hcju/isabelleps huggingface dataset:
csols_empty_iff: fixes x0 :: "'a" shows "(csols t0 x0 = {}) = (t0 \<notin> T \<or> x0 \<notin> X)"
resolve_on_def: fixes A :: "'a"   and D :: "'a literal list"   and CA :: "'a literal list" shows "resolve_on A D CA = map (resolvent D A CA) (filter (resolvable A D CA) (subseqs CA))"
not_add_mem_right: fixes x :: "hf"   and y :: "hf" shows "x @+ y \<^bold>\<notin> x"
has_bochner_integral_divide: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x / y) (x / y)"
Sup_Continuous_closed: fixes H :: "'d upred \<Rightarrow> 'd upred"   and A :: "'e set"   and P :: "'e \<Rightarrow> 'd upred" assumes "Continuous H"   and "\<And>i. i \<in> A \<Longrightarrow> P i is H"   and "\<not> A = {}" shows "\<Or>(P ` A) is H"
fst_proj_def: fixes M :: "('a \<times> 'b) set" shows "fst_proj M \<equiv> {A. \<exists>B. (A, B) \<in> M}"
simps: fixes v1 :: "nat"   and p1 :: "(form, nat) out_port"   and v2 :: "nat"   and p2 :: "(form, nat) antecedent" shows "task.edge_begin ((v1, p1), v2, p2) = v1"
zero_diff: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
fv_eqvt: fixes \<pi> :: "perm"   and e :: "'b" shows "\<pi> \<bullet> fv e = fv (\<pi> \<bullet> e)"
DistinctNetsAllowAllow: fixes b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and poo :: "int"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and po :: "int" assumes "AllowPortFromTo b c poo \<in> set p"   and "AllowPortFromTo a d po \<in> set p"   and "allNetsDistinct p"   and "dom (C (AllowPortFromTo b c poo)) \<inter> dom (C (AllowPortFromTo a d po)) \<noteq> {}" shows "b = a \<and> c = d \<and> poo = po"
sg_mult_closed: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "G \<guillemotright> H "   and "x \<in> H"   and "y \<in> H" shows "x \<cdot> y \<in> H"
lambda_one: shows "(\<lambda>x. x) = (*) (1::'d)"
regular_edge: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c \<in> set conclusions"   and "is \<in> it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))" shows "Solved_Task.edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is \<in> Solved_Task.edges freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"
pochhammer_of_nat_eq_0_lemma: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "pochhammer (- of_nat n) k = (0::'b)"
vimage_eq_rcoset_2: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set"   and a :: "'a" assumes "weak_group_morphism f H G"   and "a \<in> carrier G" shows "{b \<in> carrier G. f b = f a} = a <# H"
pmaxsimpchain: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.pmaxsimpchain X xs" shows "SimplicialComplex.maxsimpchain X xs"
SOUND_K: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and K :: "nat" assumes "G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I"   and "G_validn K G"   and "MST_validn K" shows "\<Turnstile>\<^sub>Suc K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I "
intros_2: fixes x :: "'b"   and y :: "'c"   and r :: "('b \<times> 'c) set"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> r"   and "(xs, ys) \<in> listrel r" shows "(x # xs, y # ys) \<in> listrel r"
simps: fixes P :: "'b prog"   and E :: "('c \<times> ('a, 'c) obs_event action) llist"   and a :: "nat"   and ad :: "'a"   and al :: "addr_loc" shows "value_written P E a (ad, al) = value_written_aux P (action_obs E a) al"
matrix_left_invertible_injective: fixes A :: "(('b, 'd) vec, 'c) vec" shows "(\<exists>B. B ** A = mat (1::'b)) = inj ((*v) A)"
distinct_31: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PFn x41 x42 x43 \<noteq> PPair x51 x52"
lookup_minus: fixes f :: "'e \<Rightarrow>\<^sub>0 'd"   and g :: "'e \<Rightarrow>\<^sub>0 'd"   and k :: "'e" shows "lookup (f - g) k = lookup f k - lookup g k"
abs_le_square_iff: fixes 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 times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs x) (abs y) = less_eq (power.power one times x 2) (power.power one times y 2)"
gen_lossless_gpv_Fail: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>" shows "\<not> gen_lossless_gpv co \<I> Fail"
RETURN_RES_refine_iff: fixes x :: "'a"   and R :: "('a \<times> 'b) set"   and Y :: "'b set" shows "(RETURN x \<le> \<Down> R (RES Y)) = (\<exists>y\<in>Y. (x, y) \<in> R)"
Gromov_completion_converge_to_boundary_aux: fixes x :: "'a Gromov_completion"   and v :: "nat \<Rightarrow> 'a" assumes "x \<in> Gromov_boundary"   and "abs_Gromov_completion v = x"   and "Gromov_completion_rel v v" shows "(\<lambda>n. extended_Gromov_product_at basepoint (to_Gromov_completion (v n)) x) \<longlonglongrightarrow> \<infinity>"
K_def: fixes \<K> :: "'a \<Rightarrow> 'a pmf"   and \<O> :: "'a \<Rightarrow> 'b pmf" shows "HMM_defs.K \<K> \<O> \<equiv> \<lambda>(s\<^sub>1, o\<^sub>1). \<K> s\<^sub>1 \<bind> (\<lambda>s\<^sub>2. map_pmf (Pair s\<^sub>2) (\<O> s\<^sub>2))"
diverge_lfilter_LNil: fixes xs :: "'a llist"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>lset xs. \<not> P x" shows "lfilter P xs = []\<^sub>l"
r_dovedelay_prim: shows "prim_recfn 3 r_dovedelay"
perfect_imp_quotient_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "perfect_map X Y f" shows "quotient_map X Y f"
hash_rt_F\<^sub>m_def: fixes h :: "'c \<Rightarrow> 'e"   and rhm :: "'d \<Rightarrow> 'f" shows "hash_rt_F\<^sub>m h rhm \<equiv> hash_blindable id \<circ> map_blindable\<^sub>m (map_prod h (map rhm)) id"
assms_1: fixes i :: "'b" assumes "i \<in> I" shows "{x \<in> \<Omega>. P i x} \<in> M"
triangle_in_hom_1: shows "\<guillemotleft>(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
sigma_sets_subset: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and a :: "'b set set" assumes "sigma_algebra \<Omega> M"   and "a \<subseteq> M" shows "sigma_sets \<Omega> a \<subseteq> M"
assms_2: shows "cs\<^bsup>\<pi>\<^esup> n = [\<pi> n]"
ex_y: fixes x :: "'a" assumes "count (mset ys) x < count (mset xs) x" shows "\<exists>y. gt (f y) (f x) \<and> count (mset (map f xs)) (f y) < count (mset (map f ys)) (f y)"
foo98: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and aa :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators"   and r :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x (aa # p) = \<lfloor>a\<rfloor>"   and "x \<in> dom (C r)"   and "r \<in> set p" shows "a \<in> set p"
u: fixes y :: "'a" assumes "y \<in> cball x d0" shows "s (flow0 y (u y)) = 0"
continuous_on_extension_sequentially: fixes T :: "'a set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "T \<subseteq> closure S"   and "\<And>u b. \<lbrakk>\<forall>n. u n \<in> S; b \<in> T; u \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b" shows "continuous_on T f"
defUse_path_ex: fixes v :: "'val"   and g :: "'g"   and m :: "'node" assumes "v \<in> allUses g m"   and "m \<in> set (\<alpha>n g)" obtains ns :: "'node list" where "g \<turnstile> defNode g v-ns\<rightarrow>m" and "EntryPath g ns"
subdivision_step_rev: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b"   and I :: "('a, 'b) pre_digraph"   and rev_I :: "'b \<Rightarrow> 'b" assumes "subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)"   and "subdivision (H, rev_H) (I, rev_I)" shows "subdivision (G, rev_G) (I, rev_I)"
set_nn_integral_mono: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<inter> A \<Longrightarrow> f x \<le> g x" shows "set_nn_integral M A f \<le> set_nn_integral M A g"
Cons_4: shows "\<forall>var\<in>V. var < start"
DiffI: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A\<setminus>B"
hoare_triple_preI: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn" assumes "\<And>h. h \<Turnstile> P \<Longrightarrow> <P> c <Q>" shows "<P> c <Q>"
when_commute: fixes a :: "'a"   and Q :: "bool"   and P :: "bool" shows "(a when Q when P) = (a when P when Q)"
elem_le_sum_list: fixes k :: "nat"   and ns :: "'a multiset list" assumes "k < length ns" shows "ns ! k \<subseteq># monoid_add.sum_list (+) {#} ns"
measure_preserving_prob_space: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "prob_space M = prob_space N"
intros_3: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and y :: "Eg1.var"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Load y) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 y)) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "y \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> \<C>"   and "y \<notin> \<C>"   and "\<forall>mds\<^sub>A'' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A'' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A'', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A'', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
induct: fixes G :: "prog"   and x1 :: "ty"   and x2 :: "ty"   and P :: "ty \<Rightarrow> ty \<Rightarrow> bool" assumes "G\<turnstile>x1\<preceq>x2"   and "\<And>T. P T T"   and "\<And>I J. G\<turnstile>I\<preceq>I J \<Longrightarrow> P (Iface I) (Iface J)"   and "\<And>I. P (Iface I) (Class Object)"   and "\<And>C D. G\<turnstile>C\<preceq>\<^sub>C D \<Longrightarrow> P (Class C) (Class D)"   and "\<And>C I. G\<turnstile>C\<leadsto>I \<Longrightarrow> P (Class C) (Iface I)"   and "\<And>R. P NT (RefT R)"   and "\<And>T. P (T.[]) (Class Object)"   and "\<And>S T. \<lbrakk>G\<turnstile>RefT S\<preceq>RefT T; P (RefT S) (RefT T)\<rbrakk> \<Longrightarrow> P (RefT S.[]) (RefT T.[])" shows "P x1 x2"
nprv_clear4_2: fixes \<phi>1 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<phi>4 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>2 :: "'fmla" assumes "nprv {\<phi>1, \<phi>3, \<phi>4} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
assms_1: shows "a \<le> mod_type_class.from_nat i"
alphaAbs_ex_qFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_ex_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
DW_meas: fixes n :: "nat" shows "DW n \<in> sets M"
assms_4: shows "\<exists>x. Domainp A x \<and> f x"
size_jump1: fixes l :: "nat"   and cs :: "instr list" shows "length (jump l cs) < Suc (length cs)"
Inf_inf_dual: fixes f :: "'b \<Rightarrow> 'c" assumes "Inf_dual f" shows "inf_dual f"
reachable_reaches: fixes a :: "'a"   and b :: "'a" assumes "G.reachable a"   and "a \<rightarrow>* b" shows "G.reachable b"
add_diff_cancel_left': fixes a :: "'a"   and b :: "'a" shows "a \<oplus>\<^sub>a b \<ominus>\<^sub>a a = b"
closed_insert: fixes S :: "'a set"   and a :: "'a" assumes "closed S" shows "closed (insert a S)"
extended_ord_pm_powerprod_def: fixes ord :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool" shows "extended_ord_pm_powerprod ord ord_strict \<equiv> pm_powerprod ord ord_strict"
g3up_upRules: shows "g3up \<subseteq> upRules"
no_local_consts_in_consequences: shows "\<forall>xs\<in>sset rules. \<Union> (lconsts ` set (consequent xs)) = {}"
le_infE: fixes x :: "'g multiset"   and a :: "'g multiset"   and b :: "'g multiset" assumes "x \<subseteq># a \<inter># b" obtains "x \<subseteq># a" and "x \<subseteq># b"
equivp_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "equivp R"   and "R x y"   and "R y z" shows "R x z"
1: assumes "ord s t"   and "ord t u" shows "ord s u"
sets_pair_measure_cong: fixes M1 :: "'d measure"   and M1' :: "'d measure"   and M2 :: "'e measure"   and M2' :: "'e measure" assumes "sets M1 = sets M1'"   and "sets M2 = sets M2'" shows "sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
two_func_imageTr: fixes h :: "nat \<Rightarrow> 'e"   and n :: "nat"   and B :: "'e set"   and f :: "nat \<Rightarrow> 'e"   and m :: "nat" assumes "h \<in> {j. j \<le> Suc n} \<rightarrow> B"   and "f \<in> {j. j \<le> m} \<rightarrow> h ` {j. j \<le> Suc n}"   and "h (Suc n) \<notin> f ` {j. j \<le> m}" shows "f \<in> {j. j \<le> m} \<rightarrow> h ` {j. j \<le> n}"
AndR_sim: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula"   and G :: "'b formula" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> H"   and "\<^bold>\<not> G \<triangleright> \<Gamma> \<turnstile> H" shows "\<^bold>\<not> (F \<^bold>\<and> G) \<triangleright> \<Gamma> \<turnstile> H"
rel_option_None2: fixes P :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'l option" shows "rel_option P x None = (x = None)"
add_commute: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a +\<^sub>o\<^sub>w b = b +\<^sub>o\<^sub>w a"
renaming_preserves_cardinality: fixes A :: "'at set"   and C :: "'at Literal set" shows "card (rename_clause A C) = card C"
dvd_div_eq_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times a b" shows "(divide b a = c) = (b = times c a)"
inject_1: fixes x1 :: "name"   and y1 :: "name" shows "(Var x1 = Var y1) = (x1 = y1)"
fls_shift_zero: fixes m :: "int" shows "fls_shift m 0 = 0"
commute: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<union># b = b \<union># a"
exhaust: fixes y :: "'e option" obtains "y = None"   | x2 :: "'e" where "y = \<lfloor>x2\<rfloor>"
cs_in_initial_state_implies_not_snapshotted: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "snd (cs (S t i) cid) = NotStarted"   and "channel cid = Some (p, q)" shows "\<not> ps (S t i) q \<noteq> None"
eventually_nonneg_ln_const: fixes b :: "real" assumes "0 < b" shows "eventually_nonneg at_top (\<lambda>x. ln (b * x))"
abs_eq: fixes xa :: "'f \<Rightarrow> 'g"   and x :: "'g \<Rightarrow> 'e" shows "xa \<dagger> mk\<^sub>e x = mk\<^sub>e (\<lambda>b. x (xa b))"
openin_cases: fixes y :: "'d set \<Rightarrow> bool" assumes "y \<in> {L. istopology L}" obtains x :: "'d topology" where "y = openin x"
simps_1: fixes uu :: "'a" shows "repeat 0 uu = []"
r_lazyifzero: fixes xs :: "nat list"   and n :: "nat"   and j\<^sub>1 :: "nat"   and f\<^sub>1 :: "recf"   and j\<^sub>2 :: "nat"   and f\<^sub>2 :: "recf"   and c :: "nat" assumes "length xs = Suc n"   and "j\<^sub>1 = encode f\<^sub>1"   and "j\<^sub>2 = encode f\<^sub>2"   and "recfn (Suc n) f\<^sub>1"   and "recfn (Suc n) f\<^sub>2" shows "eval (r_lazyifzero n j\<^sub>1 j\<^sub>2) (c # xs) = (if c = 0 then eval f\<^sub>1 xs else eval f\<^sub>2 xs)"
safe: fixes loc1 :: "'loc"   and t :: "'t"   and s :: "'sum"   and loc2 :: "'loc" assumes "0 < zcount (c_pts c loc1) t"   and "s \<in>\<^sub>A path_summary loc1 loc2"   and "results_in t s \<le> b" shows "\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c loc2)"
mdecl_visible: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and S :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a" assumes "wf_prog wf_md P"   and "(C, S, fs, ms) \<in> set P"   and "(M, b, Ts, T, m) \<in> set ms" shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C"
comp_inv_arr: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.inverse_arrows f g" shows "CC.comp g f = CC.dom f"
arrE: fixes f :: "'a" assumes "S.arr f" shows "arr f \<and> src f = a \<and> trg f = a"
infinite_Card_order_limit: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "Card_order r"   and "infinite (Field r)"   and "a \<in> Field r" shows "\<exists>b\<in>Field r. a \<noteq> b \<and> (a, b) \<in> r"
seq_if_composable: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "src \<nu> = trg \<mu>"
cut_greater_all_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>> t = I) = (\<forall>x\<in>I. t < x)"
A4: shows "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"
simps_10: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm"   and q :: "fm" shows "Ifm vs bs (Imp p q) = (Ifm vs bs p \<longrightarrow> Ifm vs bs q)"
lincomb_distrib: fixes U :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and c :: "'a" assumes "U \<subseteq> carrier V"   and "a \<in> U \<rightarrow> carrier K"   and "c \<in> carrier K" shows "c \<odot>\<^bsub>V\<^esub> lincomb a U = lincomb (\<lambda>u. c \<otimes>\<^bsub>K\<^esub> a u) U"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "0 < f e" shows "e \<in> support_flow f"
grun_elim: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and s :: "(nat \<times> 'b) stream"   and v :: "nat \<times> 'b" assumes "grun A w V s v" obtains r :: "'b stream" and k :: "nat" and p :: "'b" where "s = fromN (Suc k) ||| r" and "v = (k, p)"
bij_betw_f_the_inv_into_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and y :: "'b" assumes "bij_betw f A B"   and "y \<in> B" shows "f (the_inv_into A f y) = y"
preserves_reflects_arr: fixes f :: "'c" shows "C.arr (trg\<^sub>C f) = C.arr f"
insert_if: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
supp_2: fixes name :: "name" shows "supp (DBVar name) = supp name"