hcju/isabelleps huggingface dataset:
cases: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('a, 'c, 'e) gpv"   and a2 :: "('b, 'd, 'f) gpv" assumes "rel_gpv'' A C R a1 a2" obtains gpv :: "('a, 'c, 'e) gpv" and gpv' :: "('b, 'd, 'f) gpv" where "a1 = gpv" and "a2 = gpv'" and "rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R))) (the_gpv gpv) (the_gpv gpv')"
path_concat: fixes x :: "'obj"   and y :: "'obj"   and p :: "'arr list"   and z :: "'obj"   and q :: "'arr list" assumes "G.path x y p"   and "G.path y z q" shows "G.path x z (p @ q)"
index_append_vec_2: fixes v :: "'b vec"   and w :: "'b vec" shows "dim_vec (v @\<^sub>v w) = dim_vec v + dim_vec w"
eqClosed: fixes P :: "pi"   and Q :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<sim>\<^sup>s Q" shows "perm \<bullet> P \<sim>\<^sup>s perm \<bullet> Q"
I01: fixes s :: "'d"   and z :: "'e nres" assumes "s \<in> S"   and "SPEC (I (S - {s})) \<le> z" shows "f s \<le> z"
assms_1: fixes n :: "nat" shows "A n \<in> sets M"
permute_Abs_set: fixes p :: "perm"   and as :: "atom set"   and x :: "'a" shows "p \<bullet> ([as]set. x) = [p \<bullet> as]set. p \<bullet> x"
multiplicative_norm_octo: fixes x :: "octo"   and y :: "octo" shows "norm (x * y) = norm x * norm y"
fmrel_acyclic: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "acyclicP_on (fmran' xm) R"   and "fmrel R\<^sup>+\<^sup>+ xm ym"   and "fmrel R ym xm" shows "xm = ym"
fvi_And_Not: fixes b :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "fvi b (And_Not \<phi> \<psi>) = fvi b \<phi> \<union> fvi b \<psi>"
scale_left_distrib: fixes a :: "'c"   and b :: "'c"   and x :: "'b \<Rightarrow>\<^sub>0 'c" shows "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
transfer: shows "rel_fun cr_uint64 (rel_fun cr_uint64 cr_uint64) (+) (+)"
npowf_inv: fixes x :: "'a"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>"   and "0 \<le> j" shows "x\<^bsub>K\<^esub>\<^bsup>j\<^esup> = x\<^bsup>\<hyphen> K\<^esup>\<^bsub>K\<^esub>\<^bsup>- j\<^esup>"
Rep_lifted_boolean_algebra_inverse: fixes x :: "('c, 'd) lifted_boolean_algebra" shows "Abs_lifted_boolean_algebra (Rep_lifted_boolean_algebra x) = x"
diff_diff_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a b) c = minus a (plus b c)"
eventually_returns_to_continuousI: fixes P :: "'a set"   and x :: "'a"   and S :: "'a set" assumes "returns_to P x"   and "closed P"   and "continuous (at x within S) (return_time P)" shows "eventually (returns_to P) (at x within S)"
elims: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and xb :: "lock_action"   and y :: "('b \<times> nat) option" assumes "upd_lock x xa xb = y" obtains l :: "('b \<times> nat) option" and t :: "'b" where "x = la__" and "xa = tb__" and "xb = Lock" and "y = lock_lock la__ tb__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lb__" and "xa = tc__" and "xb = Unlock" and "y = unlock_lock lb__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lc__" and "xa = td__" and "xb = UnlockFail" and "y = lc__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = l" and "xa = t" and "xb = ReleaseAcquire" and "y = release_all l t"
return_time_eqI: fixes P :: "'a set"   and t :: "real"   and x :: "'a" assumes "closed P"   and "0 < t"   and "t \<in> existence_ivl0 x"   and "flow0 x t \<in> P"   and "\<And>s. \<lbrakk>0 < s; s < t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P" shows "return_time P x = t"
elims_1: fixes x :: "'b trm"   and y :: "bool" assumes "is_a_constant x = y" obtains xa :: "'b" where "x = Var xa__" and "\<not> y"   | xa :: "'b" where "x = Const xaa__" and "y"   | xa :: "'b trm" and ya :: "'b trm" where "x = xa \<cdot> ya" and "\<not> y"
trees_of_size_Suc: fixes n :: "nat" shows "trees_of_size (Suc n) = (\<lambda>(l, r). Node l r) ` (\<Union>k\<le>n. trees_of_size k \<times> trees_of_size (n - k))"
accept\<^sub>G\<^sub>R_simp: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>G\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> Fin = {} \<and> (\<forall>I\<in>Inf. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {}))"
bigstepT_the_state: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "nat"   and s' :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> t \<Down> s'" shows "\<down>\<^sub>s (c, s) = s'"
encode_kind_Pr: fixes f :: "recf" assumes "encode_kind (encode f) = 4" shows "\<exists>n f' g. f = Pr n f' g"
simps_2: fixes m :: "'b" shows "normalize_match (Match m) = [Match m]"
Struct_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "CU.Struct wtFsym wtPsym arOf parOf D intF intP \<equiv> U.Struct wtFsym wtPsym arOf parOf D intF intP"
invmfun_r_inv: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and n :: "'g" assumes "R module N"   and "bijec\<^bsub>M,N\<^esub> f"   and "n \<in> carrier N" shows "f (invmfun R M N f n) = n"
sup_commute: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> \<bool>"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> \<bool>"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
a_l_coset_swap: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <+ H"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "x \<in> y <+ H"
gcd_eucl: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm" shows "normalization_euclidean_semiring.gcd zero modulo normalize = gcd"
orbit_def: fixes m :: "'a set" shows "conj.orbit m = conj.same_orbit_rel `` {m}"
checkpoint_s_def: fixes s :: "('b, 'c) state" shows "checkpoint_s s = \<B>\<^sub>i s"
entries_balance: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "entries (balance l k v r) = entries l @ (k, v) # entries r"
simps: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "hchild \<langle>l, uu, r\<rangle> = (if height l \<le> height r then r else l)"
elims_1: fixes x :: "ty"   and y :: "bool" assumes "is_Class x = y" obtains C :: "String.literal" where "x = Class C" and "y"   | "x = Void" and "\<not> y"   | "x = Boolean" and "\<not> y"   | "x = Integer" and "\<not> y"   | "x = NT" and "\<not> y"   | v :: "ty" where "x = v\<lfloor>\<rceil>" and "\<not> y"
0: shows "deg R (f \<oplus>\<^bsub>P\<^esub> g) \<le> n"
rejects_trace_simulation: fixes e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list \<times> value list) list"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and f :: "nat \<Rightarrow> nat" assumes "rejects_trace e2 s2 r2 t"   and "accepts_trace e1 s1 r1 t" shows "\<not> trace_simulation f e1 s1 r1 e2 s2 r2 t"
rel_fun_map_fun1: fixes h :: "'e \<Rightarrow> 'd"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> A f g" shows "rel_fun (=) A (map_fun h id f) g"
sorted_wrt_hd_less_take: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f (x # xs)"   and "\<And>x. f x x" shows "\<forall>y\<in>set (take n (x # xs)). f x y"
rep_eq: fixes x :: "'b \<Rightarrow> 'a upred" shows "\<lbrakk>ushAll x\<rbrakk>\<^sub>e = (\<lambda>A. \<forall>xa. \<lbrakk>x xa\<rbrakk>\<^sub>e A)"
post_map_acom: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom" shows "post (ACom.map_acom f c) = f (post c)"
transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (list_all2 R) (rel_list_R1 R) list_to_list_R1 list_to_list_R1"
Lcm_UNIV: 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 Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<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" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm UNIV = zero"
lr_of_tran_fbs_acceptD: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and ifs :: "'d"   and p :: "('b, 'c) simple_packet_scheme"   and r :: "'b simple_match"   and oif :: "char list" assumes "valid_prefixes rt"   and "has_default_route rt"   and "no_oif_match fw"   and "generalized_sfw (lr_of_tran_fbs rt fw ifs) p = Some (r, oif, Accept)" shows "simple_linux_router_nol12 rt fw p = Some (p\<lparr>p_oiface := oif\<rparr>)"
list_all2_iff: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
ord0_le_def: fixes x :: "ord0"   and y :: "ord0" shows "(x \<le> y) = ((x, y) \<in> ord0_leq)"
simps_1: shows "X3d X3.Xd1 = D1"
subgame_E': fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G"   and "V' \<subseteq> V\<^bsub>G\<^esub>" shows "E\<^bsub>ParityGame.subgame G V'\<^esub> = Restr E\<^bsub>G\<^esub> V\<^bsub>ParityGame.subgame G V'\<^esub>"
epi_implies_arr: fixes f :: "'j" assumes "J.epi f" shows "J.arr f"
conductor_eq_1_iff_principal: shows "(\<Phi>.principal.conductor = 1) = (principal_dchar c = principal_dchar c)"
real_sqrt_diff_squares_triangle_ineq: fixes a :: "real"   and c :: "real"   and b :: "real"   and d :: "real" shows "sqrt ((a - c)\<^sup>2 + (b - d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
orecvmsg_def: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'c"   and a :: "'d seq_action" shows "orecvmsg P \<sigma> a \<equiv> case a of receive m \<Rightarrow> P \<sigma> m | _ \<Rightarrow> True"
rel_set_mono: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "A \<le> B" shows "rel_set A \<le> rel_set B"
some_eq_ex: fixes P :: "'e \<Rightarrow> bool" shows "P (SOME x. P x) = (\<exists>x. P x)"
iprev_nth_surj_aux1: fixes I :: "nat set" assumes "finite I" shows "{x \<in> I. \<nexists>n. I \<leftarrow> n = x} = {}"
weakenStatImpE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q" obtains Q' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'" and "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi>" and "(\<Psi>, P, Q') \<in> Rel"
alg_dfs_def: fixes aOps :: "('c, 'b, 'e list) MapOps"   and tOps :: "('d, 'b \<times> 'f, 'b) MapOps"   and simObs :: "'b \<Rightarrow> 'f"   and simTrans :: "'b \<Rightarrow> 'b list"   and simAction :: "'b \<Rightarrow> 'e list" shows "alg_dfs aOps tOps simObs simTrans simAction \<equiv> let k_empt = \<lparr>aActs = MapOps.empty aOps, aTrans = MapOps.empty tOps\<rparr>; k_memb = \<lambda>s A. isSome (MapOps.lookup aOps (aActs A) s); k_succs = simTrans; actsUpdate = \<lambda>ec A. update aOps ec (simAction ec) (aActs A); transUpdate = \<lambda>ec ec'. update tOps (ec, simObs ec') ec'; k_ins = \<lambda>ec A. \<lparr>aActs = actsUpdate ec A, aTrans = foldr (transUpdate ec) (k_succs ec) (aTrans A)\<rparr> in gen_dfs k_succs k_ins k_memb k_empt"
Pair_eqvt: fixes p :: "perm"   and x :: "'d"   and y :: "'e" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
bilinear_form_def: fixes v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "bilinear_form v1 v2 H = vec_cnj v1 *\<^sub>v\<^sub>m H *\<^sub>v\<^sub>v v2"
set_restriction_not_in_imp: fixes f :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and A :: "'a set" assumes "set_restriction f"   and "x \<notin> A" shows "x \<notin> f A"
negligible_hyperplane: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a) \<or> b \<noteq> 0" shows "negligible {x. a \<bullet> x = b}"
ot_eq: fixes X :: "'a set" assumes "X \<subseteq> U" shows "tp (cv X) = ordertype X r"
i_refsto_j: shows "(i::'a::refinement_lattice) \<sqsubseteq> (j::'a::refinement_lattice)"
top_lifted_pair_def: shows "top \<equiv> Abs_lifted_pair (\<lambda>f. (top, Filters.filter.Abs_filter {top}))"
get_env_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) envT" shows "get_env get f = EnvT (\<lambda>r. get (\<lambda>s. run_env (f s) r))"
obj_ty_def: fixes obj :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" shows "obj_ty obj \<equiv> Class (fst obj)"
dickson_gradingD2: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat" assumes "dickson_grading d" shows "almost_full_on (adds) {x. d x \<le> n}"
set_iterator_image_filter_def: fixes g :: "'f \<Rightarrow> 'e option"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "set_iterator_image_filter g it \<equiv> \<lambda>c f. it c (\<lambda>x \<sigma>. case g x of None \<Rightarrow> \<sigma> | Some x' \<Rightarrow> f x' \<sigma>)"
finprod_all1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (0::'d)" shows "sum f A = (0::'d)"
univalent_conv_injective: fixes x :: "'a" shows "univalent x = injective (x\<^sup>T)"
assm_3: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> s" shows "K \<subseteq> i"
shiftl_1: fixes n :: "nat" shows "1 << n = 2 ^ n"
var_to_dimacs_pos: fixes v :: "nat"   and f :: "sat_plan_variable formula" assumes "v \<in> atoms (map_formula var_to_dimacs f)" shows "0 < v"
subprob_space_dens: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "subprob_space (dens_ctxt_measure \<Y> \<rho>)"
lpred_conj_def: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "P land Q \<equiv> \<lambda>x. P x \<and> Q x"
simps_2: fixes head :: "'b" shows "succ_rel [head] = {(head, None)}"
verts3_in_slim_G: fixes x :: "'a" assumes "x \<in> verts3 (with_proj G)" shows "x \<in> pverts slim"
simps_11: shows "get_axiom AdPlus = diff_plus_axiom"
inf_sup_ord_3: fixes x :: "'b"   and y :: "'b" shows "x \<le> x \<sqinter> y"
that: fixes e :: "real" assumes "0 < e"   and "cball x e \<times> {0..t + e} \<subseteq> Sigma X existence_ivl0" shows "thesis"
one_false_all_valid_false: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "\<exists>g\<in>set G. gval g s = false"   and "\<forall>g\<in>set G. gval g s \<noteq> invalid" shows "gval (fold gAnd G (Bc True)) s = false"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha> set['a]" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = abs_Tree\<^sub>\<alpha> (tConj (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
fps_of_poly_diff: fixes p :: "'b poly"   and q :: "'b poly" shows "fps_of_poly (p - q) = fps_of_poly p - fps_of_poly q"
word_div_def: fixes a :: "'a word"   and b :: "'a word" shows "a div b = word_of_int (uint a div uint b)"
Aheap_nonrec_def: fixes x :: "Vars.var"   and e :: "exp" shows "Aheap_nonrec x e = (\<Lambda> i. esing x\<cdot>(ABind_nonrec x e\<cdot>i))"
mem_lclss_of_state_3: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "((C, Old) \<in> lclss_of_state (N, P, Q)) = (C \<in> Q)"
is_group: shows "Group.group (G \<times>\<times> DirProd_list Gs)"
coherent_Assoc'_Ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src\<^sub>C :: "'a \<Rightarrow> 'a"   and trg\<^sub>C :: "'a \<Rightarrow> 'a"   and V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i> :: "'d \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and E :: "'a \<Rightarrow> 'd"   and a :: "'a local.term"   and b :: "'a local.term"   and c :: "'a local.term" assumes "Coherence.evaluation_map C src\<^sub>C trg\<^sub>C V H \<a> \<i> src trg E"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C a"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C b"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C c"   and "bicategorical_language.Src src\<^sub>C a = bicategorical_language.Trg trg\<^sub>C b"   and "bicategorical_language.Src src\<^sub>C b = bicategorical_language.Trg trg\<^sub>C c" shows "V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Cod C src\<^sub>C trg\<^sub>C \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>]))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>]) = V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Nmlize C \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>])) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Dom C src\<^sub>C trg\<^sub>C \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>])))"
shleg_poly_def: shows "local.shleg_poly = local.gen_shleg_poly n div [:fact n:]"
ne_matchers_imp_empty_or_matchrel: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) multiset" assumes "matchers_map \<sigma> \<inter> matchers (set_mset P) \<noteq> {}" shows "P = {#} \<or> (\<exists>y. ((P, \<sigma>), y) \<in> matchrel)"
UN: fixes \<Omega> :: "'d set"   and M :: "'d set set"   and A :: "nat \<Rightarrow> 'd set" assumes "Dynkin_system \<Omega> M"   and "disjoint_family A"   and "range A \<subseteq> M" shows "\<Union> (range A) \<in> M"
step_1: fixes xs :: "'a llist"   and ys :: "'b llist" assumes "X xs ys" shows "lnull xs = lnull ys"
distinct_1: fixes x2 :: "'b" shows "Deq \<noteq> Enq x2"
runiq_wrt_eval_rel: fixes R :: "('a \<times> 'b) set" shows "runiq R = (\<forall>x. R `` {x} \<subseteq> {R ,, x})"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "Ring_Hom.ring_hom hom" shows "comm_ring_hom hom"
ot_def: fixes \<G> :: "'d cyclic_group" shows "ot \<G> \<equiv> ot_base \<G> \<and> Cyclic_Group.cyclic_group \<G>"
boundedI: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "semilattice_order f less_eq less"   and "less_eq a b"   and "less_eq a c" shows "less_eq a (f b c)"
hcomplex_of_hypreal_add_HComplex: fixes r :: "real star"   and x :: "real star"   and y :: "real star" shows "hcomplex_of_hypreal r + HComplex x y = HComplex (r + x) y"
urel64_mult: fixes x :: "uint64"   and y :: "int"   and x' :: "uint64"   and y' :: "int" assumes "urel64 x y"   and "urel64 x' y'" shows "urel64 (mult_p64 pp x x') (mult_p p y y')"
interp_ins_append_forward: fixes xs :: "('a \<times> 'a option) list"   and ys :: "('a \<times> 'a option) list"   and oid :: "'a" assumes "insert_ops (xs @ ys)"   and "oid \<in> set (interp_ins (xs @ ys))"   and "oid \<in> set (map fst xs)" shows "oid \<in> set (interp_ins xs)"
dom_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" shows "VV.dom (VV.comp g f) = VV.dom f"
Nil_sp: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 [] v2 subs = (sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> (v1 = v2 \<or> (v1, v2) \<in> subs\<^sup>+))"
connex_def: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "connex A less_eq \<equiv> \<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> less_eq x y \<or> less_eq y x"
game_1_def: fixes \<A> :: "(bool, bool list, bool list) gpv" shows "local.prf.game_1 \<A> = exec_gpv prf.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, uu_). return_spmf b)"
index_Basis_list_axis1: fixes i :: "'a" shows "index Basis_list (axis i 1) = index enum_class.enum i"
card_inj_on_le: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "card A \<le> card B"
seq_char: fixes g :: "'a"   and f :: "'a" shows "hom.seq g f = (hom.arr f \<and> hom.arr g \<and> seq g f)"
induct: fixes x1 :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and x2 :: "IMP"   and x3 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and P :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set \<Rightarrow> IMP \<Rightarrow> ((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> bool" assumes "x1 \<rhd> x2 : x3"   and "\<And>G. P G Skip (\<lambda>s t. t = s)"   and "\<And>G x e. P G (Assign x e) (\<lambda>s t. t = update s x (evalE e s))"   and "\<And>G c1 A1 c2 A2. \<lbrakk>G \<rhd> c1 : A1; P G c1 A1; G \<rhd> c2 : A2; P G c2 A2\<rbrakk> \<Longrightarrow> P G (Comp c1 c2) (\<lambda>s t. \<exists>r. A1 s r \<and> A2 r t)"   and "\<And>G c1 A c2 B b. \<lbrakk>G \<rhd> c1 : A; P G c1 A; G \<rhd> c2 : B; P G c2 B\<rbrakk> \<Longrightarrow> P G (Iff b c1 c2) (\<lambda>s t. (evalB b s \<longrightarrow> A s t) \<and> (\<not> evalB b s \<longrightarrow> B s t))"   and "\<And>G c B b A. \<lbrakk>G \<rhd> c : B; P G c B; \<forall>s. \<not> evalB b s \<longrightarrow> A s s; \<forall>s r t. evalB b s \<longrightarrow> B s r \<longrightarrow> A r t \<longrightarrow> A s t\<rbrakk> \<Longrightarrow> P G (While b c) (\<lambda>s t. A s t \<and> \<not> evalB b t)"   and "\<And>A G. \<lbrakk>({A} \<union> G) \<rhd> body : A; P ({A} \<union> G) body A\<rbrakk> \<Longrightarrow> P G Call A"   and "\<And>A G. A \<in> G \<Longrightarrow> P G Call A"   and "\<And>G c A B. \<lbrakk>G \<rhd> c : A; P G c A; \<forall>s t. A s t \<longrightarrow> B s t\<rbrakk> \<Longrightarrow> P G c B" shows "P x1 x2 x3"
simps_1: fixes x1 :: "int option"   and x2 :: "int option"   and y1 :: "int option"   and y2 :: "int option" shows "(I x1 x2 = I y1 y2) = (x1 = y1 \<and> x2 = y2)"