hcju/isabelleps huggingface dataset:
less_fun_asym: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "less_fun f g" shows "\<not> less_fun g f"
ord_isom_inj_on: fixes E :: "('g, 'h) Order_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "Order E"   and "ord_isom D E f" shows "inj_on f (carrier D)"
set_intros_2: fixes x23 :: "'a"   and x21 :: "color"   and x22 :: "('a, 'b) rbt"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "x23 \<in> set1_rbt (Branch x21 x22 x23 x24 x25)"
composition_series_def: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" shows "composition_series G \<GG> \<equiv> normal_series G \<GG> \<and> composition_series_axioms G \<GG>"
Gamma_series'_LIMSEQ: fixes z :: "'a" shows "Gamma_series' z \<longlonglongrightarrow> Gamma z"
oreachable_pair_induct: fixes \<sigma> :: "'d"   and p :: "'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma> p"   and "\<And>\<sigma> p \<sigma>'. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; P \<sigma> p; U \<sigma> \<sigma>'\<rbrakk> \<Longrightarrow> P \<sigma>' p"   and "\<And>\<sigma> p \<sigma>' p' a. \<lbrakk>(\<sigma>, p) \<in> oreachable A S U; P \<sigma> p; ((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A; S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P \<sigma>' p'" shows "P \<sigma> p"
propos_foldl: fixes f :: "'c ltl \<Rightarrow> 'c ltl \<Rightarrow> 'c ltl"   and i :: "'c ltl"   and xs :: "'c ltl list" assumes "\<And>x y. propos (f x y) = propos x \<union> propos y" shows "\<Union> {propos y |y. y = i \<or> y \<in> set xs} = propos (foldl f i xs)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> otway"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> otway; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> otway; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> otway; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Nonce NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>\<rbrace> # evs1)"   and "\<And>evs2 NB B NA A X. \<lbrakk>evs2 \<in> otway; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Nonce NA, Agent A, Agent B, X\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Nonce NA, Agent A, Agent B, X, Nonce NB, Crypt (shrK B) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB NA A B NB. \<lbrakk>evs3 \<in> otway; P evs3; Key KAB \<notin> used evs3; Gets Server \<lbrace>Nonce NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>, Nonce NB, Crypt (shrK B) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server B \<lbrace>Nonce NA, Crypt (shrK A) \<lbrace>Nonce NA, Key KAB\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 B NA A X' NB X K. \<lbrakk>evs4 \<in> otway; P evs4; B \<noteq> Server; Says B Server \<lbrace>Nonce NA, Agent A, Agent B, X', Nonce NB, Crypt (shrK B) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs4; Gets B \<lbrace>Nonce NA, X, Crypt (shrK B) \<lbrace>Nonce NB, Key K\<rbrace>\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A \<lbrace>Nonce NA, X\<rbrace> # evs4)"   and "\<And>evso B NA X NB K. \<lbrakk>evso \<in> otway; P evso; Says Server B \<lbrace>Nonce NA, X, Crypt (shrK B) \<lbrace>Nonce NB, Key K\<rbrace>\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
state_rank_step_foldl: fixes n :: "nat" shows "(\<lambda>q. state_rank q n) = foldl \<delta>\<^sub>\<R> q\<^sub>\<R> (map w [0..<n])"
ii_in_unit_circle_set: shows "ii\<^sub>h \<in> circline_set unit_circle"
bind_is_OK_pure_I: fixes f :: "('m, 'n, 'o) prog"   and h :: "'m"   and g :: "'o \<Rightarrow> ('m, 'n, 'p) prog" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
mat_extension_carrier: fixes dims :: "nat list"   and vars :: "nat set"   and m :: "'a mat" shows "mat_extension dims vars m \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
elims_1: fixes x :: "'b trm"   and y :: "bool" assumes "is_a_variable x = y" obtains xa :: "'b" where "x = Var xa__" and "y"   | xa :: "'b" where "x = Const xaa__" and "\<not> y"   | xa :: "'b trm" and ya :: "'b trm" where "x = xa \<cdot> ya" and "\<not> y"
minus_vec_def: fixes v\<^sub>1 :: "'d Matrix.vec"   and v\<^sub>2 :: "'d Matrix.vec" shows "v\<^sub>1 - v\<^sub>2 \<equiv> Matrix.vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $v i - v\<^sub>2 $v i)"
cong: fixes k :: "'a"   and v :: "'b" assumes "k \<in> Mapping.keys m" shows "f k v = f' k v"
munionMT_def: fixes uu :: "'b itself"   and m1 :: "'a"   and m2 :: "'a" shows "munionMT uu m1 m2 = Monad_Overloading.bind m1 (\<lambda>A. Monad_Overloading.bind m2 (\<lambda>B. Monad_Overloading.return (A + B)))"
UN_extend_simps_10: fixes B :: "'g \<Rightarrow> 'e set"   and f :: "'f \<Rightarrow> 'g"   and A :: "'f set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
q_def: shows "q \<equiv> 2 ^ nat \<bar>bitlen (int q')\<bar>"
Cons_1: fixes thesis :: "bool" assumes "\<And>f g. \<lbrakk>(f, g) \<in> set ps \<or> (g, f) \<in> set ps; h = monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)\<rbrakk> \<Longrightarrow> thesis"   and "h \<in> set (pdata_pairs_to_list ps)" shows "thesis"
x_1: shows "(x, \<tau>) \<notin> FV \<Gamma>"
valid_Cn_imp: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<longrightarrow>. q) = (Cn {q} \<subseteq> Cn {p})"
halloc: shows "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1"
norm_cblinfun: fixes f :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and x :: "'b" shows "norm (cblinfun_apply f x) \<le> norm f * norm x"
when_add_distrib: fixes a :: "'a"   and b :: "'a"   and P :: "bool" shows "(a + b when P) = (a when P) + (b when P)"
induct: fixes x1 :: "nat"   and x2 :: "(nat \<times> form) list"   and s :: "(nat \<times> form) list"   and P :: "nat \<Rightarrow> (nat \<times> form) list \<Rightarrow> bool" assumes "(x1, x2) \<in> deriv s"   and "P 0 s"   and "\<And>n x y. \<lbrakk>(n, x) \<in> deriv s; P n x; y \<in> set (subs x)\<rbrakk> \<Longrightarrow> P (Suc n) y" shows "P x1 x2"
orth_at_distincts: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p" assumes "X OrthAt A B C U V" shows "A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C \<and> U \<noteq> V"
monoid_isomorphism_axioms_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" shows "monoid_isomorphism_axioms \<eta> M composition unit composition' unit' \<equiv> (\<forall>x y. x \<in> M \<longrightarrow> y \<in> M \<longrightarrow> composition' (\<eta> x) (\<eta> y) = \<eta> (composition x y)) \<and> \<eta> unit = unit'"
degree_le_iff: fixes p :: "'a mpoly"   and x :: "nat"   and k :: "nat" shows "(MPoly_Type.degree p x \<le> k) = (\<forall>m\<in>monomials p. lookup m x \<le> k)"
DiffE: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A\<setminus>B" obtains "c \<in> A" and "c \<notin> B"
spaceN_diff: fixes x :: "'a"   and N :: "'a quasinorm"   and y :: "'a" assumes "x \<in> space\<^sub>N N"   and "y \<in> space\<^sub>N N" shows "x - y \<in> space\<^sub>N N"
bij_betw_quotient_range_permutation_eq_empty: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // range_permutation A B = {}"
nn_integral_mono_AE: fixes u :: "'c \<Rightarrow> ennreal"   and v :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
1: shows "[\<^bold>\<forall>x. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>x\<^sup>P,G\<rbrace> in v]"
rank_eq_card_iarrays': fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = (let A' = Gauss_Jordan_iarrays (matrix_to_iarray A) in card {row_iarray (mod_type_class.to_nat i) A' |i. \<not> is_zero_iarray (A' !! mod_type_class.to_nat i)})"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "\<Phi> \<langle>l, x, r\<rangle> = log 2 (real (size \<langle>l, x, r\<rangle>)) + \<Phi> l + \<Phi> r"
assms_3: shows "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> (S\<lparr>ArrVal\<rparr>)"
pred_spmf_def: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c spmf" shows "pred_spmf P p = (\<forall>x\<in>set_spmf p. P x)"
pmap_\<Gamma>_def: shows "pmap_\<Gamma> \<equiv> map_fun id (map_fun Rep_prefix Abs_prefix) (\<lambda>f. map (\<lambda>(x, i). (f x, i)))"
gen_ideal_monoid: fixes x :: "'a" assumes "x \<in> R" shows "Group_Theory.monoid \<langle>x\<rangle> (+) \<zero>"
distinct_of_odd: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "\<And>i. p i * i \<le> n"   and "\<And>i. p i \<noteq> 0 \<Longrightarrow> odd i"   and "distinct_of_odd p i \<noteq> 0" shows "1 \<le> i \<and> i \<le> n"
hcomp_obj_arr_1: fixes \<mu> :: "'a"   and b :: "'a" assumes "arr \<mu>"   and "obj b"   and "b = trg \<mu>" shows "b \<star> \<mu> = \<l>\<^sup>-\<^sup>1[cod \<mu>] \<cdot> \<mu> \<cdot> \<l>[local.dom \<mu>]"
cases: fixes \<Delta> :: "('a, 'b) network_scheme"   and a1 :: "'a antiparallel_edges.vertex"   and a2 :: "'a antiparallel_edges.vertex" assumes "antiparallel_edges \<Delta>"   and "antiparallel_edges.edg \<Delta> a1 a2" obtains x :: "'a" and y :: "'a" where "a1 = antiparallel_edges.Vertex x__" and "a2 = antiparallel_edges.Edge x__ y__" and "edge \<Delta> x__ y__"   | x :: "'a" and y :: "'a" where "a1 = antiparallel_edges.Edge x y" and "a2 = antiparallel_edges.Vertex y" and "edge \<Delta> x y"
bounded_WHILE'_eq: fixes n :: "nat"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and m :: "'c nres" shows "bounded_WHILE' n cond step m = bounded_WHILE n cond step m"
is_Proj_algebraic: fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "is_Proj P = (P o\<^sub>C\<^sub>L P = P \<and> P = P*)"
indRelR_reflects_binary_pred_iff_indRelL_preserves_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_reflects_binary_pred indRelR Pred = rel_preserves_binary_pred indRelL Pred"
strict_border_simps_1: fixes xs :: "'a list" shows "strict_border xs [] = False"
seqI: fixes f :: "'c"   and g :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.dom g = C.cod f" shows "C.seq g f"
cSum2: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and C :: "'a"   and P :: "pi" assumes "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "\<And>C. F C Q a x Q'" shows "F C (P \<oplus> Q) a x Q'"
setprod_one_le: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<forall>i\<le>n. f i \<le> 1" shows "prod f {..n} \<le> 1"
alpha_Tree_relI: fixes x :: "('b, 'c, 'd) Tree"   and y :: "('b, 'c, 'd) Tree" assumes "x =\<^sub>\<alpha> y" shows "(x, y) \<in> Validity.alpha_Tree_rel"
comp_proj_def: fixes X :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and n :: "nat"   and y :: "'b" shows "comp_proj X n y = {z \<in> range (proj_stoch_proc X n). z !! n = y}"
subset'_code_2: fixes ys :: "'a list"   and B :: "'a set" shows "local.subset' (set ys) B = (\<forall>y\<in>set ys. y \<in> B)"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
T: shows "(\<Sum>a\<in>(\<Union>x. set_pmf (f x)). a * pmf (f True) a) = E (f True)"
Sent_axioms_def: fixes x :: "hf"   and y :: "hf"   and z :: "hf"   and w :: "hf" shows "Sent_axioms x y z w \<equiv> x = q_Imp y y \<or> x = q_Imp y (q_Disj y z) \<or> x = q_Imp (q_Disj y y) y \<or> x = q_Imp (q_Disj y (q_Disj z w)) (q_Disj (q_Disj y z) w) \<or> x = q_Imp (q_Disj y z) (q_Imp (q_Disj (q_Neg y) w) (q_Disj z w))"
intros_12: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,R"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Catch c\<^sub>1 c\<^sub>2 Q,A"
conv_abscissa_leI: fixes c :: "ereal"   and f :: "'a fds" assumes "\<And>c'. c < ereal c' \<Longrightarrow> \<exists>s. s \<bullet> (1::'a) = c' \<and> fds_converges f s" shows "conv_abscissa f \<le> c"
add_cancel_left_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
system_result_correct: shows "proof_result SystemProof = ([], [], [(Geq (Var vid1) (Const 0) && Geq (f0 fid1) (Const 0)) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (Var vid1) (Const 0)])"
fmdom'_drop: fixes a :: "'c"   and m :: "('c, 'd) fmap" shows "fmdom' (fmdrop a m) = fmdom' m - {a}"
assms: shows "A \<subseteq> eball 0 (fps_conv_radius f)"
su3l: shows "flow0 x su3 \<in> {a<--<flow0 x su2}"
that_7: fixes CTa :: "nat \<Rightarrow> classDef option"   and Ca :: "nat"   and ea :: "exp list"   and e' :: "exp list"   and eia :: "exp"   and ei' :: "exp"   and el :: "exp list"   and er :: "exp list" assumes "x = CTa"   and "xa = New Ca ea"   and "xb = New Ca e'"   and "CTa \<turnstile> eia \<rightarrow> ei'"   and "Execute.append el (eia # er) ea"   and "Execute.append el (ei' # er) e'" shows "thesis"
euler_maclaurin_strong_int: fixes b :: "int" assumes "int a \<le> b" shows "(\<Sum>k = int a..b. f (real_of_int k)) = F (real_of_int b) + C + (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real_of_int b)) - EM_remainder N (fs N) b"
gcd_add_mult: fixes m :: "'a"   and k :: "'a"   and n :: "'a" shows "gcd m (k * m + n) = gcd m n"
w_strictly_prefers_def: fixes w :: "'w"   and \<mu> :: "('m \<times> 'w) set" shows "w_strictly_prefers w \<mu> = {m' \<in> Field (Pw w). \<forall>m\<in>{m. (m, w) \<in> \<mu>}. (m, m') \<in> Pw w \<and> (m', m) \<notin> Pw w}"
sorted_ss_hdw_n: shows "w \<noteq> [] \<longrightarrow> Sorted_Less.sorted (rev (ss ! hd w))"
that: fixes y :: "'a" assumes "x \<noteq> y"   and "A = {x, y}" shows "thesis"
nn_integral_IntVal: fixes f :: "val \<Rightarrow> ennreal" shows "integral\<^sup>N (stock_measure INTEG) f = \<integral>\<^sup>+ x. f (IntVal x) \<partial>count_space UNIV"
module_def: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme" shows "module R M \<equiv> cring R \<and> abelian_group M \<and> module_axioms R M"
huntington: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> - y) \<squnion> - (- x \<squnion> y) = x"
cases: fixes a1 :: "ccs"   and a2 :: "ccs" assumes "a1 \<approx> a2" obtains P :: "ccs" and Q :: "ccs" where "a1 = P" and "a2 = Q" and "P \<leadsto>\<^sup>^<weakBisimulation> Q" and "Q \<approx> P"
interp_gc_def: fixes gc :: "(nat, local_state, global_state) Gen_Scheduler.global_config" shows "interp_gc gc \<equiv> interp_gs (global_config.state gc)"
proper_part_eq: shows "PP x y = (P x y \<and> x \<noteq> y)"
filt_equiv_imp_subalgebra: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure" assumes "filt_equiv F M N" shows "subalgebra N M"
rel_open_def: fixes S :: "'b set" shows "rel_open S = (rel_interior S = S)"
record_call_simp: fixes x :: "'b"   and f :: "'b \<Rightarrow> one\<^sub>\<bottom>"   and x' :: "'b" shows "(record_call x\<cdot>f) x' = (if x = x' then two_pred\<cdot>(f x') else f x')"
assms_3: shows "p.dvdm u (poly_of_vec g)"
is_subset_2: fixes TA :: "('a, 'b) tree_automaton_rec"   and c :: "'a" assumes "tree_automaton TA"   and "c \<in> \<delta>_states (ta_rules TA)" shows "c \<in> ta_rstates TA"
core_eq_open: fixes a :: "'b set" assumes "core\<^bsub>S\<^esub> a = a" shows "a open\<^bsub>S\<^esub>"
that: fixes b :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
S_def: fixes x :: "'a" shows "S x = {x} \<times> Y"
simps: fixes a :: "'s2" shows "a -\<tau>2\<rightarrow> \<infinity> = (\<exists>s s'. a = s \<and> s -\<tau>2\<rightarrow> s' \<and> s' -\<tau>2\<rightarrow> \<infinity>)"
orthogonal_split: fixes Q :: "(char list \<Rightarrow> int option) \<Rightarrow> bool"   and n :: "nat"   and P :: "(char list \<Rightarrow> int option) \<Rightarrow> bool"   and m :: "nat" assumes "(embP Q \<and>* $ n) = (embP P \<and>* $ m)" shows "Q = P \<and> n = m \<or> Q = (\<lambda>s. False) \<and> P = (\<lambda>s. False)"
sorted_list_of_pdevs_domain_eq: fixes X :: "'a pdevs" shows "sorted_list_of_set (pdevs_domain X) = filter ((\<noteq>) (0::'a) \<circ> pdevs_apply X) [0..<degree X]"
assms_2: assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>"   and "src \<mu> = a"   and "trg \<nu> = c" shows "T"
simps_2: fixes x :: "nat"   and y :: "nat"   and ps :: "(nat \<times> nat) list" shows "ExcessNotAtRec ((x, y) # ps) = (\<lambda>g. max (ExcessNotAtRec ps g) (y + ExcessNotAtRec (deleteAround g x ps) g))"
ccval_def: fixes cc :: "('d, 'e) cconstraint" shows "\<lbrace>cc\<rbrace> \<equiv> {v. v \<turnstile> cc}"
semiring_norm_28: fixes n :: "num" shows "Num.BitM (num.Bit1 n) = num.Bit1 (num.Bit0 n)"
quotient_eq: shows "{f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_and_range_permutation A B = {F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B. univ (\<lambda>f. inj_on f A) F}"
\<rho>_ide_simp: fixes a :: "'d" assumes "ide a" shows "\<rho> a = \<r>[a]"
suc_AboveS: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}" shows "suc B \<in> local.AboveS B"
arg_uminus_opposite_sign: fixes z :: "complex" assumes "z \<noteq> 0" shows "(0 < arg z) = (\<not> 0 < arg (- z))"
assms_2: shows "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
cinner_sum_right: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "\<langle>x, sum f A\<rangle> = (\<Sum>y\<in>A. \<langle>x, f y\<rangle>)"
fusion: fixes \<phi> :: "i \<Rightarrow> bool" assumes "GM"   and "\<exists>x. \<phi> x" shows "\<exists>z. \<forall>y. O y z = (\<exists>x. \<phi> x \<and> O y x)"
boundedE: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "A \<noteq> {}"   and "signed.Max A \<le>s x" obtains "\<And>a. a \<in> A \<Longrightarrow> a \<le>s x"
cs: shows "cs \<equiv> THE ds. length ds = length vs \<and> v + v' = ds \<bullet>\<cdot> vs"
induct: fixes P :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "'b mat" assumes "\<And>val l i A. P val l i 0 A"   and "\<And>val l i k A. P val (l - 1) (i - 1) k (add_col_sub_row val i l A) \<Longrightarrow> P val l i (Suc k) A" shows "P a0 a1 a2 a3 a4"
monofun_cfun_arg: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<rightarrow> 'b" assumes "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> f\<cdot>y"
induct: fixes P :: "('c, 'd) strand_step list \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step list" assumes "P []"   and "\<And>x S. P S \<Longrightarrow> P (x # S)" shows "P a0"
sets_bind_measurable: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and B :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets B"
br'_\<alpha>_def: shows "br'_\<alpha> = (\<lambda>(Q, W, rcm). (Q, W))"
sum_cartesian_product': fixes f :: "'b \<times> 'c \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'c set" shows "sum f (A \<times> B) = (\<Sum>x\<in>A. \<Sum>y\<in>B. f (x, y))"
assms_2: shows "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"
fimage_eqI: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b fset" assumes "b = f x"   and "x |\<in>| A" shows "b |\<in>| f |`| A"
emptyE: fixes a :: "'c"   and P :: "bool" assumes "a \<in> {}" shows "P"
split_float_interval_realD: fixes X :: "float interval"   and A :: "float interval"   and B :: "float interval"   and x :: "real" assumes "split_float_interval X = (A, B)"   and "x \<in>\<^sub>r X" shows "x \<in>\<^sub>r A \<or> x \<in>\<^sub>r B"
bigE: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Omega>[F](g)" obtains c :: "real" where "0 < c" and "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)"
reachable_def: fixes x :: "'v" shows "reachable x = (edge G)\<^sup>*\<^sup>* x a"
simps_22: shows "OUTfromCh data22 = {data20}"
assms_2: shows "recfn (length xs) (Pr n f g)"
Rep_cases: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and y :: "'e" assumes "type_definition Rep Abs A"   and "y \<in> A" obtains x :: "'d" where "y = Rep x"
simps_7: fixes p :: "'e" shows "Language.normalize (Call p) = Call p"
finite_vars_of: fixes t :: "'b trm" shows "finite (vars_of t)"
invs_imp_InvGlobalInfoAtBetaUpright: fixes s :: "('c, 'd) configuration stream" assumes "holds InvInfoAtBetaUpright s"   and "holds InvGlobalIncomingInfoUpright s"   and "holds InvIncomingInfoUpright s" shows "holds InvGlobalInfoAtBetaUpright s"
cleanChans_def: fixes dchans :: "integer list"   and cs :: "channel list" shows "cleanChans dchans cs = snd (foldl (\<lambda>(i, cs) c. if List.member dchans i then (i + 1, cs @ [InvChannel]) else (i + 1, cs @ [c])) (0, []) cs)"
outgoing_alt: fixes u :: "nat" shows "cf.outgoing u = Pair u ` cf.E `` {u}"
extend_quasinorm: fixes F :: "'b set"   and C :: "real"   and N :: "'b \<Rightarrow> ennreal" assumes "quasinorm_on F C N" shows "quasinorm_on UNIV C (\<lambda>x. if x \<in> F then N x else \<infinity>)"
recurrence_time_restricts_compact_flow': fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and t' :: "real"   and y :: "'b" assumes "auto_ll_on_open f X"   and "t < t'"   and "t \<in> auto_ll_on_open.existence_ivl0 f X y"   and "t' \<in> auto_ll_on_open.existence_ivl0 f X y"   and "auto_ll_on_open.flow0 f X y t = auto_ll_on_open.flow0 f X y t'" shows "range (auto_ll_on_open.flow0 f X y) = auto_ll_on_open.flow0 f X y ` {t..t'}"
Suc: shows "graeffe_poly_impl_main ((- (1::'a)) ^ degree f) f m = graeffe_poly c as m"
mset_del_min: fixes h :: "'b hp option" assumes "h \<noteq> None" shows "mset_heap (del_min h) = mset_heap h - {#get_min h#}"
R: fixes u :: "'a set"   and v :: "'a set" assumes "u \<in> nhds x"   and "v \<in> nhds y"   and "z \<notin> u \<or> z \<notin> v" shows "R"
hyps_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and s' :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>2"   and "s' \<leadsto> s" shows "\<Gamma> \<turnstile> s' is t : T\<^sub>2"
a1: shows "fst (case load_word_mem s1 addra 10 of None \<Rightarrow> (None, s1) | Some w \<Rightarrow> (Some w, add_data_cache s1 addra w 15)) = fst (case load_word_mem s2 addra 10 of None \<Rightarrow> (None, s2) | Some w \<Rightarrow> (Some w, add_data_cache s2 addra w 15))"
g_def: fixes z :: "complex" shows "g z = deriv f z * complex_of_real (r - cmod (z - a))"
roots_of_2_irr_def: fixes p :: "int poly" shows "roots_of_2_irr p = (if degree p = 1 then [Rational (Rat.Fract (- coeff p 0) (coeff p 1))] else let ri = root_info p; cr = root_info.l_r ri; B = root_bound p in roots_of_2_main p ri cr [(- B, B)] [])"