hcju/isabelleps huggingface dataset:
nodeToDigit_list: fixes nd :: "('c, 'd) Node" shows "digitToList (nodeToDigit nd) = nodeToList nd"
freshAbs_psubstAbs_updEnv: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodAbs A"   and "goodEnv rho"   and "freshAbs xs x A" shows "A $[rho [x \<leftarrow> X]_xs] = A $[rho]"
eval_var: fixes \<rho> :: "(nat \<times> bval) list"   and x :: "nat"   and v :: "bval" assumes "lookup \<rho> x = Some v" shows "\<rho> \<turnstile> EVar x \<Down> v"
withG: fixes Q :: "'a \<Rightarrow> real"   and s :: "'b" assumes "unitary Q"   and "G s" shows "wp A Q (\<phi> s) = wp B (Q \<circ> \<phi>) s"
bv_to_nat_qinj: fixes xs :: "bit list"   and ys :: "bit list" assumes "bv_to_nat xs = bv_to_nat ys"   and "length xs = length ys" shows "xs = ys"
tangent_space_dim_eq: fixes X :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "X \<subseteq> sub.tangent_space p" shows "sub.tangent_space.dim p X = dim X"
mon_n_same_proc: fixes fg :: "('g, 'f, 'h, 'i, 'j) flowgraph_rec_scheme"   and n :: "'g"   and n' :: "'g" assumes "proc_of fg n = proc_of fg n'" shows "mon_n fg n = mon_n fg n'"
GatewayReq_L2: fixes req :: "nat \<Rightarrow> reqType list"   and stop :: "nat \<Rightarrow> stopType list"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "GatewayReq req dt a stop lose d ack i vc"   and "req (t + 3 + k) = [send]"   and "inf_last_ti dt t \<noteq> []"   and "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "\<forall>m\<le>k. ack (t + 2 + m) = [connection_ok]" shows "i (t + 3 + k + d) \<noteq> []"
nxtAct_def: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.nxtAct active c t n \<equiv> THE n'. n \<le> n' \<and> active c (t n') \<and> \<not> (\<exists>k\<ge>n. k < n' \<and> active c (t k))"
tendsto_obtains_N: fixes f :: "nat \<Rightarrow> 'a"   and f0 :: "'a"   and S :: "'a set" assumes "f \<longlonglongrightarrow> f0"   and "open S"   and "f0 \<in> S" obtains N :: "nat" where "\<forall>n\<ge>N. f n \<in> S"
sup_ge2: fixes y :: "'b"   and x :: "'b" shows "y \<le> x + y"
rupture_one_not_zero: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)"   and "0 < degree p" shows "\<one>\<^bsub>Rupt\<^bsub>A [X]\<^esub> K p\<^esub> \<noteq> \<zero>\<^bsub>Rupt\<^bsub>A [X]\<^esub> K p\<^esub>"
tauLaw2CongSimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<tau>.P \<leadsto>\<guillemotleft>Rel\<guillemotright> P \<oplus>\<^sub>\<top> \<tau>.P"
rsp: shows "eq_onp (\<lambda>v. v \<noteq> vec_zero) 0\<^sub>v 0\<^sub>v"
assms: shows "X \<in> M \<rightarrow>\<^sub>M N"
ord_resolve_rename_sound: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and I :: "'a set"   and \<eta> :: "'s" assumes "ord_resolve_rename S CAs DA AAs As \<sigma> E"   and "\<And>\<sigma>. is_ground_subst \<sigma> \<Longrightarrow> I \<TTurnstile>m (mset CAs + {#DA#}) \<cdot>cm \<sigma>"   and "is_ground_subst \<eta>" shows "I \<TTurnstile> E \<cdot> \<eta>"
finite_ucycles: fixes G :: "nat set \<times> nat set set" assumes "finite (uverts G)" shows "finite (ucycles G)"
kpkk: shows "ket_plus_k k = Matrix.vec (2 ^ Suc k) (\<lambda>l. 1 / csqrt (2 ^ Suc k))"
type_definition_num1: shows "type_definition Rep_num1 Abs_num1 UNIV"
iRelease_iWeakUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<R> t I. Q t) = (Q t'. t' \<W> t I. Q t \<and> P t)"
const_lt_component_subset: fixes B :: "('t \<Rightarrow>\<^sub>0 'e) set" shows "const_lt_component ` (B - {0}) - {None} \<subseteq> Some ` component_of_term ` Keys B"
assms: shows "\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"
sorted_list_of_multiset_singleton: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq {#x#} = [x]"
ereal_diff_gr0: fixes a :: "ereal"   and b :: "ereal" assumes "a < b" shows "0 < b - a"
map_val_conv: fixes xs :: "expr list"   and ys :: "val list" shows "(xs = map Val ys) = map_val xs ys"
c_eq: shows "column j (Gram_Schmidt_upt_k A (mod_type_class.to_nat k)) = column i (Gram_Schmidt_upt_k A (mod_type_class.to_nat k))"
scaleRat_scaleRat_ns_constraint: fixes a :: "rat"   and b :: "rat"   and c :: "'a ns_constraint" assumes "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0" shows "a *R b *R c = (a * b) *R c"
dom_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "local.dom (local.dom f) = local.dom f"
exhaust_sel: fixes option :: "'b option" obtains "option = \<bottom>"   | "option = \<lfloor>the option\<rfloor>"
min_elems_no: fixes x :: "'c list"   and A :: "'c list set"   and t :: "'c list" assumes "x \<in> min_elems A"   and "t \<in> A"   and "t \<le> x" shows "x = t"
share_append_Read\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat"   and S :: "nat \<Rightarrow> bool option" shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Read\<^sub>s\<^sub>b volatile a t v])) S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"
empty_neutral_1: fixes x :: "'d multiset" shows "{#} + x = x"
numgcd_nz: fixes t :: "MIR.num" assumes "nozerocoeff t"   and "numgcd t = 0" shows "t = C 0"
trm_subst_disj: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<cdot> \<theta> = t" shows "fv t \<inter> subst_domain \<theta> = {}"
BinOp1Reds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and bop :: "bop"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s'\<rangle>"
smooth_compat_refl: fixes k :: "enat"   and x :: "('b, 'c) chart" shows "k-smooth_compat x x"
less_infinityE: fixes n :: "enat" assumes "n < \<infinity>" obtains k :: "nat" where "n = enat k"
e2: shows "edges G - (F - E) = edges G - F \<union> E"
antisym: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "b \<le> a" shows "a = b"
edges_restrict_edges: fixes g :: "'a ugraph"   and E :: "('a \<times> 'a) set" shows "edges (restrict_edges g E) = edges g \<inter> (E \<union> E\<inverse>)"
subst_clause'_cons: fixes xs :: "'b ltln list"   and x :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause' (xs @ [x]) m = subst_clause' xs m \<otimes>\<^sub>l list_dnf (subst x m)"
t_FIX_inversion: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "lam"   and T :: "ty" assumes "\<Gamma> \<turnstile> FIX [x].t : T"   and "x \<sharp> \<Gamma>" shows "(x, T) # \<Gamma> \<turnstile> t : T"
prime_elem_imp_irreducible: 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 p :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p" shows "comm_semiring_1.irreducible times one zero p"
set_mode_get_child_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" assumes "l_set_mode_get_child_nodes type_wf set_mode set_mode_locs known_ptr get_child_nodes get_child_nodes_locs" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
p_fun_distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_p_fun x" shows "x ; (y \<cdot> z) = x ; y \<cdot> x ; z"
inres_simps_2: fixes X :: "'a set" shows "inres (RES X) = (\<lambda>x. x \<in> X)"
encD: fixes v :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list" assumes "v \<in> enc (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "v = map2 (enc_atom I) [0..<length v] (stake (length v) (w @- sconst any))"
nsum_mem_in_Vr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v"   and "\<forall>j\<le>n. f j \<in> carrier K"   and "\<forall>j\<le>n. 0 \<le> v (f j)" shows "\<Sigma>\<^sub>e K f n \<in> carrier (Vr K v)"
rprodl_def: shows "rprodl = (\<lambda>((a, b), c). (a, b, c))"
ennreal_mult_le_self2I: fixes y :: "ennreal"   and x :: "ennreal" assumes "0 < y \<Longrightarrow> x \<le> 1" shows "x * y \<le> y"
oalist_eq_iff: fixes xs :: "('a, 'b) oalist"   and ys :: "('a, 'b) oalist" shows "(xs = ys) = (list_of_oalist xs = list_of_oalist ys)"
sum_upto_mangoldt_over_id_asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>d. mangoldt d / real d) x - ln x) \<in> O(\<lambda>_. 1)"
pre: shows "wf_until_aux \<sigma> V R args \<phi> \<psi> aux ne"
trm_def: shows "trm \<equiv> \<lambda>(X, mrk). 2 * card (- mrk) + card X"
2: shows "NoSmallSeparationsInduct G v0 v1 n"
iterate_additive: fixes t\<^sub>1 :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and t\<^sub>2 :: "nat"   and z :: "'a" assumes "iterate t\<^sub>1 f x = y"   and "iterate t\<^sub>2 f y = z" shows "iterate (t\<^sub>1 + t\<^sub>2) f x = z"
simps_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a" shows "ins2 n d x \<langle>\<rangle> = (if bal_i (n + 1) (d + 1) then Bal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> else Unbal2 \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> 1 1)"
fg: fixes y :: "'a" assumes "y \<in> V" shows "f (g y) = y"
2202: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (y \<squnion> x) = - (y \<squnion> x)"
Iff_reduce': fixes p :: "'a pfm"   and r :: "'a pfm" assumes "static p" shows "\<turnstile>\<^sub>! ([r]\<^sub>! p \<^bold>\<longleftrightarrow>\<^sub>! reduce' r p)"
IH: fixes f :: "'a fds"   and s :: "'a" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "eval_fds ((fds_deriv ^^ n) f) s = (deriv ^^ n) (eval_fds f) s"
mzeromap_mHom: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'i) Module_scheme" assumes "R module M"   and "R module N" shows "mzeromap M N \<in> mHom R M N"
SINK_TER_in'': fixes x :: "'v" assumes "x \<in> \<E> (TER f \<union> TER g)"   and "\<And>x. x \<notin> RF (TER g) \<Longrightarrow> d_OUT g x = 0" shows "x \<in> SINK g"
pprt_add: fixes A :: "'a matrix"   and B :: "'a matrix" assumes "disj_matrices A B" shows "pprt (A + B) = pprt A + pprt B"
cltn2_transpose_transpose: fixes A :: "cltn2" shows "cltn2_transpose (cltn2_transpose A) = A"
homotopic_with_subset_right: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and Z :: "'b set" assumes "homotopic_with_canon P X Y f g"   and "Y \<subseteq> Z" shows "homotopic_with_canon P X Z f g"
obj_def': fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "horizontal_homs V src trg" shows "horizontal_homs.obj V src a = (partial_magma.arr V a \<and> trg a = a)"
dvd_div_mult_self: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times (divide b a) a = b"
ih: fixes b :: "trm" assumes "t \<Down> b" shows "a = b"
closest_is_everything: fixes s :: "('a, 'b) IEEE.float set"   and v :: "('a, 'b) IEEE.float \<Rightarrow> real"   and x :: "real"   and p :: "('a, 'b) IEEE.float \<Rightarrow> bool" assumes "finite s"   and "s \<noteq> {}" shows "is_closest v s x (closest v p s x) \<and> ((\<exists>b. is_closest v s x b \<and> p b) \<longrightarrow> p (closest v p s x))"
EX_MR_aux: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x (DenyAll # p) \<noteq> \<lfloor>DenyAll\<rfloor>" shows "\<exists>y. applied_rule_rev C x p = \<lfloor>y\<rfloor>"
new_element_get_disconnected_nodes_is_l_new_element_get_disconnected_nodes: shows "l_new_element_get_disconnected_nodes get_disconnected_nodes_locs"
assms: shows "t \<sqsubseteq> s \<cdot> \<theta>"
f_Exec_Stream_Acc_Output_nth_eq_i_nth: fixes k :: "nat"   and n :: "nat"   and n' :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" assumes "0 < k"   and "n < n'" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (input \<Down> n') c ! n = i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c n"
F'_def: shows "F' = (\<lambda>(XD, XH). (- Rf CH (undual XH), dual (- Rf (CD_on (ds - {d'})) XD)))"
swapEnv_commute: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and y :: "'var"   and var :: "'varSort \<Rightarrow> 'var set"   and zs :: "'varSort"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var" assumes "goodEnv rho"   and "{x, y} \<subseteq> var zs"   and "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}" shows "rho &[x \<and> y]_zs &[x' \<and> y']_zs' = rho &[x' \<and> y']_zs' &[x \<and> y]_zs"
refl: fixes a :: "'k multiset" shows "a \<subseteq># a"
action_loc_aux_cases: fixes adal :: "'a \<times> addr_loc"   and P :: "'b prog"   and obs :: "('a, 'c) obs_event action" assumes "adal \<in> action_loc_aux P obs" obtains hT :: "htype" and F :: "String.literal" and T :: "ty" and fm :: "fmod" and D :: "String.literal" and ad :: "'a" where "obs = NormalAction (NewHeapElem ad__ hT)" and "adal = (ad__, CField D F)" and "P \<turnstile> class_type_of hT has F:T__ (fm) in D"   | n :: "nat" and n' :: "nat" and ad :: "'a" and T :: "ty" where "obs = NormalAction (NewHeapElem ada__ (Array_type T n'))" and "adal = (ada__, ACell n)" and "n < n'"   | ad :: "'a" and al :: "addr_loc" and v :: "'a val" where "obs = NormalAction (WriteMem adb__ al__ v__)" and "adal = (adb__, al__)"   | ad :: "'a" and al :: "addr_loc" and v :: "'a val" where "obs = NormalAction (ReadMem ad al v)" and "adal = (ad, al)"
add_diff_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
sumPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim>\<^sup>s Q" shows "P \<oplus> R \<sim>\<^sup>s Q \<oplus> R"
project_Increasing_D: fixes F :: "'a program"   and G :: "'c program"   and func :: "'a \<Rightarrow> 'e" assumes "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Increasing func" shows "extend h F \<squnion> G \<in> Increasing (func \<circ> f)"
integral_norm_bound_ennreal: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "integrable M f" shows "ennreal (norm (integral\<^sup>L M f)) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
att_and_nonemptyD2: fixes x2 :: "'a attree list"   and x1 :: "'a attree"   and s :: "'a set \<times> 'a set" assumes "x2 \<noteq> []"   and "\<turnstile>x1 # x2 \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>" shows "\<turnstile>x2 \<oplus>\<^sub>\<and>\<^bsup>(snd (attack x1), snd s)\<^esup>"
rep_eq: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "FL_Formula.formula.Rep_formula (FL_Formula.Not x) = FL_Formula.Not\<^sub>\<alpha> (FL_Formula.formula.Rep_formula x)"
is_TComp_var_instance_closedD': fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and x :: "('fun, 'atom) Term.term \<times> nat"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"   and "Fun f T \<sqsubseteq> \<Gamma> (Var x)"   and "is_TComp_var_instance_closed \<Gamma> M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)" shows "\<exists>g U. Fun g U \<in> set M \<and> \<Gamma> (Fun g U) = Fun f T \<and> (\<forall>u\<in>set U. is_Var u) \<and> distinct U"
executable_edgeSet: fixes prog :: "program"   and g :: "gState\<^sub>I" assumes "gState_inv prog g"   and "program_inv prog"   and "states prog = states prog" shows "executable (states prog) g \<le> SPEC (\<lambda>cs. \<forall>(e, p)\<in>set cs. e \<in> edgeSet (states prog !! pState.idx p) \<and> pState_inv prog p \<and> cl_inv (g, p))"
extendRule2_def: fixes S1 :: "('a, 'b) sequent"   and S2 :: "('a, 'b) sequent"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" shows "extendRule2 S1 S2 r \<equiv> (map (extend S1) (fst r), extend S2 (snd r))"
failures_traces: fixes xs :: "'a list"   and X :: "'a set"   and P :: "'a process" assumes "(xs, X) \<in> failures P" shows "xs \<in> traces P"
Suc_1: assumes "t j \<le> T"   and "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (T - t 0) + 1) - 1))" shows "dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j) \<le> B / L * (exp (L * (t j - t 0) + 1) - 1) * max_stepsize j ^ p"
assms_2: shows "\<forall>i. f i \<in> A"
PL_Indis_axioms: shows "PL_Indis cval indis"
set_to_map_inverse: fixes S :: "('e \<times> 'f) set" assumes "inj_on fst S" shows "map_to_set (set_to_map S) = S"
finite_nonzero_positions: fixes A :: "'b matrix" shows "finite (nonzero_positions (Rep_matrix A))"
rel_refl_strong: fixes x :: "('a, 'b, 'c) gpv"   and R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> results'_gpv x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> outs'_gpv x \<Longrightarrow> R2a z2 z2" shows "rel_gpv R1a R2a x x"
h_eq_0: fixes t :: "real" assumes "2 \<le> t" shows "h t = 0"
assms_2: shows "(x, y) \<in> A\<^sup>\<down>"
bl: shows "bounded_linear (if x \<in> S then f' x else g' x)"
steps_z_alpha_closure_involutive: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<subseteq> V" shows "\<exists>Z''. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z''\<rangle> \<and> Closure\<^sub>\<alpha> Z' \<subseteq> Closure\<^sub>\<alpha> Z'' \<and> Z'' \<subseteq> Z'"
Int_absorb1: fixes B :: "'a set"   and A :: "'a set" assumes "order.greater_eq A B" shows "A \<inter> B = B"
assms_5: shows "preds (slice_kinds S (a # asx)) s'"
wfp_on_UNIV: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wfp_on P UNIV = wfP P"
uv: shows "w \<le>p (u \<cdot> v) \<cdot> w"
wt_nat: fixes \<Gamma> :: "(nat \<times> ty) list"   and n :: "nat" shows "\<Gamma> \<turnstile> ENat n : TNat n"
qorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.qorder less_eq less \<equiv> quasi_ordering UNIV less_eq less"
fold_multiset_equiv: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and ys :: "'a list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"   and "mset xs = mset ys" shows "fold f xs = fold f ys"
word_ao_absorbs_3: fixes x :: "'a word"   and y :: "'a word" shows "x AND (x OR y) = x"
Eq\<^sub>ES: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>E y in v] = (\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2)"
binop_axioms: shows "binop \<real>\<^sub>\<circ> vreal_mult"
approx_hcmod_approx_zero: fixes x :: "complex star"   and y :: "complex star" shows "(x \<approx> y) = (hcmod (y - x) \<approx> 0)"
order_antisym_conv: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "(x \<le> y) = (x = y)"
isCont_cot: fixes x :: "'a" assumes "sin x \<noteq> (0::'a)" shows "isCont cot x"
zero_le_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (numeral.numeral one plus n)"
finite_chamber: fixes x :: "'a set" assumes "domain.chamber x" shows "finite x"
h1: shows "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and> (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
simps_4: fixes C :: "('a, 'b, 'c) psiCase" shows "extractFrame (Case C) = \<bottom>\<^sub>F"