hcju/isabelleps huggingface dataset:
simps: fixes a :: "'b \<Rightarrow> real"   and X :: "'b set"   and I :: "'b \<Rightarrow> intv"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r :: "('b \<times> 'b) set" shows "(a \<in> region X I J r) = (\<exists>u X\<^sub>0. a = u \<and> (\<forall>x\<in>X. 0 \<le> u x) \<and> (\<forall>x\<in>X. intv_elem x u (I x)) \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d} \<and> (\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> intv'_elem x y u (J x y)))"
wellformed_items_\<I>_k_if_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k 0)" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<I> \<TT> \<RR> \<SS> Lex Sel Doc k)"
linear_homeomorphic_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "linear f"   and "inj f" shows "S homeomorphic f ` S"
r_last_def: shows "r_last \<equiv> Cn 1 r_hd [r_rev]"
that_1: fixes ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok' s t ta'"   and "actions_subset ta' ta" shows "thesis"
reduction_pair_def: fixes P :: "('a \<times> 'a) set \<times> ('a \<times> 'a) set" shows "reduction_pair P = (wf (fst P) \<and> fst P O snd P \<subseteq> fst P)"
simps_2: fixes uv :: "'b"   and h :: "'a"   and c :: "('a, 'c) antecedent" shows "labelsOut uv (Hyp h c) = h"
star_sup_one: fixes x :: "'a" shows "((1::'a) \<squnion> x)\<^sup>\<star> = x\<^sup>\<star>"
existence_ivl_cong: fixes Y :: "'a set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and t0 :: "real"   and x0 :: "'a" assumes "Y = Y"   and "S = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> g t x = g t x" shows "Y.existence_ivl t0 x0 = ll_on_open.existence_ivl S g Y t0 x0"
dup_simps_1: fixes a :: "'a" assumes "ide a" shows "arr \<d>[a]"
one_member: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "nat" assumes "reflp R" shows "{k} \<in> {S. S \<subseteq> {0..k} \<and> gen_mono_on f R S \<and> k \<in> S}"
urel_integer_plus: fixes x :: "integer"   and y :: "int"   and x' :: "integer"   and y' :: "int" assumes "urel_integer x y"   and "urel_integer x' y'" shows "urel_integer (plus_p_integer pp x x') (plus_p p y y')"
Nil_min_elems: fixes A :: "'a list set" assumes "[] \<in> A" shows "[] \<in> min_elems A"
emeasure_lborel_cbox_finite: fixes a :: "'a"   and b :: "'a" shows "emeasure lborel (cbox a b) < \<infinity>"
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_inject: fixes x :: "(_) node_ptr"   and y :: "(_) node_ptr" shows "(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 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) = (x = y)"
vp_slp_Append: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n'' :: "'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.valid_path' n as n''"   and "CFGExit.same_level_path' n'' as' n'" shows "CFGExit.valid_path' n (as @ as') n'"
simps_4: fixes f1 :: "'b"   and f2 :: "'b" shows "(case Drop of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2) = f2"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and \<alpha> :: "'g" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)"
assms_1: shows "a \<bullet> x = b \<bullet> y"
eval_fds_newman: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds fds_newman s = newman s"
degree_aform_approx_bin: fixes p :: "nat"   and ivl :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and Z :: "(real \<times> real pdevs) \<times> real"   and m :: "nat" assumes "approx_bin p ivl X Y = Some Z"   and "degree_aform_err X \<le> m"   and "degree_aform_err Y \<le> m" shows "degree_aform_err Z \<le> m"
assms_1: shows "y \<in> FvarsT (rawpsubstT r txs)"
distinct_20: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "PRODUCT x51 x52 \<noteq> REAL"
that: fixes k :: "'a\<cdot>'b writer\<cdot>'c" assumes "y = WriterT\<cdot>k" shows "thesis"
Prog\<^sub>s\<^sub>b: shows "r = Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is"
g_def: shows "g = (\<lambda>x. max (f x) 0)"
le_diff_conv2: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "(c \<subseteq># b - a) = (c + a \<subseteq># b)"
inc_below_below_pred: fixes a :: "Arity"   and b :: "Arity" assumes "inc\<cdot>a \<sqsubseteq> b" shows "a \<sqsubseteq> pred\<cdot>b"
subset_PiE: fixes I :: "'f set"   and S :: "'f \<Rightarrow> 'g set"   and T :: "'f \<Rightarrow> 'g set" shows "(Pi\<^sub>E I S \<subseteq> Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. S i \<subseteq> T i))"
map_def: fixes f :: "'c" shows "C.map f = (if C.arr f then f else C.null)"
CHORun_eq: fixes A :: "('e, 'f, 'g, 'h) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'e \<Rightarrow> 'f"   and HOs :: "nat \<Rightarrow> 'e \<Rightarrow> 'e set"   and coords :: "nat \<Rightarrow> 'e \<Rightarrow> 'e" shows "CHORun A rho HOs coords = (CHOinitConfig A (rho 0) (coords 0) \<and> (\<forall>r. CHOnextConfig A r (rho r) (HOs r) (coords (Suc r)) (rho (Suc r))))"
set_sel_1: fixes a :: "'c stream" shows "shd a \<in> sset a"
d2: shows "mat 1 (dim_vec v) (\<lambda>(i, y). conjugate v $ y) \<in> carrier_mat 1 n"
tendsto_minus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and F :: "'c filter" assumes "class.topological_group_add minus plus zero uminus open"   and "filterlim f (topological_space.nhds open a) F" shows "LIM x F. uminus (f x) :> topological_space.nhds open (uminus a)"
root_trans: fixes v :: "'b list"   and u :: "'b list"   and t :: "'b list" assumes "v \<in> u*"   and "u \<in> t*" shows "v \<in> t*"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c list_R1"   and b :: "'d list_R1" assumes "rel_list_R1 R a b" obtains xa :: "unit + 'c \<times> 'c list_R1" and ya :: "unit + 'd \<times> 'd list_R1" where "a = list_R1 xa" and "b = list_R1 ya" and "rel_sum (=) (rel_prod R (rel_list_R1 R)) xa ya"
discounted_cong': fixes X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and w :: "'b"   and Y :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and r :: "real" assumes "X n w = Y n w" shows "discounted_value r X n w = discounted_value r Y n w"
return_state_def: fixes return :: "'h \<times> 'f \<Rightarrow> 'g"   and x :: "'h" shows "return_state return x = StateT (\<lambda>s. return (x, s))"
prems_1: fixes e :: "exp" assumes "e \<in> snd ` set heap2" shows "aexp1 e = aexp2 e"
subset_\<I>: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "\<I> i \<subseteq> \<I> j"
abs_eq: fixes xb :: "nat"   and xa :: "nat"   and x :: "nat \<times> nat \<Rightarrow> 'a" shows "mat xb xa x = Abs_mat (xb, xa, mk_mat xb xa x)"
evaluation_functor_axioms: shows "evaluation_functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)"
vsv_axioms: shows "vsv [a, b]\<^sub>\<circ>"
sign_changes_neg_number_rootat_def: fixes ps :: "rat poly list" shows "sign_changes_neg_number_rootat ps = length (remdups_adj (list_neq (map poly_neg_number_rootat ps) 0)) - 1"
Tensor_in_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "\<F>C.arr f"   and "\<F>C.arr g"   and "t \<in> f"   and "u \<in> g" shows "t \<^bold>\<otimes> u \<in> f \<otimes> g"
subset_\<Q>Suc: fixes k :: "nat" shows "\<Q> k \<subseteq> \<Q> (Suc k)"
RedInitBlock: fixes V :: "char list"   and T :: "ty"   and v :: "val"   and u :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; Val u},s\<rangle> \<rightarrow> \<langle>Val u,s\<rangle>"
h3: shows "h2 \<turnstile> set_mode new_shadow_root_ptr shadow_root_mode \<rightarrow>\<^sub>h h3"
eval_elim_cases_2: fixes G :: "prog"   and s :: "st"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r Skip\<succ>\<rightarrow> (x, s')" obtains "x = \<diamondsuit>" and "s' = Norm s"
vars1: shows "{k} \<union> (vars1 - {k}) = vars1"
fun_permute_def: fixes perm1 :: "'b perm \<Rightarrow> 'c \<Rightarrow> 'c"   and perm2 :: "'b perm \<Rightarrow> 'd \<Rightarrow> 'd" shows "fun_permute perm1 perm2 \<equiv> permute perm1 \<and> permute perm2"
d_IN_lift_source: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "d_IN (lift f) (Inner (source \<Delta>)) = value_flow \<Delta> f + d_IN f (source \<Delta>)"
find_exponent_def: fixes p :: "int"   and bnd :: "int" shows "find_exponent p bnd = find_exponent_main p p 1 bnd"
inter_set_filter: fixes A :: "'b set"   and xs :: "'b list" shows "A \<inter> set xs = set (filter (\<lambda>x. x \<in> A) xs)"
assms_1: fixes x :: "real" assumes "x \<le> b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y"
preservesI: fixes F :: "'d program"   and v :: "'d \<Rightarrow> 'c" assumes "\<And>z. F \<in> stable {s. v s = z}" shows "F \<in> preserves v"
OrdNotEqP_imp_Neg_Eq: fixes x :: "tm"   and y :: "tm" shows "{x NEQ y} \<turnstile> Neg (x EQ y)"
failures_snd: shows "failures (ts_process {[], [Some b]}) = {(xs, X). xs \<in> {[], [Some b]} \<and> (\<forall>x\<in>X. xs @ [x] \<notin> {[], [Some b]})}"
rep_eq: fixes x :: "('b, 'c) \<I>" shows "responses_\<I> x = Rep_\<I> x"
Ints_vec_def: shows "\<int>\<^sub>v = {x. \<forall>i<dim_vec x. x $ i \<in> \<int>}"
root_nat_floor: fixes p :: "nat"   and x :: "nat" shows "root_nat_floor p x = \<lfloor>root p (real x)\<rfloor>"
FrP: shows "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
16: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
subgraph_node: fixes H :: "('a, 'b) graph"   and G :: "('a, 'b) graph"   and v :: "'a" assumes "subgraph H G" shows "(v \<in> nodes G) = (v \<in> nodes H)"
to_nat_plus_one_less_card: shows "\<forall>a. a + (1::'a) \<noteq> (0::'a) \<longrightarrow> mod_type_class.to_nat a + 1 < CARD('a)"
face_of_imp_subset: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S" shows "T \<subseteq> S"
iso_is_arr: fixes f :: "'a \<times> 'a" assumes "S.VV.iso f" shows "S.VV.arr f"
simps: fixes w :: "'a list"   and I :: "(nat + nat set) list" shows "enc (w, I) = map_index (enc_atom I) w"
While: fixes pre :: "'c set"   and rely :: "('c \<times> 'c) set"   and b :: "'c set"   and post :: "'c set"   and P :: "'c com"   and guar :: "('c \<times> 'c) set" assumes "stable pre rely"   and "pre \<inter> - b \<subseteq> post"   and "stable post rely"   and "\<turnstile> P sat [pre \<inter> b, rely, guar, pre]"   and "\<forall>s. (s, s) \<in> guar" shows "\<turnstile> While b P sat [pre, rely, guar, post]"
induct: fixes P :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list \<Rightarrow> ('a \<times> ('b, 'a) Term.term) list \<Rightarrow> bool"   and a0 :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and a1 :: "('a \<times> ('b, 'a) Term.term) list" assumes "\<And>bs. P [] bs"   and "\<And>f ss g ts E bs. (\<And>x2. decompose (Fun f ss) (Fun g ts) = Some x2 \<Longrightarrow> P (x2 @ E) bs) \<Longrightarrow> P ((Fun f ss, Fun g ts) # E) bs"   and "\<And>x t E bs. \<lbrakk>t = Var x \<Longrightarrow> P E bs; \<lbrakk>t \<noteq> Var x; x \<notin> fv t\<rbrakk> \<Longrightarrow> P (subst_list (subst x t) E) ((x, t) # bs)\<rbrakk> \<Longrightarrow> P ((Var x, t) # E) bs"   and "\<And>v va x E bs. (x \<notin> fv (Fun v va) \<Longrightarrow> P (subst_list (subst x (Fun v va)) E) ((x, Fun v va) # bs)) \<Longrightarrow> P ((Fun v va, Var x) # E) bs" shows "P a0 a1"
cq_ff: shows "set (coeffs cq) \<subseteq> range to_fract"
inv_drop_map_eq: fixes i :: "nat" shows "inv (drop_map i) = lift_map i"
Err_any_conv: fixes P :: "'b prog"   and T :: "ty err" shows "P \<turnstile> Err \<le>\<^sub>\<top> T = (T = Err)"
cet_sub_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL" shows "\<Gamma> \<turnstile>\<^sub>c e1 -\<^sub>c e2 : REAL"
prj_simps_16: shows "C.cod trg_trg_f.Prj\<^sub>1 = trg.apex"
mult_scalar_zero_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair p 0 = 0"
TE_Wit: fixes \<sigma> :: "'tp" assumes "\<not> isRes \<sigma>"   and "protFw \<sigma>" shows "eintF (Tag \<sigma>) [eintF (Wit \<sigma>) []] = eintF (Wit \<sigma>) []"
pref_take: fixes p :: "'b list"   and w :: "'b list" assumes "p \<le>p w" shows "take \<^bold>|p\<^bold>| w = p"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>n. P 0 n"   and "\<And>v. P (Suc v) 0"   and "\<And>v va. P (Suc v + 1) (H2N (Suc v + 2) (N2H (Suc v + 1) (Suc va)) - 1) \<Longrightarrow> P (Suc v) (Suc va)" shows "P a0 a1"
square_free_m_factor_1: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.square_free_m m (f * g)" shows "poly_mod.square_free_m m f"
heap_del_min: fixes t :: "('b \<times> nat) tree" assumes "Leftist_Heap.linorder_class.heap t" shows "Leftist_Heap.linorder_class.heap (del_min t)"
towards_idem: shows "\<bottom> = \<bottom> \<squnion> \<bottom>"
simps_1: fixes l :: "'b sort_key extended"   and i :: "'b"   and u :: "'b sort_key extended" assumes "l < \<lbrakk>Final i\<rbrakk> \<and> \<lbrakk>Final i\<rbrakk> < u" shows "\<Psi> (l, u) i = Final i"
larger_is_join1: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x" shows "x \<squnion> y = x"
cases: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'d" assumes "r\<^sup>+\<^sup>+ a1 a2" obtains a :: "'d" and b :: "'d" where "a1 = a__" and "a2 = b__" and "r a__ b__"   | a :: "'d" and b :: "'d" and c :: "'d" where "a1 = a" and "a2 = c" and "r\<^sup>+\<^sup>+ a b" and "r b c"
pnet_tau_single_node: fixes p :: "net_tree"   and s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and s' :: "'c net_state" assumes "wf_net_tree p"   and "s \<in> reachable (pnet np p) TT"   and "(s, \<tau>, s') \<in> automaton.trans (pnet np p)" shows "\<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np \<tau> i p s s'"
fg_aezfun: shows "fg \<in> aezfun_set"
elims_3: fixes x :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "('a, 'c) flow_entry_match list" assumes "\<not> no_overlaps x xa" obtains \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool" and a :: "('a, 'c) flow_entry_match" and as :: "('a, 'c) flow_entry_match list" where "x = \<gamma>" and "xa = a # as" and "\<not> (no_overlaps \<gamma> as \<and> (\<forall>b\<in>set as. ofe_prio a = ofe_prio b \<longrightarrow> \<not> (\<exists>p\<in>UNIV. \<gamma> (ofe_fields a) p \<and> \<gamma> (ofe_fields b) p)))"
zero: fixes x :: "'c" shows "0 *\<^sub>R x = (0::'c)"
deleteMin_def: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.deleteMin bq = (let min = SkewBinomialHeapStruc.getMinTree bq in SkewBinomialHeapStruc.insertList (filter (\<lambda>t. SkewBinomialHeapStruc.rank t = 0) (SkewBinomialHeapStruc.children min)) (SkewBinomialHeapStruc.meld (rev (filter (\<lambda>t. 0 < SkewBinomialHeapStruc.rank t) (SkewBinomialHeapStruc.children min))) (SkewBinomialHeapStruc.remove1Prio (SkewBinomialHeapStruc.prio min) bq)))"
the_RES_inv: fixes m :: "'c nres" assumes "nofail m" shows "RES (the_RES m) = m"
symrun_interp_neutral2: fixes \<Gamma> :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma> @ [] \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
a_gla: fixes x :: "'a"   and y :: "'a" shows "(ad x \<cdot> y = (zero_class.zero::'a)) = (ad x \<le> ad y)"
boundedE: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
mem_samp_primes_excl_gt_2: fixes prime_set :: "nat set"   and x :: "nat"   and y :: "nat" assumes "rsa_base prime_set"   and "x \<in> set_spmf (sample_set_excl prime_set {y})" shows "2 < x"
content_eq_0_interior: fixes a :: "'a"   and b :: "'a" shows "(content (cbox a b) = 0) = (interior (cbox a b) = {})"
maps_simps_1: fixes f :: "'c \<Rightarrow> 'b list"   and x :: "'c"   and xs :: "'c list" shows "List.maps f (x # xs) = f x @ List.maps f xs"
prems_2: shows "(tab'(x # xs [\<mapsto>] ys)) vn = None"
strictly_precedes_e: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>"
cond_spmf_fst_try1: fixes x :: "'a"   and q :: "('a \<times> 'b) spmf"   and p :: "('a \<times> 'b) spmf" assumes "x \<notin> fst ` set_spmf q" shows "cond_spmf_fst (TRY p ELSE q) x = cond_spmf_fst p x"
module_with_def: fixes plus\<^sub>M :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>M :: "'c"   and minus\<^sub>M :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus\<^sub>M :: "'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "module_with plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M scale \<equiv> class.ab_group_add plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M \<and> module_with_axioms plus\<^sub>M scale"
heads_append: fixes xs :: "bool list"   and ys :: "bool list" shows "heads (xs @ ys) = heads xs + heads ys"
midpoint_eq_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(midpoint a b = c) = (a + b = c + c)"
chop: shows "R_Chop(ext v,l1,l2) \<and> 0 < \<parallel>l1\<parallel> \<and> 0 < \<parallel>l2\<parallel>"
msg_fresh_2: fixes \<sigma> :: "nat \<Rightarrow> state"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and oip :: "nat"   and sip :: "nat" shows "msg_fresh \<sigma> (Rrep hops dip dsn oip sip) = (1 \<le> dsn \<and> (sip \<noteq> dip \<longrightarrow> dip \<in> kD (rt (\<sigma> sip)) \<and> dsn \<le> nsqn (rt (\<sigma> sip)) dip \<and> (nsqn (rt (\<sigma> sip)) dip = dsn \<longrightarrow> the (dhops (rt (\<sigma> sip)) dip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) dip) = Aodv_Basic.inv)))"
l2_when_same: fixes l :: "nat"   and i :: "int" shows "l2_\<phi> (l, i) (l, i) = 1 / 3 / 2 ^ l"
notin_Dag_update_l: fixes q :: "ref"   and t :: "dag"   and p :: "ref"   and l :: "ref \<Rightarrow> ref"   and y :: "ref"   and r :: "ref \<Rightarrow> ref" assumes "q \<notin> set_of t" shows "Dag p (l(q := y)) r t = Dag p l r t"
interior_Iic: fixes x :: "'b"   and b :: "'b" assumes "x < b" shows "interior {..x} = {..<x}"
cinner_eq_zero_iff: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(cinner x x = 0) = (x = zero)"
contract_inclusion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "AGM_Contraction Cn contraction"   and "K = Cn A" shows "contraction K \<phi> \<subseteq> K"
pw_flat_le_iff: fixes S :: "'b nres"   and S' :: "'b nres" shows "flat_le S S' = ((\<exists>x. inres S x) \<longrightarrow> nofail S = nofail S' \<and> (\<forall>x. inres S x = inres S' x))"
Inf_atLeast: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (ord.atLeast less_eq x) = x"
poincare_collinear_def: fixes S :: "complex_homo set" shows "poincare_collinear S = (\<exists>p. is_poincare_line p \<and> S \<subseteq> circline_set p)"
simps_1: fixes z :: "gauss_int" shows "Re (gauss2complex z) = real_of_int (ReZ z)"
assms_3: shows "has_subprob_density M (embed_measure N f) \<delta>"
closed_orbit_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.closed_orbit f X x = (\<exists>T\<in>auto_ll_on_open.existence_ivl0 f X x. T \<noteq> 0 \<and> auto_ll_on_open.flow0 f X x T = x)"
deg_aboveD: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and m :: "nat" assumes "UP_ring R"   and "deg R p < m"   and "p \<in> carrier (UP R)" shows "coeff (UP R) p m = \<zero>\<^bsub>R\<^esub>"
weak_action_modality_not_tau: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and \<alpha> :: "'b"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition"   and "\<alpha> \<noteq> \<tau>" shows "indexed_weak_nominal_ts.weak_action_modality \<tau> \<alpha> x = indexed_weak_nominal_ts.weak_tau_modality \<tau> (Act \<alpha> (indexed_weak_nominal_ts.weak_tau_modality \<tau> x))"
FINITE: shows "\<forall>x\<in>{1..m}. finite (A x)"
list_rel_simp_4: fixes x :: "'a"   and l :: "'a list"   and x' :: "'b"   and l' :: "'b list"   and R :: "('a \<times> 'b) set" shows "((x # l, x' # l') \<in> \<langle>R\<rangle>list_rel) = ((x, x') \<in> R \<and> (l, l') \<in> \<langle>R\<rangle>list_rel)"
const_periodic_arithmetic: fixes y :: "complex"   and k :: "nat" shows "periodic_arithmetic (\<lambda>x. y) k"