hcju/isabelleps huggingface dataset:
ins_invariant: fixes ks :: "int list"   and l :: "int"   and x :: "int"   and S :: "nat option list \<times> int list" assumes "dioph_is_node ks l x"   and "dioph_invariant ks l S"   and "\<not> dioph_memb x S" shows "dioph_invariant ks l (dioph_ins x S)"
leadsto_def: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "F \<leadsto> G \<equiv> \<box>(Init F \<longrightarrow> \<diamond>G)"
Y_Z: fixes n :: "nat" assumes "1 \<le> n" shows "Y n \<subseteq> Z n"
complement_finite: shows "finite_incidence_system \<V> \<B>\<^sup>C"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "Pds'.stable_on ds X" shows "inj_on Xd X"
dord_fun_def: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "dord_fun ord s t \<equiv> let d1 = deg_fun s; d2 = deg_fun t in d1 < d2 \<or> d1 = d2 \<and> ord s t"
split_beta': fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
ennreal_SUP: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "(\<Squnion>a\<in>A. ennreal (f a)) \<noteq> \<top>"   and "A \<noteq> {}" shows "ennreal (\<Squnion> (f ` A)) = (\<Squnion>a\<in>A. ennreal (f a))"
assms_3: shows "\<not> (\<exists>(u, v)\<in>Minimal_Solutions a b. u @ v <\<^sub>v x @ y)"
lex_take_index: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> lex r" obtains i :: "nat" where "length ys = length xs" and "i < length xs" and "take i xs = take i ys" and "(xs ! i, ys ! i) \<in> r"
zero_order_1: fixes x :: "'e multiset" shows "{#} \<subseteq># x"
enum_all_option_def: fixes P :: "'a option \<Rightarrow> bool" shows "enum_all P = (P None \<and> enum_all (\<lambda>x. P (Some x)))"
assoc_simps_1: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.arr \<a>\<^sub>D[f, g, h]"
list_all2_opt_False: fixes n :: "nat"   and A_zs :: "nat set list"   and A_x :: "nat set"   and A_xs :: "nat set list"   and A_ws :: "nat set list"   and A_y :: "nat set"   and A_ys :: "nat set list"   and zs :: "'b option list set list"   and x :: "'b option list set"   and xs :: "'b option list set list"   and ws :: "'b option list set list"   and y :: "'b option list set"   and ys :: "'b option list set list" assumes "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs) @ A_ws @ A_y # A_ys) ((zs @ x # xs) @ ws @ y # ys)"   and "length A_ws = length ws"   and "length A_xs = length xs"   and "length A_ys = length ys"   and "length A_zs = length zs"   and "A_y \<subseteq> A_x" shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs) @ A_ws @ A_ys) ((zs @ join x False y # xs) @ ws @ ys)"
rep_eq: fixes x :: "'a"   and xa :: "'a poly" shows "poly.coeff (Polynomial.smult x xa) = (\<lambda>n. x * poly.coeff xa n)"
exists_fresh_2: fixes x :: "'a" assumes "finite (supp x)" obtains c :: "coname" where "c \<sharp> x"
IH: fixes c\<^sub>2 :: "com" assumes "\<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>2,s\<^sub>2\<rangle>" shows "c\<^sub>1' = c\<^sub>2 \<and> s' = s\<^sub>2"
disc_compl_def: fixes H :: "ocircline" shows "disc_compl H = {z. out_ocircline H z}"
enc_atom_max: fixes I :: "(nat + nat set) list"   and n :: "nat"   and n' :: "nat"   and a :: "'b" assumes "\<forall>x\<in>set I. case x of Inl p \<Rightarrow> p \<le> n | Inr P \<Rightarrow> \<forall>p\<in>P. p \<le> n"   and "n \<le> n'" shows "enc_atom I (Suc n') a = (a, replicate (length I) False)"
WellDefined: fixes \<phi> :: "('t, 'f) Language"   and i :: "('o, 'm) IType" assumes "L\<lbrakk>\<phi>\<rbrakk> \<rightarrow> i" shows "Sig iS \<triangleright> \<phi>"
Image_Id_on: fixes A :: "'g set"   and B :: "'g set" shows "Id_on A `` B = A \<inter> B"
log_base_pow: fixes a :: "real"   and n :: "nat"   and x :: "real" assumes "0 < a" shows "log (a ^ n) x = log a x / real n"
cring_simprules_22: shows "- (0::'b) = (0::'b)"
small_digraph: fixes \<alpha> :: "V" shows "small {\<CC>. digraph \<alpha> \<CC>}"
inf_sup_aci_2: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "inf (inf x y) z = inf x (inf y z)"
set_iterator_emp_def: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma>0 :: "'c" shows "set_iterator_emp c f \<sigma>0 = \<sigma>0"
normal_form_diff_in_ideal: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "p - normal_form F p \<in> ideal F"
v: fixes x :: "'a" assumes "x \<in> space M" shows "f x = v (g x)"
0: shows "prv (neg (cnj (PP \<langle>\<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>))))"
replicate_once: shows "\<exists>rn. [Bk] = Bk \<up> rn"
SimIncrEnvironment_axioms_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'h"   and simRels :: "'b \<Rightarrow> ('h \<times> 'h) set"   and simVal :: "'h \<Rightarrow> 'c \<Rightarrow> bool" shows "SimIncrEnvironment_axioms jkbp envInit envAction envTrans envVal jview simf simRels simVal \<equiv> sim (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview)) (mkKripke (simf ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview) simRels simVal) simf"
inj_imp_inv_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "inj f"   and "\<forall>x. f (g x) = x" shows "inv f = g"
noteq_def: fixes r :: "'c array"   and s :: "'d array" shows "r =!!= s = (TYPEREP('c) \<noteq> TYPEREP('d) \<or> addr_of_array r \<noteq> addr_of_array s)"
le_sum_list_mono: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<le>\<^sub>v ys" shows "sum_list xs \<le> sum_list ys"
to_fun_lin_part: fixes f :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "f \<in> carrier P"   and "b \<in> carrier R" shows "to_fun (lin_part f) b = f 0 \<oplus> f 1 \<otimes> b"
lcm_least_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(lcm a b dvd c) = (a dvd c \<and> b dvd c)"
projJ_def: shows "projJ = (\<lambda>x n. U ((T ^^ n) x))"
Abs_type': fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and s :: "trm"   and U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda>:S. s) : U \<rightarrow> U'" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
swap_3: assumes "remove1 c (u @ [a] @ [b] @ v) =\<^sub>S remove1 c (u @ [b] @ [a] @ v)" shows "thesis"
NNotInTrace: shows "\<exists>n<length exec. \<forall>n'\<ge>n. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg"
cl_inv_def: fixes gp :: "'b gState_scheme \<times> pState" shows "cl_inv gp = (case gp of (g, p) \<Rightarrow> length (pState.channels p) \<le> length (gState.channels g))"
changes_filter_eq: fixes xs :: "'b list" shows "changes (filter (\<lambda>x. x \<noteq> (0::'b)) xs) = changes xs"
isomorphic_sum_integer_group: fixes I :: "'b set" shows "sum_group I (\<lambda>i. integer_group) \<cong> free_Abelian_group I"
subst2: fixes x :: "name"   and M :: "'b"   and xvec :: "name list"   and Tvec :: "'a list" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> Tvec" shows "x \<sharp> substAssert M xvec Tvec"
eq_add_iff2: fixes a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" shows "(a * e + c = b * e + d) = (c = (b - a) * e + d)"
RETURN_RES_refine_iff: fixes x :: "'e"   and R :: "('e \<times> 'f) set"   and Y :: "'f set" shows "(RETURN x \<le> \<Down> R (RES Y)) = (\<exists>y\<in>Y. (x, y) \<in> R)"
wi_homs_2: fixes a :: "int"   and b :: "int" shows "word_of_int a - word_of_int b = word_of_int (a - b)"
induct: fixes P :: "Reflection_Examples.num \<Rightarrow> Reflection_Examples.num \<Rightarrow> bool"   and a0 :: "Reflection_Examples.num"   and a1 :: "Reflection_Examples.num" assumes "\<And>n1 c1 r1 n2 c2 r2. \<lbrakk>\<And>x. \<lbrakk>n1 = n2; x = c1 + c2; x = 0\<rbrakk> \<Longrightarrow> P r1 r2; \<And>x. \<lbrakk>n1 = n2; x = c1 + c2; x \<noteq> 0\<rbrakk> \<Longrightarrow> P r1 r2; \<lbrakk>n1 \<noteq> n2; n1 \<le> n2\<rbrakk> \<Longrightarrow> P r1 (CN n2 c2 r2); \<lbrakk>n1 \<noteq> n2; \<not> n1 \<le> n2\<rbrakk> \<Longrightarrow> P (CN n1 c1 r1) r2\<rbrakk> \<Longrightarrow> P (CN n1 c1 r1) (CN n2 c2 r2)"   and "\<And>n1 c1 r1 v. P r1 (C v) \<Longrightarrow> P (CN n1 c1 r1) (C v)"   and "\<And>n1 c1 r1 v va. P r1 (Add v va) \<Longrightarrow> P (CN n1 c1 r1) (Add v va)"   and "\<And>n1 c1 r1 v va. P r1 (num.Mul v va) \<Longrightarrow> P (CN n1 c1 r1) (num.Mul v va)"   and "\<And>n1 c1 r1 v. P r1 (num.Var v) \<Longrightarrow> P (CN n1 c1 r1) (num.Var v)"   and "\<And>v n2 c2 r2. P (C v) r2 \<Longrightarrow> P (C v) (CN n2 c2 r2)"   and "\<And>v va n2 c2 r2. P (Add v va) r2 \<Longrightarrow> P (Add v va) (CN n2 c2 r2)"   and "\<And>v va n2 c2 r2. P (num.Mul v va) r2 \<Longrightarrow> P (num.Mul v va) (CN n2 c2 r2)"   and "\<And>v n2 c2 r2. P (num.Var v) r2 \<Longrightarrow> P (num.Var v) (CN n2 c2 r2)"   and "\<And>b1 b2. P (C b1) (C b2)"   and "\<And>v va vb. P (C v) (Add va vb)"   and "\<And>v va vb. P (C v) (num.Mul va vb)"   and "\<And>v va. P (C v) (num.Var va)"   and "\<And>v va vb. P (Add v va) (C vb)"   and "\<And>v va vb vc. P (Add v va) (Add vb vc)"   and "\<And>v va vb vc. P (Add v va) (num.Mul vb vc)"   and "\<And>v va vb. P (Add v va) (num.Var vb)"   and "\<And>v va vb. P (num.Mul v va) (C vb)"   and "\<And>v va vb vc. P (num.Mul v va) (Add vb vc)"   and "\<And>v va vb vc. P (num.Mul v va) (num.Mul vb vc)"   and "\<And>v va vb. P (num.Mul v va) (num.Var vb)"   and "\<And>v va. P (num.Var v) (C va)"   and "\<And>v va vb. P (num.Var v) (Add va vb)"   and "\<And>v va vb. P (num.Var v) (num.Mul va vb)"   and "\<And>v va. P (num.Var v) (num.Var va)"   and "\<And>va vb v. P (Add va vb) (C v)"   and "\<And>va vb v. P (num.Mul va vb) (C v)"   and "\<And>va v. P (num.Var va) (C v)"   and "\<And>vb v va. P (C vb) (Add v va)"   and "\<And>vb vc v va. P (Add vb vc) (Add v va)"   and "\<And>vb vc v va. P (num.Mul vb vc) (Add v va)"   and "\<And>vb v va. P (num.Var vb) (Add v va)"   and "\<And>vb v va. P (C vb) (num.Mul v va)"   and "\<And>vb vc v va. P (Add vb vc) (num.Mul v va)"   and "\<And>vb vc v va. P (num.Mul vb vc) (num.Mul v va)"   and "\<And>vb v va. P (num.Var vb) (num.Mul v va)"   and "\<And>va v. P (C va) (num.Var v)"   and "\<And>va vb v. P (Add va vb) (num.Var v)"   and "\<And>va vb v. P (num.Mul va vb) (num.Var v)"   and "\<And>va v. P (num.Var va) (num.Var v)" shows "P a0 a1"
smooth_on_add: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x + g x)"
S11: shows "upb = c_pair (c_pair n x) (upb1 + upb2) + 1"
ran_eq: shows "ran (process_mru vs |` Q) = Max_by fst ` (\<lambda>a. \<Union>a\<in>{a} \<inter> voters vs. vote_set vs {a}) ` (Q \<inter> voters vs)"
NSconvergentD: fixes X :: "nat \<Rightarrow> 'a" assumes "NSconvergent X" shows "\<exists>L. X \<longlonglongrightarrow>\<^sub>N\<^sub>S L"
zinrel: shows "z \<in> rel_interior (\<Inter> {S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F \<and> z \<in> S \<and> a h \<bullet> z = b h})"
real_of_float_inverse: fixes x :: "float" shows "float_of (real_of_float x) = x"
step_z_dbm_sound: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and v :: "'f \<Rightarrow> nat"   and n :: "nat"   and l' :: "'e"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))" shows "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto> \<langle>l', [D']\<^bsub>v,n\<^esub>\<rangle>"
sum_list_simps_1: shows "sum_list [] = (0::'b)"
nprv_mp: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F (imp \<phi> \<psi>)"   and "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
transitive_closure_def: fixes rel :: "(nat \<times> nat) set" shows "transitive_closure rel = steps rel (SOME n. is_bound rel n)"
boundedI: fixes b :: "'a \<times> 'b"   and a :: "'a \<times> 'b"   and c :: "'a \<times> 'b" assumes "less_eq_prod b a"   and "less_eq_prod c a" shows "less_eq_prod (sup_prod b c) a"
core_idem: fixes T :: "'b set set"   and a :: "'b set" shows "core\<^bsub>T\<^esub> (core\<^bsub>T\<^esub> a) = core\<^bsub>T\<^esub> a"
isEmpty_def: fixes t :: "('c, 'd) FingerTree" shows "isEmpty t \<equiv> FingerTreeStruc.isEmpty (Rep_FingerTree t)"
wellformed_mk_symmetric: shows "pair_wf_digraph (mk_symmetric G)"
SI_implies_BSI_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "SI \<V> Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "BSI \<V>' Tr\<^bsub>ES\<^esub>"
lcm_proj2_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(lcm m n = normalize n) = dvd.dvd times m n"
sub_conj_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "sub_conj t \<equiv> \<forall>P Q. sound P \<and> sound Q \<longrightarrow> t P && t Q \<tturnstile> t (P && Q)"
pt_fresh_bij1: fixes a :: "'b"   and x :: "'a"   and pi :: "('b \<times> 'b) list" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "a \<sharp> x" shows "pi \<bullet> a \<sharp> pi \<bullet> x"
simps: fixes a1 :: "super_binop + boolean_binop + numeric_binop + string_binop + collection_binop"   and a2 :: "call_kind"   and a3 :: "'a OCL_Types.type"   and a4 :: "'a OCL_Types.type"   and a5 :: "'a OCL_Types.type" shows "binop_type a1 a2 a3 a4 a5 = ((\<exists>op \<tau> \<sigma> \<rho>. a1 = Inl op \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> super_binop_type op \<tau> \<sigma> \<rho>) \<or> (\<exists>op \<tau> \<sigma> \<rho>. a1 = Inr (Inl op) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> boolean_binop_type op \<tau> \<sigma> \<rho>) \<or> (\<exists>op \<tau> \<sigma> \<rho>. a1 = Inr (Inr (Inl op)) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> numeric_binop_type op \<tau> \<sigma> \<rho>) \<or> (\<exists>op \<tau> \<sigma> \<rho>. a1 = Inr (Inr (Inr (Inl op))) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> string_binop_type op \<tau> \<sigma> \<rho>) \<or> (\<exists>op \<tau> \<sigma> \<rho>. a1 = Inr (Inr (Inr (Inr op))) \<and> a2 = ArrowCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> collection_binop_type op \<tau> \<sigma> \<rho>))"
sleq_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "sleq x y"   and "sleq y z" shows "sleq x z"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "color \<Rightarrow> ('c, 'd) rbt \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> ('c, 'd) rbt \<Rightarrow> 'b"   and rbt :: "('c, 'd) rbt" shows "P (case rbt of rbt.Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = ((rbt = rbt.Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24 x25. rbt = Branch x21 x22 x23 x24 x25 \<longrightarrow> P (f2 x21 x22 x23 x24 x25)))"
j: shows "j < dim_col (submatrix (submatrix A I UNIV) UNIV J)"
sum_unroll: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum f {..n} = (if n = 0 then f 0 else f n + sum f {..n - 1})"
pmf_map_inj': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c pmf"   and x :: "'c" assumes "inj f" shows "pmf (map_pmf f M) (f x) = pmf M x"
prems_2: shows "set (a # u) \<inter> set v = {}"
ac: shows "absolutely_continuous (restr_to_subalg M F) (restr_to_subalg (density M g) F)"
Bfun_inverse: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'b)" shows "Bfun (\<lambda>x. inverse (f x)) F"
returns_heap_eq: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog"   and h' :: "'aq"   and h'' :: "'aq" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
prv_imp_mp: fixes \<phi> :: "fm"   and \<chi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV"   and "{} \<turnstile> \<phi> IMP \<chi>"   and "{} \<turnstile> \<phi>" shows "{} \<turnstile> \<chi>"
rem_initial_length: fixes b :: "bit"   and w :: "bit list" shows "length (rem_initial b w) \<le> length w"
unit_def: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one" shows "Dmonoid.unit prod one x = (\<exists>y. prod x y = one \<and> prod y x = one)"
lsum_list_llist_of: fixes xs :: "'a list" shows "lsum_list (llist_of xs) = sum_list xs"
cases: fixes x :: "('fsym, 'tp) efsym" obtains f :: "'fsym" where "x = Oldf f"   | \<sigma> :: "'tp" where "x = Wit \<sigma>"
B'_def: fixes x :: "'a" shows "B' x = {y \<in> B x. (0::'c) \<noteq> f (x, y)}"
pairwise_image: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b set" shows "Compiler_Utils.pairwise P (f ` M) = Compiler_Utils.pairwise (\<lambda>x y. P (f x) (f y)) M"
list_of_vec_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "list_of_vec (vec n f) = map f [0..<n]"
convergence_G_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and t :: "'a"   and z :: "'b" shows "MC_syntax.convergence_G K s t z = summable (\<lambda>n. MC_syntax.p K s t n * norm z ^ n)"
atLeastAtMost_iff: fixes i :: "'a"   and l :: "'a"   and u :: "'a" shows "(i \<in> {l..u}) = (l \<le> i \<and> i \<le> u)"
IH: fixes \<nu> :: "variable \<Rightarrow> real" assumes "Uvariation \<nu> (F 0) {RVar x, DVar x}" shows "term_sem I (the (usubstappt \<sigma> {RVar x, DVar x} \<theta>)) \<nu> = term_sem (USubst.adjoint \<sigma> I (F 0)) \<theta> \<nu>"
g': fixes s :: "real" assumes "s \<in> {0..t}" shows "(g has_derivative g' s) (at s within {0..t})"
subject_reduction: fixes e :: "trm"   and e' :: "trm"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes "e \<Down> e'"   and "\<Gamma> \<turnstile> e : T" shows "\<Gamma> \<turnstile> e' : T"
Un_Int_eq_3: fixes S :: "'d set"   and T :: "'d set" shows "S \<inter> (S \<union> T) = S"
bangPres: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<approx> Q" shows "!P \<approx> !Q"
rel_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a stream"   and b :: "'b stream" shows "stream_all2 R\<inverse>\<inverse> a b = stream_all2 R b a"
monadify_simps_2: fixes x :: "'j" shows "EVAL $ x \<equiv> RETURN $ x"
var_order_rel': shows "(\<le>) = (\<lambda>x y. x = y \<or> (x, y) \<in> var_order_rel)"
dfc_implies_dwfc: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set" assumes "d_future_consistent P D R" shows "d_weakly_future_consistent P I D R"
mon_c_alt: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and c :: "'g list multiset" shows "mon_c fg c \<equiv> \<Union> (mon_s fg ` set_mset c)"
permute_set_def: fixes p :: "perm"   and X :: "'c set" shows "p \<bullet> X = {p \<bullet> x |x. x \<in> X}"
intensional_rews_3: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'d \<Rightarrow> 'b"   and y :: "'d \<Rightarrow> 'c"   and w :: "'d" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
before_forall: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "\<turnstile> $(\<forall>x. P x) = (\<forall>x. $P x)"
pw_le_iff: fixes S :: "'e nres"   and S' :: "'e nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
prems: shows "c\<^sub>1 \<sqsubseteq> mds\<^sub>1"
diffeomorphismD_1: fixes k :: "enat"   and S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "diffeomorphism k S T f g" shows "k-smooth_on S f"
attach_parametric: shows "rel_fun (rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>) (rel_fun (rel_resource (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>) (rel_resource (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g))) attach attach"
case_prod_conv: fixes f :: "'y \<Rightarrow> 'z \<Rightarrow> 'x"   and a :: "'y"   and b :: "'z" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
impl_of_ahm_delete: fixes k :: "'a"   and hm :: "('a, 'b) hashmap" shows "impl_of (ahm_delete k hm) = ArrayHashMap_Impl.ahm_delete k (impl_of hm)"
Pf_quot_imp_is_proved: fixes e :: "(name, hf) finfun"   and \<alpha> :: "fm" assumes "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e" shows "{} \<turnstile> \<alpha>"
encode_all_frame_axioms_set: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "set (map (\<lambda>(k, v). encode_negative_transition_frame_axiom \<Pi> k v) (List.product [0..<t] (\<Pi>\<^sub>\<V>)) @ map (\<lambda>(k, v). encode_positive_transition_frame_axiom \<Pi> k v) (List.product [0..<t] (\<Pi>\<^sub>\<V>))) = {encode_negative_transition_frame_axiom \<Pi> k v |k v. (k, v) \<in> {0..<t} \<times> set (\<Pi>\<^sub>\<V>)} \<union> {encode_positive_transition_frame_axiom \<Pi> k v |k v. (k, v) \<in> {0..<t} \<times> set (\<Pi>\<^sub>\<V>)}"
all_zero: shows "\<forall>m>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
dres_Inf_cases: fixes S :: "'b dres set" obtains "S \<subseteq> {dFAIL}" and "Inf S = dFAIL"   | "dSUCCEED \<in> S" and "Inf S = dSUCCEED"   | a :: "'b" and b :: "'b" where "a__ \<noteq> b" and "dRETURN a__ \<in> S" and "dRETURN b \<in> S" and "dSUCCEED \<notin> S" and "Inf S = dSUCCEED"   | a :: "'b" where "S \<subseteq> {dFAIL, dRETURN a}" and "dRETURN a \<in> S" and "Inf S = dRETURN a"
select_convs_4: fixes g_V :: "'b set"   and g_E :: "('b \<times> 'b) set"   and g_V0 :: "'b set"   and more :: "'a" shows "graph_rec.more \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = more"
w\<^sub>f'_is_map: shows "is_left_adjoint w\<^sub>f'"
vector_component_5: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec"   and i :: "'d" shows "(x - y) $ i = x $ i - y $ i"
lnk: shows "length (nat_to_bv n) \<le> k"
axioms_2: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc_diff_face G M a" shows "bidel_arc_diff_face_axioms M a"