hcju/isabelleps huggingface dataset:
lookup_phiNodes_of: fixes g :: "'g"   and v :: "'val" assumes "finite (dom (Mapping.lookup (phis g)))" shows "Mapping.lookup (uninst_code.ssa.phiNodes_of phis g) v = (if \<exists>n\<in>dom (Mapping.lookup (phis g)). v \<in> set (the (Mapping.lookup (phis g) n)) then Some {n \<in> dom (Mapping.lookup (phis g)). v \<in> set (the (Mapping.lookup (phis g) n))} else None)"
J: shows "(\<Sum>j<a * N. u N ((T ^^ j) x)) \<le> (\<Sum>j<n - 2 * N. u N ((T ^^ j) x)) + real N * E2"
assms: shows "is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"
poly_deg_monom_mult: fixes c :: "'c"   and p :: "('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 'e" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "poly_deg (MPoly_Type_Class.punit.monom_mult c t p) = deg_pm t + poly_deg p"
snoc_single_Cons: fixes xs :: "'c list"   and x :: "'c"   and y :: "'a"   and z :: "'b"   and zs :: "'b list" assumes "as = xs @ [x]"   and "bs = [y]"   and "cs = z # zs" shows "P"
bij_betw_byWitness: fixes A :: "'d set"   and f' :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e"   and A' :: "'e set" assumes "\<forall>a\<in>A. f' (f a) = a"   and "\<forall>a'\<in>A'. f (f' a') = a'"   and "f ` A \<subseteq> A'"   and "f' ` A' \<subseteq> A" shows "bij_betw f A A'"
induct: fixes P :: "('b \<Rightarrow> nat \<Rightarrow> 'b nres) \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> nat \<Rightarrow> 'b nres"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>f a. P f a 0"   and "\<And>f a n. P f a n \<Longrightarrow> P f a (Suc n)" shows "P a0 a1 a2"
finite_mult_of: assumes "finite (carrier R)" shows "finite (carrier (mult_of R))"
assms: fixes v :: "i" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
emeasure_measure_of_conv: fixes \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "emeasure (measure_of \<Omega> A \<mu>) = (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)"
r2f_f2r_galois: fixes f :: "'a \<Rightarrow> 'b set"   and R :: "('a \<times> 'b) set" shows "(\<R> f = R) = (\<F> R = f)"
add_left_mono: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "c + a \<subseteq># c + b"
constant_def: fixes f :: "'c \<Rightarrow> 'd" shows "constant f = (\<forall>x y. f x = f y)"
valid_graph_rep_code: fixes l :: "('c \<times> 'd \<times> 'e) list" shows "valid_graph_rep l = distinct (map (\<lambda>(u, v, uu_). (u, v)) l)"
simps: fixes p :: "real mpoly" shows "get_if_const p = (if is_constant p then Some (MPoly_Type.coeff p 0) else None)"
trm_rep_is_lower_aux: fixes t :: "'a trm"   and S :: "'a eclause set" assumes "\<forall>y. (y, t) \<in> trm_ord \<longrightarrow> y \<noteq> trm_rep y S \<longrightarrow> (trm_rep y S, y) \<in> trm_ord"   and "subterm_reduction_applicable S t" shows "(trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S, t) \<in> trm_ord"
finite_prod_map_of_rat_poly_hom: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> rat poly"   and s :: "('a \<times> 'b) set"   and y :: "real" shows "rpoly (\<Prod>(a, b)\<in>s. f a b) y = (\<Prod>(a, b)\<in>s. rpoly (f a b) y)"
p_not_inf: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and s :: "'a" assumes "Reachability_Problem K S S1 S2"   and "s \<in> S" shows "Reachability_Problem.p K S S1 S2 s \<noteq> \<top>"
rep_eq: fixes x :: "circline" shows "circline_type x = circline_type_clmat (rep_circline x)"
rot_circle_cube_is_type_II: shows "typeII_twoCube rot_circle_cube"
abs_powr_nat_le: fixes x :: "real"   and y :: "real" shows "\<bar>powr_nat x y\<bar> \<le> powr_nat \<bar>x\<bar> y"
insert_ran_image: fixes j :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "j < Suc n"   and "f ` {0..<n} = {0..<n}" shows "insert_ran f j ` {0..<n} = {0..<Suc n} - {j}"
prefix_slength_strengthen: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" shows "prefix\<cdot>xs\<cdot>ys = (le\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>ys) andalso prefix\<cdot>xs\<cdot>ys)"
downset_Sup_id: shows "id \<le> ord.downset (\<lambda>x y. y \<le> x) \<circ> Inf"
strict_iff_order: fixes a :: "'b"   and b :: "'b" shows "(a < b) = (a \<le> b \<and> a \<noteq> b)"
mono_comp_fun: fixes f :: "'b \<Rightarrow> 'c" assumes "mono f" shows "mono ((\<circ>) f)"
imp_igSwapInpIPresIGWlsInp: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "igSwapIPresIGWls MOD" shows "igSwapInpIPresIGWlsInp MOD"
Nil_eq_concat_conv: fixes xss :: "'g list list" shows "([] = concat xss) = (\<forall>xs\<in>set xss. xs = [])"
cons_exec_def: fixes e :: "'d \<times> 'e list"   and p :: "'e" shows "cons_exec e p \<equiv> (fst e, Cons p (snd e))"
simple_arg_pos_weight: fixes f :: "'f"   and n :: "nat"   and i :: "nat" shows "simple_arg_pos weight_NS (f, n) i"
encode_not_Z_or_S: fixes f :: "recf"   and a :: "nat"   and b :: "nat" assumes "encode f = prod_encode (a, b)"   and "1 < a" shows "f \<noteq> Z \<and> f \<noteq> S"
fres_canc2: fixes y :: "'a"   and x :: "'a" shows "y \<le> y \<cdot> x \<leftarrow> x"
length_insert_wrt: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "length (insert_wrt R x xs) = Suc (length xs)"
produces_imp_Max_literal: fixes C :: "'a literal multiset"   and A :: "'a" assumes "production C = {A}" shows "A = atm_of (Max_mset C)"
T_LET: fixes x :: "W.var"   and \<Gamma> :: "(W.var \<times> tyS) list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "ty"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and " \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 "   and " (x, close \<Gamma> T\<^sub>1) # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 "   and "set (ftv T\<^sub>1 - ftv \<Gamma>) \<sharp>* T\<^sub>2" shows " \<Gamma> \<turnstile> Let x be t\<^sub>1 in t\<^sub>2 : T\<^sub>2 "
gen_wf_inEdges_def: shows "gen_wf_inEdges \<equiv> map_fun Rep_gen_cfg_wf id gen_inEdges"
4: shows "((F \<circ> real_of_ereal) \<longlongrightarrow> F 1) (at_left (ereal 1))"
assms_3: fixes n :: "nat" shows "0 \<le> Re (g n)"
length_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>xs. \<forall>ys. order.greater (length xs) (length ys) \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
induce_eq_iff_induced: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "induced_subgraph H G" shows "G \<restriction> verts H = H"
Red: shows "s -t\<triangleright>ta\<rightarrow> s'"
dlo_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "dlo le \<equiv> dpo le \<and> dlo_axioms le"
subst_closed_preserved: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c"   and env :: "(name, 'c) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "pre_term.closed_except frees t {||}" shows "pre_term.closed_except frees (subst t env) {||}"
prems_3: shows "hd (a # as) = hd (b # bs)"
IH: shows "vars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> vars\<^sub>s\<^sub>t S \<union> range_vars \<delta>"
simps_1: fixes uu :: "trilean" shows "invalid \<or>? uu = invalid"
intros_4: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a"   and SRel :: "('b \<times> 'b) set" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelST Enc SRel TRel"
update_convs_2: fixes trans' :: "('d \<times> 'e \<times> 'd) set \<Rightarrow> ('d \<times> 'e \<times> 'd) set"   and init :: "'d set"   and trans :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f" shows "trans_update trans' \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = \<lparr>init = init, trans = trans' trans, \<dots> = more\<rparr>"
minf_7: fixes t :: "'a" shows "\<exists>z. \<forall>x<z. (t < x) = False"
assms: shows "a \<bullet> c \<noteq> d \<or> b \<bullet> c \<noteq> d"
vertices_def: fixes g :: "('c, 'd) rgraph_scheme" shows "Graph.vertices g = {root g} \<union> src ` edges g \<union> tgt ` edges g"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>D f : D a \<rightarrow> D b\<guillemotright>"
f11: shows "\<one> \<otimes> (s'' \<otimes> s \<otimes> (s'' \<otimes> s') \<otimes> (r'' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) \<ominus> s'' \<otimes> (s \<otimes> s') \<otimes> (s'' \<otimes> s' \<otimes> (r'' \<otimes> r) \<oplus> s'' \<otimes> s \<otimes> (r'' \<otimes> r'))) = \<zero>"
assms_3: shows "sigma_finite_measure (restr_to_subalg M F)"
Bag_s_mul_ext: fixes xs :: "('a, nat) alist"   and ys :: "('a, nat) alist"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" shows "((Bag xs, Bag ys) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)}) = fst (mul_ext f (ass_list_to_single_list (alist.impl_of xs)) (ass_list_to_single_list (alist.impl_of ys)))"
splitAt_drop: fixes ls :: "'c list"   and i :: "nat" assumes "distinct ls"   and "i < |ls|" shows "snd (splitAt (ls ! i) ls) = drop (Suc i) ls"
inv_into_inv_into_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and a :: "'a" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A' (inv_into A f) a = f a"
y: fixes r :: "real" assumes "0 < r" shows "\<exists>no. \<forall>n\<ge>no. norm ((\<Sum>(x, K)\<in>p n. content K *\<^sub>R f x) - y) < r"
prems: shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # S) \<inter> subst_domain \<theta> = {}"
strict_implies_order: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
FG_advice_insert: fixes \<psi> :: "'b ltln"   and Y :: "'b ltln set"   and \<phi> :: "'b ltln" assumes "\<psi> \<notin> Y"   and "size \<phi> < size \<psi>" shows "\<phi>[insert \<psi> Y]\<^sub>\<mu> = \<phi>[Y]\<^sub>\<mu>"
rep_eq: fixes x :: "'b \<Longrightarrow> 'a" shows "Rep_scene \<lbrakk>x\<rbrakk>\<^sub>\<sim> = (\<lambda>s\<^sub>1 s\<^sub>2. if mwb_lens x then s\<^sub>1 \<triangleleft>\<^bsub>x\<^esub> s\<^sub>2 else s\<^sub>1)"
D_cycle_free_len_dest: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and m' :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "cycle_free m n"   and "\<forall>i\<le>n. \<forall>j\<le>n. D m i j n = m' i j"   and "i \<le> n"   and "j \<le> n"   and "set xs \<subseteq> {0..n}" shows "\<exists>ys. set ys \<subseteq> {0..n} \<and> len m' i j xs = len m i j ys"
cod_inv: fixes f :: "'d" assumes "D.iso f" shows "D.cod (D.inv f) = D.dom f"
hyps_1: shows "n'' -as\<rightarrow>* (_Low_)"
residue_mult_group_kernel_partition_5: fixes m :: "nat"   and n :: "nat"   and b :: "nat set" assumes "1 < m"   and "0 < n"   and "m dvd n"   and "b \<in> rcosets\<^bsub>residue_mult_group n\<^esub> kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m)" shows "card (kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m)) = card b"
SDG_edge_valid_SDG_node_2: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and n' :: "'node SDG_node" assumes "SDG_edge n Vopt popt n'" shows "valid_SDG_node n'"
rep_number_unfold_set: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and design_rep_number :: "int"   and x :: "'a" assumes "constant_rep_design point_set block_collection design_rep_number"   and "x \<in> point_set" shows "int (size (filter_mset ((\<in>) x) block_collection)) = design_rep_number"
wellformed_transaction_cases_4: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and l :: "'h strand_label"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "(l, x) \<in> set (transaction_updates T)" shows "(\<exists>t s. x = insert\<langle>t,s\<rangle>) \<or> (\<exists>t s. x = delete\<langle>t,s\<rangle>)"
assms: shows "\<not> (a = \<infinity> \<and> b = - \<infinity> \<or> a = - \<infinity> \<and> b = \<infinity>)"
has_codomain_char: fixes f :: "'a1 \<times> 'a2" shows "(codomains f \<noteq> {}) = (C1.arr (fst f) \<and> C2.arr (snd f))"
RUN_unfold: fixes A :: "'a set" shows "RUN A = Mprefix A (\<lambda>z. RUN A)"
divide_pos_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less zero y" shows "less zero (divide x y)"
subgame_VV: fixes p :: "Player"   and V' :: "'a set" shows "(if p = Even then V0\<^bsub>subgame V'\<^esub> else V\<^bsub>subgame V'\<^esub> - V0\<^bsub>subgame V'\<^esub>) = V' \<inter> VV p"
\<sigma>_2: fixes w :: "'a" assumes "v0 \<rightarrow> w" shows "winning_strategy p \<sigma> w"
pell_valuation_pos_neg_aux: fixes D :: "nat"   and x :: "int"   and y :: "int" assumes "pell D"   and "0 < x"   and "0 < y"   and "pell.solution D (x, y)" shows "pell.pell_valuation D (x, - y) \<in> {0<..<1}"
diff_intersect_sym_diff: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A - B) \<inter># (B - A) = {#}"
integral_unique: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and k :: "'c set" assumes "(f has_integral y) k" shows "integral k f = y"
distance_successor_distance: fixes n :: "'node"   and n' :: "'node"   and x :: "nat" assumes "distance n n' x"   and "x \<noteq> 0" obtains a :: "'edge" where "valid_edge a" and "n = sourcenode a" and "intra_kind (kind a)" and "distance (targetnode a) n' (x - 1)" and "targetnode a = (SOME nx. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') n' (x - 1) \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = nx)"
fundfold_trivial_f\<C>_list: fixes r :: "'a permutation"   and Cs :: "'a set list" assumes "r \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "\<forall>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>" shows "fst (Spair r) \<Turnstile> Cs = Cs"
sq_norm_smult_vec: fixes c :: "'a"   and v :: "'a vec" shows "\<parallel>c \<cdot>\<^sub>v v\<parallel>\<^sup>2 = sq_norm_conjugate c * \<parallel>v\<parallel>\<^sup>2"
irreducible_cf_pos_poly: fixes p :: "int poly" assumes "irreducible p"   and "degree p \<noteq> 0" shows "irreducible (cf_pos_poly p)"
wf\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>s'_def: fixes \<S> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" shows "wf\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>s' \<S> \<A> \<equiv> (\<forall>\<A>'\<in>\<S>. wf'\<^sub>s\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>) (unlabel \<A>')) \<and> (\<forall>\<A>'\<in>\<S>. \<forall>\<A>''\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>'' = {}) \<and> (\<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> = {}) \<and> (\<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>' = {})"
bind_altc1: fixes C :: "'c cset"   and g :: "'c \<Rightarrow> 'm"   and f :: "'a option \<Rightarrow> 'm" shows "bind (altc C g) f = altc C (\<lambda>c. bind (g c) f)"
intros: fixes p :: "perm"   and x :: "'b" shows "(p \<bullet> x, x) \<in> hull_rel"
prj_chine_assoc_2: shows "\<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> \<nu>\<pi>\<rho>.chine_assoc = \<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>1"
ide_w: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b"   and u :: "'b"   and w :: "'b"   and \<theta> :: "'b" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.ide V w"
a_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> y \<in> carrier V"
simps_2: fixes n :: "nat"   and p :: "'a"   and t' :: "'b tree23" shows "dfull n (Some (p, True, t')) = full n t'"
widen_null: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list" shows "subtype P NT (Class C)"
Nmlize_Nmlize: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Nmlize V (bicategorical_language.Nmlize V t) = bicategorical_language.Nmlize V t"
Polys_closed_power: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and m :: "nat" assumes "p \<in> P[X]" shows "p ^ m \<in> P[X]"
p_residue_ring_car_memE_2: fixes m :: "int"   and k :: "nat" assumes "m \<in> carrier (residue_ring (p ^ k))" shows "0 \<le> m"
div_ceiling: fixes q :: "int"   and x :: "int" assumes "q \<noteq> 0" shows "of_int x / of_int q \<le> of_int (div_ceiling x q)"
mono_tick_count: fixes r :: "'b run"   and c :: "clock" shows "mono (tick_count r c)"
uniqueness_of_restricted_lift: fixes S :: "'a set"   and T :: "'a freeword \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "GroupHom (FreeGroup S) T"   and "\<forall>s\<in>S. T (Abs_freeletter s) = f s" shows "T = res_freeword_funlift f S"
dube_eq_4: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and j :: "nat" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "j < card X" shows "1 = 2 * (- 1) ^ (card X - Suc j) * (int (poly_deg f) - 1 gchoose (card X - Suc j)) - 1 - (\<Sum>i = Suc j..card X - 1. (- 1) ^ (i - j) * (int (pm_powerprod.aa X f F i) gchoose (i - j) + (int (pm_powerprod.bb ord X F i) gchoose (i - j))))"
HaddP_Mem_cancel_left: fixes H :: "fm set"   and x :: "tm"   and y' :: "tm"   and z' :: "tm"   and y :: "tm"   and z :: "tm" assumes "H \<turnstile> HaddP x y' z'"   and "H \<turnstile> HaddP x y z"   and "H \<turnstile> OrdP x" shows "H \<turnstile> z' IN z IFF y' IN y"
prefixes_length: fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat"   and b :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>= b" shows "t < e_length b"
list_emb_Cons_iff1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P xs ys"
apfst_convE: fixes q :: "'a \<times> 'b"   and f :: "'c \<Rightarrow> 'a"   and p :: "'c \<times> 'b" assumes "q = apfst f p" obtains x :: "'c" and y :: "'b" where "p = (x, y)" and "q = (f x, y)"
algebraic_representsI: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "algebraic x"
const_poly_dvd_iff_dvd_content: fixes c :: "'a"   and p :: "'a poly" shows "([:c:] dvd p) = (c dvd content p)"
cases: fixes a :: "msg"   and H :: "msg set" assumes "a \<in> synth H" obtains X :: "msg" where "a = X__" and "X__ \<in> H"   | A :: "agent" where "a = Messages.Agent A"   | n :: "nat" where "a = Number n__"   | n :: "nat" where "a = NonceA na__"   | n :: "nat" where "a = epubKA nb__"   | n :: "nat" where "a = epriKA n"   | X :: "msg" where "a = Hash Xa__" and "Xa__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = \<langle>Xb__, Y__\<rangle>" and "Xb__ \<in> synth H" and "Y__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = Enc Xc__ Ya__" and "Xc__ \<in> synth H" and "Ya__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = Aenc Xd__ Yb__" and "Xd__ \<in> synth H" and "Yb__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = Sign Xe__ Yc__" and "Xe__ \<in> synth H" and "Yc__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = Exp X Y" and "X \<in> synth H" and "Y \<in> synth H"
pw_INF_nofail: fixes f :: "'b \<Rightarrow> 'a nres"   and C :: "'b set" shows "nofail (Inf (f ` C)) = (\<exists>x\<in>C. nofail (f x))"
that: fixes q2 :: "'c"   and q1 :: "'d"   and tr :: "('c \<times> 'd) list" assumes "io_targets PM (initial PM) vs = {(q2, q1)}"   and "path PM (xs || tr) (q2, q1)"   and "length xs = length tr" shows "thesis"
outs: shows "outs_\<I> \<I>' = outs_\<I> \<I>"
Bot_def: shows "Bot \<equiv> Abs_process ({(s, X). front_tickFree s}, {d. front_tickFree d})"
map_cong0: fixes x :: "'a pmf"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g x"
m1_inv3_ifreshD: fixes x :: "'a m1_state_scheme"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and nl :: "atom list"   and Ra' :: "fid_t"   and A' :: "agent"   and B' :: "agent"   and nl' :: "atom list" assumes "runs x Ra = Some (Init, [A, B], aKey Kab # nl)"   and "runs x Ra' = Some (Init, [A', B'], aKey Kab # nl')"   and "A \<notin> bad"   and "B \<notin> bad"   and "Kab \<notin> Domain (m1_state.leak x)"   and "x \<in> m1_inv3_ifresh" shows "Ra = Ra'"
strongBisimWeakCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>e Q" shows "P \<simeq> Q"
observable_transition_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P' \<equiv> \<exists>Q Q'. weak_nominal_ts.tau_transition transition \<tau> P Q \<and> transition Q \<langle>\<alpha>,Q'\<rangle> \<and> weak_nominal_ts.tau_transition transition \<tau> Q' P'"
cases: fixes a1 :: "'d option"   and a2 :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option" assumes "(a1, a2) \<in> option_while' C B" obtains r :: "'d" where "a1 = Some ra__" and "a2 = Some ra__" and "\<not> C ra__"   | r :: "'d" where "a1 = Some rb__" and "a2 = None" and "C rb__" and "B rb__ = None"   | r :: "'d" and r' :: "'d" and sr'' :: "'d option" where "a1 = Some r" and "a2 = sr''" and "C r" and "B r = Some r'" and "(Some r', sr'') \<in> option_while' C B"
adm_fn: fixes r :: "('a \<Rightarrow> ValD) admS dual \<times> ('a \<Rightarrow> ValD) admS" shows "adm (\<lambda>x. x \<in> fn_lf_rep r)"