hcju/isabelleps huggingface dataset:
P2_ideal_model_end_def: fixes M :: "'a \<times> 'a"   and l :: "'a"   and bs :: "('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'e"   and \<A>3 :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'e \<Rightarrow> 'd spmf" shows "P2_ideal_model_end M l bs \<A>3 = (let (x0, x1) = M; ((h0, h1, a, b0, b1), s) = bs; \<sigma>' = if l = \<one> then False else True in mal_def.trusted_party (x0, x1) \<sigma>' \<bind> (\<lambda>(uu_, x\<sigma>). sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> (if \<sigma>' then \<one> else x\<sigma>); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> (if \<sigma>' then x\<sigma> else \<one>); e0 = (w0, z0); e1 = (w1, z1) in \<A>3 e0 e1 s \<bind> (\<lambda>out. return_spmf ((), out))))))))"
if_semi1: fixes c' :: "com"   and s :: "loc \<Rightarrow> nat"   and t :: "loc \<Rightarrow> nat"   and e :: "exp"   and c1 :: "com"   and c2 :: "com"   and c :: "com" assumes "(c', s) -[eval]-> t"   and "c' = IF e THEN c1 ELSE c2;;c" shows "(IF e THEN c1;;c ELSE c2;;c, s) -[eval]-> t"
inner_right_distrib: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
acyclic_down_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "pd_kleene_allegory_class.acyclic y" shows "pd_kleene_allegory_class.acyclic x"
sigma_UNIV: fixes A :: "'b set set" shows "UNIV \<in> sigma A"
Un_empty_right: fixes A :: "'e set" shows "A \<union> {} = A"
simps_1: fixes w\<^sub>c :: "nat"   and c :: "'a"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "swapLeaves (Leaf w\<^sub>c c) w\<^sub>a a w\<^sub>b b = (if c = a then Leaf w\<^sub>b b else if c = b then Leaf w\<^sub>a a else Leaf w\<^sub>c c)"
thd: fixes x :: "'a"   and y :: "'a" shows "dist (if x = (0::'a) then 0::'a else norm x *\<^sub>R f (x /\<^sub>R norm x)) (if y = (0::'a) then 0::'a else norm y *\<^sub>R f (y /\<^sub>R norm y)) = dist x y"
eqButGH_def: fixes i :: "nat"   and i' :: "nat" shows "eqButGH i i' \<equiv> scon {eqOnUC i i' u c |u c. (u, c) \<in> (UNIV - GH) \<times> UNIV}"
dvd_trans: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times b c" shows "dvd.dvd times a c"
cong_3421: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong C D B A"
False: shows "\<not> eNorm (\<LL> p M) f \<noteq> \<infinity>"
Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "Sup A \<le> z"
Suc_2: fixes j :: "nat" assumes "Suc i < length (seq j)" shows "thesis"
cases: fixes \<delta> :: "('a, 'b) ta_rule \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" assumes "f_succp \<delta> a1 a2" obtains q :: "'a" and l :: "'b" and qs :: "'a list" and q' :: "'a" where "a1 = q" and "a2 = q'" and "\<delta> q \<rightarrow> l qs" and "q' \<in> set qs"
intro: fixes S :: "<'d, 'f> \<Longleftrightarrow> 'e" assumes "vwb_lens \<V>\<^bsub>S\<^esub>"   and "vwb_lens \<C>\<^bsub>S\<^esub>"   and "\<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub>"   and "bij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)" shows "sym_lens S"
in_orthogonal_complement_span: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n" shows "orthogonal_complement (local.span S) = orthogonal_complement S"
vc_foreach_interrupted_imp_post: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v"   and it :: "'v set"   and V' :: "'v set" assumes "fe_inv S V v it (V', True)" shows "rpost S (V, v) (V', True)"
nprv_def: fixes F :: "'fmla set"   and \<phi> :: "'fmla" shows "nprv F \<phi> \<equiv> prv (imp (scnj F) \<phi>)"
trans: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "comparator comp"   and "comp x y = Lt"   and "comp y z = Lt" shows "comp x z = Lt"
reduction_to_dis_log: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "chaum_ped_\<Sigma>_commit.rel_advantage \<A> = local.dis_log.advantage (local.dis_log_alt.adversary3 \<A>)"
vector_space_on_alt_def: fixes U\<^sub>V\<^sub>S :: "'d set" shows "vector_space_on U\<^sub>V\<^sub>S = module_on U\<^sub>V\<^sub>S"
simps_2: fixes k :: "nat" shows "ket_plus_k (Suc k) = partial_state2.ptensor_vec dims {0..<Suc k} {Suc k} (ket_plus_k k) ket_plus"
MClkInit_def: fixes rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkInit rcv cst p = (cst!p = #clkA \<and> \<not> Calling rcv p)"
refl_on_init_seg_of: fixes r :: "('a \<times> 'a) set" shows "r initial_segment_of r"
eval_def: fixes pred :: "'a pred" shows "eval pred = (case pred of Pred x \<Rightarrow> x)"
le_imp_inv: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.add_linorder plus minus zero less_eq less"   and "less_eq a b" shows "b = plus a (minus b a)"
ts_process_def: fixes T :: "'a list set" shows "ts_process T \<equiv> Abs_process ({(xs, X). xs \<in> T \<and> (\<forall>x\<in>X. xs @ [x] \<notin> T)}, {})"
map\<^sub>0_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "horizontal_homs.obj C src\<^sub>C a" shows "horizontal_homs.obj D src\<^sub>D (weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F a)"
eval_Body: fixes G :: "prog"   and s0 :: "st"   and D :: "qtname"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and res :: "val"   and s3 :: "state"   and s4 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>Init D\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"   and "res = the (locals (snd s2) Result)"   and "s3 = (if \<exists>l. fst s2 = Some (Jump (Break l)) \<or> fst s2 = Some (Jump (Cont l)) then abupd (\<lambda>x. Some (Error CrossMethodJump)) s2 else s2)"   and "s4 = abupd (absorb Ret) s3" shows "G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>res\<rightarrow> s4"
in_orbit_self: fixes a :: "'b"   and f :: "'b perm" shows "a \<in> orbit f a"
neutral_const: fixes A :: "'c set" shows "(\<Prod>_\<in>A. (1::'b)) = (1::'b)"
UNIF_range_vars_subset: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF ss E E'" shows "range_vars (compose ss) \<subseteq> vars_mset E"
Pr_def: shows "Pr = (\<lambda>x. {p. prime p \<and> real p \<le> x})"
evaln_elim_cases_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains "v = undefined3 t" and "s' = (Some xc, s)"
simps_3: fixes x1 :: "'b"   and x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "Var x1 \<noteq> Fun x21 x22"
C_def: shows "C = (\<lambda>t. {x \<in> B. t \<le> f x})"
tendsto_diff_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'a filter"   and g :: "'a \<Rightarrow> ennreal"   and y :: "ennreal" assumes "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F"   and "x \<noteq> \<top>"   and "y \<noteq> \<top>" shows "((\<lambda>z. f z - g z) \<longlongrightarrow> x - y) F"
strip_anno: fixes a :: "'c"   and c :: "com" shows "strip (anno a c) = c"
span_vertical_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "span_vertical_category C \<equiv> category C"
elims_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a option"   and xa :: "'a list"   and xb :: "'a list" assumes "\<not> proper_intrvl.set_less_eq_aux_Compl less proper_interval x xa xb" obtains ao :: "'a option" and xc :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = ao" and "xa = xc # xs" and "xb = y # ys" and "\<not> (if less xc y then proper_interval ao (Some xc) \<or> proper_intrvl.set_less_eq_aux_Compl less proper_interval (Some xc) xs (y # ys) else if less y xc then proper_interval ao (Some y) \<or> proper_intrvl.set_less_eq_aux_Compl less proper_interval (Some y) (xc # xs) ys else proper_interval ao (Some y))"
reachable_nodes_step'_2: fixes u :: "'b"   and g :: "'b ugraph"   and r :: "'b"   and v :: "'b" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges (component_of g r)"
coprime_minus_right_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "algebraic_semidom.coprime times one a (uminus b) = algebraic_semidom.coprime times one a b"
exec_gpv_inline: fixes callee :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and gpv :: "'b \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'b, 'd, 'e) gpv"   and c' :: "('a, 'f, 'g) gpv"   and s' :: "'b"   and s :: "'c" shows "exec_gpv callee (inline gpv c' s') s = map_spmf (\<lambda>(x, s', s). ((x, s'), s)) (exec_gpv (\<lambda>(s', s) y. map_spmf (\<lambda>((x, s'), s). (x, s', s)) (exec_gpv callee (gpv s' y) s)) c' (s', s))"
intro: fixes G :: "('e, 'f) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid_axioms G"
strict_prefixI: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys"   and "xs \<noteq> ys" shows "strict_prefix xs ys"
shuffle_to_top_Total: fixes Y :: "'x set"   and d :: "'d" assumes "inj_on Xd Y" shows "Total (shuffle_to_top Y d)"
unit_def: fixes a :: "'b" shows "HoF.unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : HoF.map\<^sub>0 a \<Rightarrow>\<^sub>D HoF.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[HoF.map\<^sub>0 a] = (HoF.map \<i>\<^sub>B[a] \<cdot>\<^sub>D HoF.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1"
count_digit_correct: fixes d :: "('c, 'd) Digit" shows "count_digit d = length (digitToList d)"
compl_mono: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq x y" shows "less_eq (uminus y) (uminus x)"
order_linear: fixes a :: "'b" shows "order a [:- a, 1::'b:] = Suc 0"
Sup_bot_conv_1: fixes A :: "'al set" assumes "A \<subseteq> U" shows "(\<Squnion>\<^sub>o\<^sub>w A = \<bottom>\<^sub>o\<^sub>w) = (\<forall>x\<in>A. x = \<bottom>\<^sub>o\<^sub>w)"
simps_10: fixes i :: "String.literal"   and x :: "String.literal"   and p :: "fol fm" shows "free_in i (Forall x p) = (i \<noteq> x \<and> free_in i p)"
upd_cs_empty: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.upd_cs kind cs [] = []" shows "cs = []"
pth_4_2: fixes c :: "real" shows "c *\<^sub>R (0::'c) = (0::'c)"
0: shows "prv (imp (sdsj G) (imp (scnj F) \<chi>))"
cases: fixes x :: "exetheory" obtains \<Sigma> :: "exesignature" and ax :: "Core.term list" where "x = ExeTheory \<Sigma> ax"
scalar_prod_minus_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>3 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 - v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 - v\<^sub>1 \<bullet> v\<^sub>3"
treap_rotate1: fixes l2 :: "('a \<times> 'b) tree"   and r2 :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree"   and p1 :: "'b"   and p2 :: "'b"   and k :: "'a"   and k1 :: "'a"   and p :: "'b"   and l :: "('a \<times> 'b) tree"   and k2 :: "'a" assumes "treap l2"   and "treap r2"   and "treap r"   and "\<not> p1 \<le> p2"   and "k < k1"   and "ins k p l = \<langle>l2, (k2, p2), r2\<rangle>"   and "treap (ins k p l)"   and "treap \<langle>l, (k1, p1), r\<rangle>" shows "treap \<langle>l2, (k2, p2), \<langle>r2, (k1, p1), r\<rangle>\<rangle>"
PrimRec1_def: shows "PrimRec1 \<equiv> {x. PrimRec1p x}"
simps_1: fixes z :: "gauss_int" shows "ReZ (gauss_cnj z) = ReZ z"
is_suffix_def: fixes u :: "'b list"   and v :: "'b list" shows "is_suffix u v = (\<exists>w. w @ u = v)"
intros_3: fixes t :: "('b, 'c, 'd, 'e) Tree"   and f :: "'e"   and \<alpha> :: "'d" shows "(t, tAct f \<alpha> t) \<in> Tree_wf"
column_add_row_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and n :: "'c"   and m :: "'c"   and q :: "'b"   and i :: "'d" shows "column_add_row A n m q i = (\<chi>j. if j = n then A $h i $h n + A $h i $h m * q else A $h i $h j)"
subst_pp_cong: fixes s :: "'b \<Rightarrow>\<^sub>0 nat"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and g :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "s = t"   and "\<And>x. x \<in> keys t \<Longrightarrow> f x = g x" shows "subst_pp f s = subst_pp g t"
unsat: shows "\<not> satisfiable (\<G>_Fset (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)))"
example_nodes_1: shows "nodes M\<^sub>S = {0, 1}"
simps_4: fixes \<pi> :: "char list \<Rightarrow> com option"   and l :: "char list \<Rightarrow> int \<Rightarrow> int"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (Assign_Locals l, s) = Some (SKIP, <l|s>)"
norm_diff_triangle_ineq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (plus a b) (plus c d)) \<le> norm (minus a c) + norm (minus b d)"
phiCount_def: fixes g :: "'g" shows "phiCount g = card ((\<lambda>(n, v). (n, var g v)) ` dom (phis g))"
as_needed_asses_submap_exec_viii: fixes f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "f \<subseteq>\<^sub>f g" shows "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v"
Abs_nat_term_order_inverse: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "y \<in> {cmp. comparator cmp \<and> nat_term_comp cmp}" shows "nat_term_compare (Abs_nat_term_order y) = y"
g_ab: shows "g b - g a = (b - a) * g' \<xi>"
cdis: shows "d\<inverse> O s \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
fls_lr_inverse_minus_2: fixes f :: "'a fls"   and x :: "'a" shows "fls_right_inverse (- f) (- x) = - fls_right_inverse f x"
fps_of_poly_smult: fixes c :: "'b"   and p :: "'b poly" shows "fps_of_poly (smult c p) = fps_const c * fps_of_poly p"
simps_2: fixes t :: "'a FOL_Fitting.term"   and ts :: "'a FOL_Fitting.term list" shows "paramsts (t # ts) = paramst t \<union> paramsts ts"
atomics_def: fixes as :: "'b list"   and i :: "nat"   and j :: "nat" shows "atomics as i j \<equiv> atomics' as i j 1"
eqvt_2: fixes x1 :: "binding list"   and x2 :: "ty"   and x3 :: "ty"   and pi :: "(tyvrs \<times> tyvrs) list" assumes "x1\<turnstile>x2<:x3" shows "(pi \<bullet> x1)\<turnstile>(pi \<bullet> x2)<:(pi \<bullet> x3)"
product_character: fixes x :: "i"   and y :: "i" assumes "CM"   and "O x y" shows "\<forall>w. P w (x \<otimes> y) = (P w x \<and> P w y)"
productive_not_empty: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "C \<noteq> {#}"
lookup_a1_map_keys: fixes xs :: "event_data option list"   and tp' :: "nat" assumes "Mapping.lookup a1_map xs = Some tp'" shows "tp' < tp"
not_dom_eq_not_sdom: fixes n1 :: "nat"   and n2 :: "nat" assumes "\<not> dominate n1 n2" shows "\<not> strict_dominate n1 n2"
assms_3: shows "u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k)"
vec1: fixes i :: "nat"   and nr :: "nat"   and ze :: "'b"   and on :: "'b" assumes "i < nr" shows "vec nr (vec1I ze on nr i)"
a2c_c2a_id: shows "a2c oo c2a = ID"
Id_on_set: fixes xs :: "'b list" shows "Id_on (set xs) = set (map (\<lambda>x. (x, x)) xs)"
of_nat_prod: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> nat"   and A :: "'c set" assumes "class.comm_semiring_1 times one plus zero" shows "semiring_1.of_nat one plus zero (prod f A) = comm_monoid_mult.prod times one (\<lambda>x. semiring_1.of_nat one plus zero (f x)) A"
fprod_cl1: shows "ideal (finprod (ideals_set R) I {Suc 0..Suc n}) R"
n_def: shows "n = nat (max 0 (max (- fls_subdegree f) (- fls_subdegree g)))"
etran_or_ctran2: fixes i :: "nat"   and x :: "('b com option \<times> 'b) list" assumes "Suc i < length x"   and "x \<in> cptn" shows "(x ! i -c\<rightarrow> x ! Suc i \<longrightarrow> (x ! i, x ! Suc i) \<notin> etran) \<or> (x ! i -e\<rightarrow> x ! Suc i \<longrightarrow> (x ! i, x ! Suc i) \<notin> ctran)"
array_rel_alt: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>array_rel = {(Array l, l) |l. True} O \<langle>R\<rangle>list_rel O {(l, Array l) |l. True}"
extra: shows "\<forall>f\<in>U. \<forall>n. extra U f n (\<psi> (the (s (f \<triangleright> n))))"
nsetnm_sub_mem: fixes n :: "nat"   and m :: "nat" shows "\<forall>j. j \<in> nset n (n + m) \<longrightarrow> j \<in> nset n (Suc (n + m))"
obs_r': shows "action_obs E r' = NormalAction (ReadMem ad al v)"
iMinI2_ex2: fixes I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "I \<noteq> {}"   and "\<And>x. x \<in> I \<Longrightarrow> P x" shows "P (iMin I)"
drop_bit_or: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "drop_bit n (or a b) = or (drop_bit n a) (drop_bit n b)"
iso_def: fixes f :: "('a E.term, 'a) arr" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
prems: shows "map Val (v' # vs') = map Val ws @ throw ex # es"
nat_neq_iff: fixes m :: "nat"   and n :: "nat" shows "(m \<noteq> n) = (m \<sqsubset> n \<or> n \<sqsubset> m)"
f: shows "f \<in> M \<rightarrow>\<^sub>M M'"
strategyproofness: shows "(pmf (sds R31) b + pmf (sds R31) d < pmf (sds R38) d + pmf (sds R38) b \<or> pmf (sds R31) b + (pmf (sds R31) d + pmf (sds R31) c) < pmf (sds R38) d + (pmf (sds R38) b + pmf (sds R38) a)) \<or> pmf (sds R31) b + pmf (sds R31) d = pmf (sds R38) d + pmf (sds R38) b \<and> pmf (sds R31) b + (pmf (sds R31) d + pmf (sds R31) c) = pmf (sds R38) d + (pmf (sds R38) b + pmf (sds R38) a)"
shortestPath_predecessor: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g) - {Entry g}"   and "invar g" obtains n' :: "'node" where "Suc (shortestPath g n') = shortestPath g n" and "n' \<in> set (predecessors g n)"
WHILE: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "BExpr_low b"   and "G \<rhd> c : Sec \<Phi>" shows "G \<rhd> While b c : Sec (PhiWhile b \<Phi>)"
f: shows "f \<in> A \<rightarrow> A \<rightarrow> A"
hyp: shows "column (mod_type_class.from_nat i) (Gram_Schmidt_upt_k A k) = column (mod_type_class.from_nat i) A"
ip_and_mask8_bl_drop24: shows "ip && mask 8 = of_bl (drop 24 (to_bl ip))"
simps_1: shows "extractFrame (\<zero>) = \<bottom>\<^sub>F"
bisimilar_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.bisimilar satisfies transition P Q" shows "nominal_ts.bisimilar satisfies transition (p \<bullet> P) (p \<bullet> Q)"
max_min_same_4: fixes y :: "'c"   and x :: "'c" shows "max y (min x y) = y"