dom s"   and "\<I>_full \<turnstile>g gpv \<surd>" shows "dom \<sigma> \<subseteq> dom s'"
h_split: shows "rev u @ h = [] \<or> rev u @ h = [x] \<or> (\<exists>hs. rev u @ h = [x, x] @ hs)"
DirProds_in_carrI: fixes I :: "'e set"   and x :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> carrier (G i)"   and "\<And>i. i \<notin> I \<Longrightarrow> x i = undefined" shows "x \<in> carrier (DirProds G I)"
chain3: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q"   and "a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c" shows "ch {a, b, c}"
measurable_comp: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'d \<Rightarrow> 'e"   and L :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "g \<circ> f \<in> M \<rightarrow>\<^sub>M L"
fun_def: shows "fun \<equiv> \<lambda>p b. if b then snd p else fst p"
sin_0_iff_canon: fixes \<phi> :: "real" assumes "sin \<phi> = 0"   and "- pi < \<phi>"   and "\<phi> \<le> pi" shows "\<phi> = 0 \<or> \<phi> = pi"
exhaust: fixes y :: "'b flat_complete_lattice" obtains x1 :: "'b" where "y = flat_complete_lattice.Value x1"   | "y = flat_complete_lattice.Bot"   | "y = flat_complete_lattice.Top"
ratfps_lcm: fixes f :: "'b ratfps"   and g :: "'b ratfps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "lcm f g = ratfps_of_poly (monom (1::'b) (max (ratfps_subdegree f) (ratfps_subdegree g)))"
prems: shows "generalized_sfw (generalized_fw_join (a # as) f2) p = Some (u, d1, d2)"
div_seq: fixes list :: "('b com option \<times> 'b) list"   and P :: "'b com"   and Q :: "'b com"   and s :: "'b"   and zs :: "('b com option \<times> 'b) list" assumes "list \<in> cptn_mod"   and "list = (Some (Seq P Q), s) # zs" shows "\<exists>xs. (Some P, s) # xs \<in> cptn_mod \<and> (zs = map (lift Q) xs \<or> fst (((Some P, s) # xs) ! length xs) = None \<and> (\<exists>ys. (Some Q, snd (((Some P, s) # xs) ! length xs)) # ys \<in> cptn_mod \<and> zs = map (lift Q) xs @ ys))"
of_nat_le_\<omega>: fixes n :: "nat" shows "of_nat n \<le> \<omega>"
elims_2: fixes x :: "('form \<times> nat list) \<times> ('form, 'var) out_port" assumes "valid_out_port x" obtains v :: "'form \<times> nat list" and p :: "('form, 'var) out_port" where "x = (v, p)" and "v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v)"
assms: shows "P \<rightarrow>\<^sub>S \<alpha>\<^sub>SP'"
2: shows "one_chain_line_integral F basis (one_chain2 - ps2) = one_chain_line_integral F basis subdiv"
map_poly_pCons: fixes c :: "'a"   and p :: "'a poly"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'a) \<or> p \<noteq> 0" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
D_last_index_compE\<^sub>1: fixes A :: "char list set"   and Vs :: "char list list"   and e :: "char list exp" assumes "A \<subseteq> set Vs"   and "fv e \<subseteq> set Vs"   and "\<D> e \<lfloor>A\<rfloor>" shows "\<D> (compE\<^sub>1 Vs e) \<lfloor>last_index Vs ` A\<rfloor>"
proj_arcs_H_simps_1: shows "proj_arcs_H uw = uv"
IH: shows "real (T_splay x lb) + \<Phi> lb1 + \<Phi> lb2 - \<Phi> lb \<le> 2 * \<phi> lb - 3 * \<phi> X + 1"
intro: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and H :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>H :: "'s \<times> 's \<Rightarrow> 't"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" assumes "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"   and "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H"   and "pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1"
induct: fixes x1 :: "'a com"   and x2 :: "'a option"   and x3 :: "'a option"   and P :: "'a com \<Rightarrow> 'a option \<Rightarrow> 'a option \<Rightarrow> bool" assumes "Sem x1 x2 x3"   and "\<And>f. P (Basic f) None None"   and "\<And>f s. P (Basic f) (Some s) (Some (f s))"   and "\<And>s. P Abort s None"   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>b c1 c2. P (Cond b c1 c2) None None"   and "\<And>s b c1 s' c2. \<lbrakk>s \<in> b; Sem c1 (Some s) s'; P c1 (Some s) s'\<rbrakk> \<Longrightarrow> P (Cond b c1 c2) (Some s) s'"   and "\<And>s b c2 s' c1. \<lbrakk>s \<notin> b; Sem c2 (Some s) s'; P c2 (Some s) s'\<rbrakk> \<Longrightarrow> P (Cond b c1 c2) (Some s) s'"   and "\<And>b x y c. P (While b x y c) None None"   and "\<And>s b x y c. s \<notin> b \<Longrightarrow> P (While b x y c) (Some s) (Some s)"   and "\<And>s b c s'' x y s'. \<lbrakk>s \<in> b; Sem c (Some s) s''; P c (Some 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) (Some s) s'" shows "P x1 x2 x3"
gr_implies_gr0: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "0 < n"
assms_1: shows "(Q -* R) * F \<Longrightarrow>\<^sub>A S"
distinct_209: fixes x81 :: "floatarith"   and x82 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Max x81 x82 \<noteq> x151 ^\<^sub>e x152"
geq: fixes z :: "complex" assumes "z \<in> S" shows "f z = (g z)\<^sup>2"
rep_eq: fixes x :: "complex" shows "cblinfun_apply (cblinfun_scaleC_right x) = (*\<^sub>C) x"
coprime_set_non_empty: fixes prime_set :: "nat set"   and N :: "nat" assumes "rsa_base prime_set"   and "2 < N" shows "coprime_set N \<noteq> {}"
bit_of_nat_iff_bit: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "bit (semiring_1.of_nat one plus zero m) n = bit m n"
induct: fixes P :: "aform \<Rightarrow> bool"   and a0 :: "aform" assumes "\<And>s t. P (Lt s t)"   and "\<And>s t. P (Eq s t)"   and "\<And>s t. P (Ge s t)"   and "\<And>s t. P (NEq s t)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Conj p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Disj p q)"   and "P (NEG T)"   and "P (NEG aform.F)"   and "\<And>a b. P (NEG (Lt a b))"   and "\<And>a b. P (NEG (Ge a b))"   and "\<And>a b. P (NEG (Eq a b))"   and "\<And>a b. P (NEG (NEq a b))"   and "\<And>p. P p \<Longrightarrow> P (NEG (NEG p))"   and "\<And>p q. \<lbrakk>P (NEG p); P (NEG q)\<rbrakk> \<Longrightarrow> P (NEG (Conj p q))"   and "\<And>p q. \<lbrakk>P (NEG p); P (NEG q)\<rbrakk> \<Longrightarrow> P (NEG (Disj p q))"   and "P T"   and "P aform.F" shows "P a0"
HEndPhase0_Inv2a_allBlocksRead: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv2a_inner s p"   and "Inv2c_inner s p" shows "\<forall>t\<in>mbal_update (\<lambda>_. mbal (dblock s' p)) ` allBlocksRead s p. Inv2a_innermost s p t"
aux_ack_t2: fixes k :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and t :: "nat"   and t2 :: "nat" assumes "\<forall>m\<le>k. ack (Suc (Suc (t + m))) = [connection_ok]"   and "Suc (Suc t) < t2"   and "t2 < t + 3 + k" shows "ack t2 = [connection_ok]"
dim_span: fixes S :: "'a vec set"   and U :: "'a vec set" assumes "S \<subseteq> carrier_vec n"   and "finite S"   and "maximal U (\<lambda>T. T \<subseteq> S \<and> lin_indpt T)" shows "vectorspace.dim class_ring (vs (local.span S)) = card U"
transitionWeakCongTransition: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
fresh_fin_insert: fixes X :: "'a set"   and a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X" shows "a \<sharp> insert x X = (a \<sharp> x \<and> a \<sharp> X)"
other: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<zeta> :: "'b"   and R :: "nat set"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"   and "U \<sigma> \<sigma>'"   and "P (\<sigma>, NodeS ii \<zeta> R)" shows "P (\<sigma>', NodeS ii \<zeta> R)"
evenperm_inv: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "evenperm (inv' p) = evenperm p"
11: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e * (v * v\<^sup>T \<sqinter> g) = bot"
4: fixes ys :: "nat list"   and ba :: "nat list"   and x :: "nat"   and xs :: "nat list"   and s :: "nat"   and x' :: "nat"   and s' :: "nat" assumes "ys \<in> fst ` set (c2.gen_check ba)"   and "cond1 ba ys (x # xs) s"   and "x' \<le> x"   and "s' \<le> s" shows "cond1 ba ys (x' # xs) s'"
exp_ordinal_def: shows "(**) = (\<lambda>x. if 0 < x then ordinal_rec 1 (\<lambda>p w. w * x) else (\<lambda>y. if y = 0 then 1 else 0))"
ennreal_eq_1: fixes x :: "real" shows "(ennreal x = 1) = (x = 1)"
comm_monoid_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_mult_hom hom \<equiv> monoid_mult_hom hom"
ferrack: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" assumes "qfree p" shows "qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)"
noeffect: shows "swaps (snd (Strat ! x)) (Partial_Cost_Model.steps init (take x qs) (take x Strat)) = swaps (take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) (Partial_Cost_Model.steps init (take x qs) (take x Strat))"
assms_2: fixes s :: "'a" assumes "Q s" shows "R s"
choose_mult: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m"   and "m \<le> n" shows "(n choose m) * (m choose k) = (n choose k) * (n - k choose (m - k))"
intros_1: fixes P :: "'b prog"   and T :: "ty" shows "P \<turnstile> T \<le> T"
strengthen_step_2: fixes rv :: "'a \<times> nat" assumes "rv \<in> red_vertices prb"   and "finite (pred (init_conf prb))"   and "\<forall>e\<in>pred (init_conf prb). finite (Bexp.vars e)"   and "finite_lts (black prb)" shows "feasible_subpaths_from (black prb) (confs prb rv) (fst rv) \<subseteq> RedBlack_subpaths_from prb rv"
prems: fixes k :: "nat" assumes "k < Suc (Suc n)" shows "f k \<in> carrier_mat d d"
ivl_cong: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "sum g {a..<b} = sum h {c..<d}"
bexI: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and A :: "'f set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
cset_seq_def: fixes A :: "'c cset"   and i :: "nat" shows "cset_seq A i = (if i \<in> range (cset_count A) \<and> inv_into (rcset A) (cset_count A) i \<in>\<^sub>c A then Some (inv_into (rcset A) (cset_count A) i) else None)"
add_cancel_right_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
degree_zero: fixes v :: "nat" shows "MPoly_Type.degree 0 v = 0"
right_unique_rel_set_lemma: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and X :: "'e set"   and Y :: "'f set" assumes "right_unique R"   and "rel_set R X Y" obtains f :: "'e \<Rightarrow> 'f" where "Y = f ` X" and "\<forall>x\<in>X. R x (f x)"
l_1: shows "(SUP i. integral\<^sup>S M (l i)) = \<integral>\<^sup>+ x. (SUP i. l i x) \<partial>M"
heap_is_wellformed_children_disc_nodes: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "node_ptr |\<in>| node_ptr_kinds h"   and "\<not> (\<exists>parent\<in>fset (object_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_child_nodes parent|\<^sub>r)" shows "\<exists>document_ptr\<in>fset (document_ptr_kinds h). node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r"
le_numeral_extra_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
map_comp_empty_1: fixes m :: "'e \<Rightarrow> 'd option" shows "m \<circ>\<^sub>m Map.empty = Map.empty"
get_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_tag_name element_ptr"
assms_2: fixes r :: "real"   and x :: "'a" shows "f (r *\<^sub>R x) = r *\<^sub>R f x"
relH_dist_union: fixes as :: "nat set"   and as' :: "nat set"   and h :: "heap"   and h' :: "heap" shows "relH (as \<union> as') h h' = (relH as h h' \<and> relH as' h h')"
simps_36: fixes x4 :: "('a, 'b, 'c) expr"   and x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Ite x31 x32 x33"
assms_6: shows "\<forall>x\<in>A. \<forall>y\<in>A. ((f x, f y) \<in> s) = ((x, y) \<in> r)"
simps: fixes P :: "(('a \<times> 'c \<times> 'a) set \<times> ('a \<times> 'c \<times> 'a) set) set" shows "local.combine_pairs P = (\<Union> (fst ` P), snd ` P)"
add_decreasing2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq c zero"   and "less_eq a b" shows "less_eq (plus a c) b"
bisim1Sync3: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and v :: "'addr val"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (sync\<^bsub>V\<^esub> (Val v) e2, xs) \<leftrightarrow> ([v], xs[V := v], Suc (Suc (length (compE2 e1))), None)"
simps_3: fixes l :: "nat"   and xs :: "nat option list"   and zs :: "nat list"   and i :: "nat"   and T :: "bool list list" shows "mk_eqcl (Some l # xs) zs i T = (let (xs', y) = mk_eqcl xs zs (Suc i) T in (l # xs', y))"
real_root_ge_0_iff: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "(0 \<le> root n y) = (0 \<le> y)"
path_split2: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and \<beta> :: "'b list"   and \<alpha> :: "'b list"   and s :: "'a" assumes "Unwinding SES \<V>"   and "s0\<^bsub>SES\<^esub> (\<beta> @ \<alpha>)\<Longrightarrow>\<^bsub>SES\<^esub> s" shows "\<exists>s'\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' \<alpha>\<Longrightarrow>\<^bsub>SES\<^esub> s \<and> reachable SES s'"
simps: fixes e :: "'b DBMEntry"   and t :: "'b" shows "norm_lower e t = (if e \<prec> Lt t then Lt t else e)"
induct: fixes P :: "('a \<times> 'a \<times> 'a poly) list \<Rightarrow> bool"   and a0 :: "('a \<times> 'a \<times> 'a poly) list" assumes "\<And>e1 e2 es. P (local.neville_aitken_merge (e1 # e2 # es)) \<Longrightarrow> P (e1 # e2 # es)"   and "\<And>uu_ uv_ p. P [(uu_, uv_, p)]"   and "P []" shows "P a0"
le_list_length: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<preceq> ys" shows "length xs \<le> length ys"
P_f: fixes c :: "'a acom" assumes "P c" shows "P (c \<triangle>\<^sub>c f c)"
smult_four_block_mat: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat"   and a :: "'b" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "a \<cdot>\<^sub>m four_block_mat A B C D = four_block_mat (a \<cdot>\<^sub>m A) (a \<cdot>\<^sub>m B) (a \<cdot>\<^sub>m C) (a \<cdot>\<^sub>m D)"
ltakeWhile_trans_fusion: fixes P :: "'c \<Rightarrow> bool"   and g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lunstream (takeWhile_raw P g) s = ltakeWhile P (lunstream g s)"
simps_7: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
P'''Trans: shows "P''' \<longmapsto> \<alpha> \<prec> P''"
finite: fixes m :: "('a, 'b) assoc_list" assumes "lm.invar m" shows "finite (dom (lm.\<alpha> m))"
hseq_char': fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" shows "hseq \<nu> \<mu> = (\<nu> \<star> \<mu> \<noteq> null)"