hcju/isabelleps huggingface dataset:
nres_relI: fixes c :: "'d nres"   and R :: "('d \<times> 'e) set"   and a :: "'e nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
take_append_take_minus_idem: fixes n :: "nat"   and XS :: "'c list" shows "take n XS @ map ((!) XS) [n..<length XS] = XS"
single_iteration_inv': fixes i' :: "nat"   and i :: "nat"   and j' :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i' < i"   and "j' \<le> n"   and "j \<le> n"   and "i \<le> n" shows "fw m n k i j i' j' = fw m n k i' j' i' j'"
iD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> iD rt" shows "\<exists>dsn dsk hops nhip. rt dip = Some (dsn, dsk, Aodv_Basic.inv, hops, nhip)"
read_buffer_assign_temp_onwards_RelS: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" shows "(\<langle>(temp \<leftarrow> aexp.Const 0) ;; Skip@[temp -=\<^sub>m AsmNoReadOrWrite], mds\<^sub>A_of ((case_Mode {reg\<^sub>C} {} {} {})(AsmNoWrite := {control_var\<^sub>C}, AsmNoReadOrWrite := {temp\<^sub>C, reg\<^sub>C})), mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>(temp\<^sub>C \<leftarrow> aexp\<^sub>C.Const 0) ;; Skip@[temp\<^sub>C -=\<^sub>m AsmNoReadOrWrite], (case_Mode {reg\<^sub>C} {} {} {})(AsmNoWrite := {control_var\<^sub>C}, AsmNoReadOrWrite := {temp\<^sub>C, reg\<^sub>C}), mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
Eats_Subset_Iff: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm" shows "H \<turnstile> Eats x y SUBS z IFF x SUBS z AND y IN z"
Abs_filter_inject: fixes x :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and y :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "x \<in> {F. is_filter F}"   and "y \<in> {F. is_filter F}" shows "(Filter.filter.Abs_filter x = Filter.filter.Abs_filter y) = (x = y)"
open_part_witness_def: fixes X :: "part_witness set" shows "open X = (X \<in> {{}, {a}, {b, c}, {a, b, c}})"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "Oldf x1 \<noteq> Tag x2"
mk_linear_orders_set_r_decisive_on: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "r \<in> local.mk_linear_orders C B"   and "B \<subseteq> A"   and "B \<subseteq> set r"   and "iia_on A f" shows "decisive_on B f"
m1_secrecy_def: shows "m1_secrecy \<equiv> {s. knC (runs s) \<subseteq> azC (runs s) \<union> Domain (m1_state.leak s) \<times> UNIV}"
strictly_preserves_\<alpha>: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.arr h" shows "F \<a>\<^sub>C[f, g, h] = \<a>\<^sub>D[F f, F g, F h]"
valid_frac_add: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and X :: "('c \<times> 'c) set"   and Y :: "('c \<times> 'c) set" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit"   and "quotient_ring.valid_frac S R addition multiplication zero X"   and "quotient_ring.valid_frac S R addition multiplication zero Y" shows "quotient_ring.valid_frac S R addition multiplication zero (quotient_ring.add_rel S R addition multiplication zero X Y)"
intro: fixes G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and g :: "'a" assumes "Dijkstra_Impl_Defs G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>"   and "Dijkstra_Impl_axioms G_invar g" shows "Dijkstra_Impl G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>"
left_total_cr_spmf_gpv: shows "left_total spmf_to_gpv.cr_spmf_gpv"
lexordp_into_lexordp_eq: fixes xs :: "'a list"   and ys :: "'a list" assumes "lexordp xs ys" shows "lexordp_eq xs ys"
f_last_message_hold_0: fixes xs :: "'b message_af list" shows "xs \<longmapsto> 0 = last_message_hold xs"
Subspace_Span: fixes ms :: "'b list" assumes "set ms \<subseteq> V" shows "Subspace (RSpan ms)"
simps_2: fixes uu :: "'c" shows "(Num_arc_delta uu \<le> MinInfty_delta) = False"
ide_compE: fixes g :: "'b"   and f :: "'b" assumes "J0.ide (J0 g f)" obtains "J0.seq g f" and "J0.seq f g" and "J0 g f = J0.dom f" and "J0 g f = J0.cod g"
Abs_bset_inject: fixes x :: "'d set"   and y :: "'d set" assumes "x \<in> {A. |A| \<le>o ctwo}"   and "y \<in> {A. |A| \<le>o ctwo}" shows "(Abs_bset x = Abs_bset y) = (x = y)"
Ball_set_list_all: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
absorb1: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># a" shows "a \<union># b = a"
rk2_slp_def: shows "rk2_slp = slp_of_fas (rk2_fas' D)"
newQuery_equi_dev: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and I :: "nat set"   and t :: "'a option list" shows "newQuery V Q (I, t) = projectTable V ` (\<lambda>tab. semiJoin tab (I, t)) ` Q"
ld_bounded: fixes n :: "nat" shows "Max {i. 2 ^ i \<le> Suc n} \<le> Suc n"
ipurge_fail_aux_t_inv_1_def: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and Y :: "('c, 'b) ipurge_rec" shows "ipurge_fail_aux_t_inv_1 I D U xs Y \<equiv> (case ListOp Y of None \<Rightarrow> [] | Some ys \<Rightarrow> ys) @ ipurge_tr_aux I D U (List Y) = ipurge_tr_aux I D U xs"
minus_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a - b = fsub To_nearest a b"
ivl_disj_un_one_6: fixes l :: "'c"   and u :: "'c" assumes "l < u" shows "{l<..<u} \<union> {u..} = {l<..}"
vneq0: shows "v \<noteq> 0\<^sub>v n"
sup_Inf: fixes a :: "'a"   and B :: "'a set" shows "sup a (Inf B) = Inf (sup a ` B)"
elims_1: fixes x :: "('a \<times> 'c) list \<times> 'o"   and xa :: "('a \<times> 'c) list \<times> 'o"   and y :: "bool" assumes "oalist_eq_raw x xa = y" obtains xs :: "('a \<times> 'c) list" and ox :: "'o" and ys :: "('a \<times> 'c) list \<times> 'o" where "x = (xs, ox)" and "xa = ys" and "y = (xs = sort_oalist_aux ox ys)"
conway_c3: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
borel_measurable_max: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
c_card_def: shows "c_card = (\<lambda>u. card (nat_to_set u))"
bulkload_tabulate: fixes xs :: "'a list" shows "Mapping.bulkload xs = Mapping.tabulate [0..<length xs] ((!) xs)"
cancel_pseudo_hoop_algebra_axioms_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.cancel_pseudo_hoop_algebra_axioms times \<equiv> (\<forall>a b c. times a b = times a c \<longrightarrow> b = c) \<and> (\<forall>b a c. times b a = times c a \<longrightarrow> b = c)"
degree_pderiv_less: fixes p :: "'b poly" assumes "pderiv p \<noteq> 0" shows "degree (pderiv p) < degree p"
length_sorted_list_of_set: fixes A :: "'t set" shows "length (ord_term_lin.sorted_list_of_set A) = card A"
finite_induct: fixes F :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (x \<triangleright> F)" shows "P F"
list_of_dlist_dlist: fixes xs :: "'b list" shows "list_of_dlist (dlist xs) = remdups xs"
pref_pow_ext_take: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<^sup>@ k" shows "u \<le>p take \<^bold>|r\<^bold>| u \<cdot> r \<^sup>@ k"
Cont: fixes generat :: "('a, 'b, ('a, 'b, 'c) rpv) generat"   and gpv :: "('a, 'b, 'c) gpv"   and c :: "('a, 'b, 'c) rpv"   and input :: "'c" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "c \<in> generat_conts generat"   and "x \<in> results'_gpv (c input)"   and "P (c input)" shows "P gpv"
add_carrier_vec: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 \<in> carrier_vec n"
bind_rpv_def: fixes rpv :: "('g, 'f, 'd) rpv"   and f :: "'g \<Rightarrow> ('e, 'f, 'd) gpv" shows "rpv \<bind> f = (\<lambda>input. rpv input \<bind> f)"
Cidr: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f"
run_of_lasso_def: fixes L :: "'e lasso" shows "run_of_lasso L \<equiv> lasso_reach L \<frown> (lasso_cycle L)\<^sup>\<omega>"
assms: shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
limitin_def: fixes X :: "'c topology"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter" shows "limitin X f l F \<equiv> l \<in> topspace X \<and> (\<forall>U. openin X U \<and> l \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> U))"
onlyTwoNets_def: fixes x :: "('b, 'c) Combinators" shows "onlyTwoNets x = ((\<exists>a b. sdnets x = {(a, b)}) \<or> (\<exists>a b. sdnets x = {(a, b), (b, a)}))"
distinct_4: fixes x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap"   and x11 :: "name"   and x12 :: "value list" shows "Vrecabs x31 x32 x33 \<noteq> Vconstr x11 x12"
game\<^sub>0: shows "run_gpv oracle game\<^sub>0 empty = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. coin_spmf \<bind> (\<lambda>b. exec_gpv oracle' (\<A>\<^sub>1 (\<^bold>g [^] x)) ({}, empty) \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains msg1 msg2) \<bind> (\<lambda>_. oracle s_h (\<^bold>g [^] (x * y)) \<bind> (\<lambda>(h, s_h'). let cipher = (\<^bold>g [^] y, h [\<oplus>] (if b then msg1 else msg2)) in exec_gpv oracle' (\<A>\<^sub>2 cipher \<sigma>) (s, s_h') \<bind> (\<lambda>(b', s', s_h''). return_spmf (b' = b)))) ELSE (coin_spmf \<bind> return_spmf)))))"
cos_minus: fixes x :: "'b" shows "cos (- x) = cos x"
card_of_Times_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "B \<noteq> {}"   and "|B| \<le>o |A|" shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
assms_1: shows "\<forall>t2\<le>t. a t2 = []"
case_2: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a" shows "(case \<infinity> of eint x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
\<H>: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> \<H>" shows "h holomorphic_on S"
succ_inv_4: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "\<B>\<^sub>i s = \<B>\<^sub>i s'"
inv_solve_right': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(b \<oplus> \<ominus> c = a) = (b = a \<oplus> c)"
boundedBy0: fixes A :: "(nat \<times> 'b) set" shows "boundedBy 0 A = (A = {})"
u: shows "approx_tse_form' prec t f s (snd (split_float_interval ivl)) cmp"
G1': shows "s |\<^sub>s Suc n = nextsuffix (s |\<^sub>s n)"
intro_Flow: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and s :: "nat"   and t :: "nat" assumes "\<forall>e. (0::'capacity) \<le> f e \<and> f e \<le> cf e"   and "\<forall>v\<in>cf.V - {s, t}. sum f (cf.incoming v) = sum f (cf.outgoing v)" shows "Flow cf s t f"
game_1_def: fixes rand :: "'a spmf"   and \<A> :: "(bool, 'b, 'a) gpv" shows "Pseudo_Random_Function.prf.game_1 rand \<A> = exec_gpv (random_function.random_oracle rand) \<A> Map.empty \<bind> (\<lambda>(b, uu_). return_spmf b)"
is_partial_magma_2: shows "partial_magma.null (\<star>\<^sub>B) = B.null"
set_plus_mono3_b: fixes a :: "'b"   and C :: "'b set"   and x :: "'b"   and D :: "'b set" assumes "a \<in> C"   and "x \<in> a +o D" shows "x \<in> C + D"
can_restrict_env: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>(C\<cdot>r) = (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)\<cdot>(C\<cdot>r)"
typ_of_imp_closed: fixes t :: "Core.term"   and ty :: "typ" assumes "typ_of t = Some ty" shows "is_closed t"
finfun_Ex_def: fixes P :: "'b \<Rightarrow>f bool" shows "finfun_Ex P = (\<not> finfun_All (Not \<circ>$ P))"
a_ptr_disconnected_node_rel_def: fixes h :: "(_) heap" shows "a_ptr_disconnected_node_rel h = (\<lambda>(x, y). (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y)) ` {(document_ptr, disconnected_node). document_ptr |\<in>| document_ptr_kinds h \<and> disconnected_node \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r}"
assms_2: fixes x :: "'b"   and s :: "'a" assumes "x \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g callee s x \<surd>"
bangInduct: fixes P :: "pi"   and Rs :: "residual"   and F :: "'a \<Rightarrow> pi \<Rightarrow> residual \<Rightarrow> bool"   and C :: "'a" assumes "!P \<longmapsto> Rs"   and "\<And>a x P' C. \<lbrakk>P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'; x \<sharp> P; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (a\<guillemotleft>x\<guillemotright> \<prec> P' \<parallel> !P)"   and "\<And>\<alpha> P' C. P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> F C (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"   and "\<And>a x P' C. \<lbrakk>!P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'; x \<sharp> P; x \<sharp> C; \<And>C. F C (!P) (a\<guillemotleft>x\<guillemotright> \<prec> P')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> P')"   and "\<And>\<alpha> P' C. \<lbrakk>!P \<longmapsto> \<alpha> \<prec> P'; \<And>C. F C (!P) (\<alpha> \<prec> P')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"   and "\<And>a x P' b P'' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; !P \<longmapsto> a[b] \<prec> P''; x \<sharp> C; \<And>C. F C (!P) (a[b] \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> P'[x::=b] \<parallel> P'')"   and "\<And>a b P' x P'' C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; !P \<longmapsto> a<x> \<prec> P''; x \<sharp> C; \<And>C. F C (!P) (a<x> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P''[x::=b])"   and "\<And>a x P' y P'' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; !P \<longmapsto> a<\<nu>y> \<prec> P''; y \<sharp> P; x \<sharp> C; y \<sharp> C; \<And>C. F C (!P) (a<\<nu>y> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>y>(P'[x::=y] \<parallel> P''))"   and "\<And>a y P' x P'' C. \<lbrakk>P \<longmapsto> a<\<nu>y> \<prec> P'; !P \<longmapsto> a<x> \<prec> P''; y \<sharp> P; x \<sharp> C; y \<sharp> C; \<And>C. F C (!P) (a<x> \<prec> P'')\<rbrakk> \<Longrightarrow> F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>y>(P' \<parallel> P''[x::=y]))"   and "\<And>Rs C. \<lbrakk>P \<parallel> !P \<longmapsto> Rs; \<And>C. F C (P \<parallel> !P) Rs\<rbrakk> \<Longrightarrow> F C (!P) Rs" shows "F C (!P) Rs"
sorted_list_of_set_inject: fixes A :: "'a word set"   and B :: "'a word set" assumes "signed.sorted_list_of_set A = signed.sorted_list_of_set B"   and "finite A"   and "finite B" shows "A = B"
Seq_append_app2: fixes s1 :: "hf"   and k1 :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and l :: "hf"   and j :: "hf" assumes "Seq s1 k1"   and "Seq s2 k2"   and "l = k1 @+ j" shows "app (seq_append k1 s1 s2) l = app s2 j"
is_normal_minus_float: fixes f :: "('a, 'b) IEEE.float" shows "is_normal (- f) = is_normal f"
wf_list_graph_axioms_def: fixes G :: "'b list_graph" shows "wf_list_graph_axioms G = (fst ` set (edgesL G) \<subseteq> set (nodesL G) \<and> snd ` set (edgesL G) \<subseteq> set (nodesL G))"
upred_eq_false: fixes p :: "'b upred" shows "(p =\<^sub>u false) = (\<not> p)"
scaleR_zero_left: fixes x :: "'a" shows "0 *\<^sub>R x = (0::'a)"
sg_1: shows "subgraph (S i) (S (max i j))"
Ana_vars_1: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and K :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "Ana t = (K, M)" shows "fv\<^sub>s\<^sub>e\<^sub>t (set K) \<subseteq> fv t"
tau_steps_eqvt: fixes p :: "perm"   and x :: "('idx, 'pred, 'act) formula"   and n :: "nat" shows "p \<bullet> tau_steps x n = tau_steps (p \<bullet> x) (p \<bullet> n)"
mat_of_rows_carrier_3: fixes n :: "nat"   and vs :: "'a vec list" shows "dim_col (mat_of_rows n vs) = n"
simps: fixes a1 :: "nat"   and a2 :: "hmultiset" shows "local.aligned\<^sub>e a1 a2 = (\<exists>M e. a1 = e \<and> a2 = M \<and> (\<forall>m\<in>#hmsetmset M. local.encode 0 e \<le> m))"
beta_norm_simps_4: fixes xa :: "typ"   and x :: "Core.term" shows "beta_norm (Abs xa x) = map_option (Abs xa) (beta_norm x)"
image_comp_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'a" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = f (g a)" shows "f ` A = f ` g ` A"
mreal_max_simp_3: fixes y :: "mreal" shows "mreal_max MInfty y = y"
in_code_status_status_rel_iff_3: fixes b :: "status" shows "((CSUCCESS, b) \<in> code_status_status_rel) = (b = SUCCESS)"
Scons_not_Atom: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and a :: "'a + nat" shows "Scons M N \<noteq> Atom a"
comp_assoc: fixes P :: "'c \<times> 'e \<Rightarrow> bool"   and Q :: "'e \<times> 'f \<Rightarrow> bool"   and R :: "'f \<times> 'd \<Rightarrow> bool" shows "(P ;; Q ;; R) = ((P ;; Q) ;; R)"
PAC_checker_step_def: shows "PAC_checker_step = (\<lambda>spec (stat, \<V>, A) st. case st of Add x xa xb xc \<Rightarrow> normalize_poly_spec (pac_res st) \<bind> (\<lambda>r. check_add A \<V> (pac_src1 st) (pac_src2 st) (new_id st) r \<bind> (\<lambda>eq. SPEC (\<lambda>st'. \<not> is_failed st' \<and> is_found st' \<longrightarrow> r - spec \<in> More_Modules.ideal polynomial_bool) \<bind> (\<lambda>st'. if eq then RETURN (merge_status stat st', \<V>, fmupd (new_id st) r A) else RETURN (FAILED, \<V>, A)))) | Mult x xa xb xc \<Rightarrow> normalize_poly_spec (pac_res st) \<bind> (\<lambda>r. normalize_poly_spec (pac_mult st) \<bind> (\<lambda>q. check_mult A \<V> (pac_src1 st) q (new_id st) r \<bind> (\<lambda>eq. SPEC (\<lambda>st'. \<not> is_failed st' \<and> is_found st' \<longrightarrow> r - spec \<in> More_Modules.ideal polynomial_bool) \<bind> (\<lambda>st'. if eq then RETURN (merge_status stat st', \<V>, fmupd (new_id st) r A) else RETURN (FAILED, \<V>, A))))) | Extension x xa xb \<Rightarrow> normalize_poly_spec (pac_res st - Var (new_var st)) \<bind> (\<lambda>r. check_extension A \<V> (new_id st) (new_var st) r \<bind> (\<lambda>eq. if eq then RETURN (stat, insert (new_var st) \<V>, fmupd (new_id st) r A) else RETURN (FAILED, \<V>, A))) | Del x \<Rightarrow> check_del A (pac_src1 st) \<bind> (\<lambda>eq. if eq then RETURN (stat, \<V>, fmdrop (pac_src1 st) A) else RETURN (FAILED, \<V>, A)))"
dist_triangle: fixes x :: "'b"   and z :: "'b"   and y :: "'b" shows "dist x z \<le> dist x y + dist y z"
existence_ivl_empty_iff: fixes x0 :: "'a" shows "(existence_ivl0 x0 = {}) = (0 \<notin> UNIV \<or> x0 \<notin> X)"
unit_div_mult_self: 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" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "times (divide b a) a = b"
distinct_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "distinct (map fst (RBT_Impl.entries t))"
imageI: fixes x :: "'h"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "x \<in> A" shows "f x \<in> f ` A"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option" shows "proper_intrvl.set_less_aux_Compl less proper_interval ao [] [] = proper_interval ao None"
filter_M_is_OK_I: fixes xs :: "'x list"   and h :: "'y"   and P :: "'x \<Rightarrow> ('y, 'z, bool) prog" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok P x"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> ok filter_M P xs"
mahler_measure_factor: fixes a :: "complex" shows "mahler_measure_poly (pCons (- a) 1) = max 1 (cmod a)"
add_neg_numeral_special_7: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus one) = zero"
cp_1: shows "pathstart cp = p + complex_of_real (h p)"
heap_is_wellformed_disc_nodes_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node \<in> set disc_nodes" shows "node |\<in>| node_ptr_kinds h"
domains_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "partial_magma.domains C f \<equiv> {a. partial_magma.ide C a \<and> C f a \<noteq> partial_magma.null C}"
two_comparisons_into_case_order_1: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and Q :: "'c"   and R :: "'c" shows "(if le x y then if x = y then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
measurable_iff_measure_of: fixes N :: "'d set set"   and \<Omega> :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'e measure"   and \<mu> :: "'d set \<Rightarrow> ennreal" assumes "N \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>" shows "(f \<in> M \<rightarrow>\<^sub>M measure_of \<Omega> N \<mu>) = (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
span_vertical_category_axioms: shows "span_vertical_category (\<cdot>)"
matrix_works: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and x :: "('a, 'b) vec" assumes "linear (*s) (*s) f" shows "matrix f *v x = f x"
right_options_plus: fixes u :: "game"   and v :: "game" shows "right_options (plus_game u v) = zunion (zimage (\<lambda>g. plus_game g v) (right_options u)) (zimage (plus_game u) (right_options v))"
IH_prem3: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr"   and m' :: "'a match_expr" assumes "f \<in> set fs"   and "normalized_nnf_match m"   and "f m = Some m'" shows "normalized_nnf_match m'"
invarNT_def: fixes Inv :: "'state \<Rightarrow> bool" shows "invarNT Inv \<equiv> \<forall>s a ou s'. reachNT s \<and> Inv s \<and> \<not> T (Trans s a ou s') \<and> step s a = (ou, s') \<longrightarrow> Inv s'"