hcju/isabelleps huggingface dataset:
weakSimI: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "\<And>\<Psi>'' \<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>); \<alpha> \<noteq> \<tau>\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel)"   and "\<And>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
2: shows "fold_graph And_ltln true\<^sub>n y (And\<^sub>n y)"
fbox_demodalisation3: fixes y :: "'b"   and x :: "'b"   and z :: "'b" shows "(d y \<le>  |x] d z) = (d y \<cdot> x \<le> x \<cdot> d z)"
apply_swap_simp_2: fixes a :: "'b"   and b :: "'b" shows "apply (swap a b) b = a"
lexordp_eq_simps_4: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "ord.lexordp_eq less (x # xs) (y # ys) = (less x y \<or> \<not> less y x \<and> ord.lexordp_eq less xs ys)"
module_with_axioms: shows "module_with (+\<^sub>M\<^sub>_\<^sub>1) 0\<^sub>M\<^sub>_\<^sub>1 (-\<^sub>M\<^sub>_\<^sub>1) uminus\<^sub>M\<^sub>_\<^sub>1 (*s\<^sub>w\<^sub>i\<^sub>t\<^sub>h\<^sub>_\<^sub>1)"
converse_mono: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(r\<inverse> \<subseteq> s\<inverse>) = (r \<subseteq> s)"
get_child_nodes_def: shows "get_child_nodes = (\<lambda>ptr. invoke (CD.a_get_child_nodes_tups @ local.a_get_child_nodes_tups) ptr ())"
coeff_int_prod_const: fixes xs :: "'b set"   and y :: "'b"   and f :: "'b \<Rightarrow> 'c poly"   and i :: "int" assumes "finite xs"   and "y \<notin> xs"   and "\<And>x. x \<in> xs \<Longrightarrow> degree (f x) = 0" shows "coeff_int (prod f (insert y xs)) i = (\<Prod>x\<in>xs. coeff_int (f x) 0) * coeff_int (f y) i"
vector_fraction_eq_iff: fixes u :: "real"   and v :: "real"   and a :: "'a"   and x :: "'a" shows "((u / v) *\<^sub>R a = x) = (if v = 0 then x = (0::'a) else u *\<^sub>R a = v *\<^sub>R x)"
simps: fixes uu :: "'b extB" shows "uu\<^sup>\<star> = extB.Val top"
to_fun_monom: fixes c :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp"   and "x \<in> carrier Zp" shows "monom (UP Zp) c n \<bullet> x = c \<otimes> x [^] n"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> D.CCC.arr f" shows "D.T.ToTC f = D.null"
IH: fixes Z :: "('b \<Rightarrow> 'c) set" assumes "u' \<in> Z" shows "\<exists>Z'. A \<turnstile> \<langle>l', Z\<rangle> \<leadsto>* \<langle>l'', Z'\<rangle> \<and> u'' \<in> Z'"
rel_state_elim: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "(heap, 'e) state"   and g :: "'f Heap"   and heap :: "heap" assumes "local.rel_state R f g"   and "P heap" obtains heap' :: "heap" and v :: "'e" and v' :: "'f" where "run_state f heap = (v, heap')" and "execute g heap = Some (v', heap')" and "R v v'" and "P heap'"
WhileF: fixes s :: "'b"   and b :: "'b set"   and P :: "'b com" assumes "s \<notin> b" shows "(Some (While b P), s) -c\<rightarrow> (None, s)"
inner_prod_smult_right: fixes u :: "complex vec"   and n :: "nat"   and v :: "complex vec"   and a :: "complex" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n" shows "inner_prod (a \<cdot>\<^sub>v u) v = conjugate a * inner_prod u v"
P_def: fixes t :: "'a rexp" shows "P t \<equiv> \<exists>t'. t \<sim> ro t +\<^sub>r t' \<and> ro t' = 0\<^sub>r"
facet_unique_other_chamber: fixes B :: "'a set"   and z :: "'a set"   and C :: "'a set"   and D :: "'a set" assumes "folding.chamber B"   and "z \<lhd> B"   and "folding.chamber C"   and "z \<lhd> C"   and "folding.chamber D"   and "z \<lhd> D"   and "C \<noteq> B"   and "D \<noteq> B" shows "C = D"
sum_pmf_util_commute: fixes p :: "'a pmf" shows "(\<Sum>a\<in>outcomes. pmf p a * u a) = (\<Sum>a\<in>outcomes. u a * pmf p a)"
simps_4: fixes vb :: "'a"   and va :: "'a local.term" shows "Nml (\<^bold>\<langle>vb\<^bold>\<rangle>\<^sub>0 \<^bold>\<star> va) = False"
elims_2: fixes x :: "'a word list" assumes "signed.sorted x" obtains "x = []"   | xa :: "'a word" and ys :: "'a word list" where "x = xa # ys" and "Ball (set ys) ((\<le>s) xa) \<and> signed.sorted ys"
notmult: fixes ls :: "of_action list" shows "OF_priority_match OF_match_fields_safe oft p = Action ls \<longrightarrow> length ls \<le> 1"
I_def: shows "I = - complex_of_real (pbernpoly (Suc n) (real N)) / (of_nat (Suc n) * (of_nat N + s) ^ n) + complex_of_real (bernoulli (Suc n) / real (Suc n)) / s ^ n + integral {0..real N} (\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n)"
netsDistinct_def: fixes a :: "'b set set"   and b :: "'b set set" shows "netsDistinct a b = (\<nexists>x. x \<sqsubset> a \<and> x \<sqsubset> b)"
bounded_sesquilinear_uminus: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_sesquilinear A" shows "bounded_sesquilinear (\<lambda>x y. - A x y)"
set_take_two: fixes A :: "'b set" assumes "2 \<le> card A" obtains x :: "'b" and y :: "'b" where "x \<in> A" and "y \<in> A" and "x \<noteq> y"
action_absc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" shows "action (absc cfg) = abst (action cfg)"
exhaust: fixes y :: "'b Trace" obtains x1 :: "'b" where "y = tInit x1"   | x21 :: "'b Trace" and x22 :: "'b" where "y = x21 \<leadsto> x22"
fmlookup_inject: fixes x :: "('d, 'e) fmap"   and y :: "('d, 'e) fmap" shows "(fmlookup x = fmlookup y) = (x = y)"
mono_post_fun: fixes p :: "'b" shows "mono (post_fun p)"
sim_disj: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<union> AX10 \<turnstile>\<^sub>H H"   and "G \<triangleright> \<Gamma> \<union> AX10 \<turnstile>\<^sub>H H"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<or> G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H H"
simps: fixes i :: "int"   and j :: "int" shows "[i..j] = (if i \<le> j then i # [i + 1..j] else [])"
is_balanced_sum_mod_2: assumes "is_balanced" shows "(f 0 + f 1) mod 2 = 1"
f_def: shows "f = monom (UP Zp) \<one> n \<oplus>\<^bsub>UP Zp\<^esub> monom (UP Zp) (\<ominus> a) 0"
abstract_step: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat"   and c\<^sub>a :: "('e, 'f) conf"   and prb' :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb"   and "abstract_extends prb rv c\<^sub>a prb'" shows "RedBlack prb'"
local_inj: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "inj_on (\<phi> (b, a)) (C.hom b a)"
wlp_soundness: fixes S :: "com"   and P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<forall>\<rho>\<in>density_states. trace (wlp S P * \<rho>) = trace (P * denote S \<rho>) + trace \<rho> - trace (denote S \<rho>)"
strict_from_inj_preserves_F: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> local.F_inj" shows "strict_from_inj n f \<in> local.F"
circ_decompose_6: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> * (y * x\<^sup>\<circ>)\<^sup>\<circ> = y\<^sup>\<circ> * (x * y\<^sup>\<circ>)\<^sup>\<circ>"
simps_2: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" shows "funpower f (Suc n) x = f (funpower f n x)"
J: shows "{} = (\<Union>n\<le>length (xs @ [y]). \<Union>W\<in>R n. W) \<and> (\<forall>W\<in>R 0. xs @ [y] \<notin> sentences P \<and> None \<notin> set (xs @ [y]) \<and> (xs @ [y], W) \<in> failures P \<or> xs @ [y] \<in> sentences P \<and> (\<exists>U V. (xs @ [y], U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length (xs @ [y])}. \<forall>W\<in>R n. take (length (xs @ [y]) - n) (xs @ [y]) \<in> sentences P \<and> (drop (length (xs @ [y]) - n) (xs @ [y]), W) \<in> failures Q) \<and> (\<exists>n\<in>{..length (xs @ [y])}. \<exists>W. W \<in> R n)"
c_types_agree_top2: fixes ts :: "ct list"   and ts'' :: "t list"   and ts' :: "t list" assumes "ct_list_eq ts (to_ct_list ts'')" shows "c_types_agree (TopType ts) (ts' @ ts'')"
B_trusts_YM4_newK: fixes K :: "nat"   and evs :: "event list"   and NB :: "nat" assumes "Key K \<notin> analz (knows Spy evs)"   and "evs \<in> yahalom"   and "Crypt K (Nonce NB) \<in> parts (knows Spy evs)" shows "\<exists>A B NA. Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
FM_P_module: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "R module P"
istopology_def: fixes L :: "'b set \<Rightarrow> bool" shows "istopology L \<equiv> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>\<K>. (\<forall>K\<in>\<K>. L K) \<longrightarrow> L (\<Union> \<K>))"
new_tv_TVar: fixes n :: "nat"   and m :: "nat" shows "new_tv n (TVar m) = (m < n)"
C_eq_RS1n: fixes p :: "((int \<times> int) set set, int) Combinators list" shows "C (list2FWpolicy (removeShadowRules1_alternative p)) = C (list2FWpolicy p)"
choose_good: fixes v :: "'a" assumes "v \<in> winning_region p" shows "choose v \<in> good v"
memoization_correct_1: fixes dp\<^sub>T :: "'a \<Rightarrow> 'b Heap"   and x :: "'a"   and v :: "'b"   and m :: "heap" assumes "consistentDP dp\<^sub>T"   and "execute (dp\<^sub>T x) empty = Some (v, m)" shows "dp x = v"
power2_eq_imp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times x 2 = power.power one times y 2"   and "less_eq zero x"   and "less_eq zero y" shows "x = y"
tfr_subset_3: fixes B :: "('fun, 'var) Term.term set"   and A :: "('fun, 'var) Term.term set" assumes "tfr\<^sub>s\<^sub>e\<^sub>t B"   and "SMP A \<subseteq> SMP B" shows "tfr\<^sub>s\<^sub>e\<^sub>t A"
restrict_res: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "restrict v (res ts) c \<le> res ts c"
nat_filtration_borel_measurable_simple: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "simple_function M f"
cmp_simps'_3: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "trg\<^sub>D (\<Phi>\<^sub>F (\<mu>, \<nu>)) = F.map\<^sub>0 (trg\<^sub>C \<mu>)"
eq: fixes U :: "'b set" shows "{x \<in> topspace X. f x \<in> U} = {x \<in> topspace X. f x \<in> U \<and> f x \<in> S}"
finite_ReZ_segments_imp_jumpFs: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "finite_ReZ_segments g z"   and "path g" shows "finite_jumpFs (\<lambda>t. Im (g t - z) / Re (g t - z)) 0 1"
UINF_Suc_shift: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> i \<in> {Suc 0..<Suc n} \<bullet> P i) = (\<Sqinter> i \<in> {0..<n} \<bullet> P (Suc i))"
simps: fixes n :: "nat"   and S :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set"   and NS :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set"   and prc :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool \<times> bool"   and prl :: "'a \<times> nat \<Rightarrow> bool"   and \<sigma>\<sigma> :: "'a status"   and c :: "'a \<times> nat \<Rightarrow> order_tag"   and ssimple :: "bool"   and large :: "'a \<times> nat \<Rightarrow> bool"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "wpo.wpo n S NS prc prl \<sigma>\<sigma> c ssimple large s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> NS \<and> status \<sigma>\<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> S then (True, True) else if (s, t) \<in> NS then if \<exists>i\<in>set (status \<sigma>\<sigma> (f, length ss)). snd (wpo.wpo n S NS prc prl \<sigma>\<sigma> c ssimple large (ss ! i) t) then (True, True) else case t of Var x \<Rightarrow> (False, ssimple \<and> large (f, length ss)) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (status \<sigma>\<sigma> (g, length ts)). fst (wpo.wpo n S NS prc prl \<sigma>\<sigma> c ssimple large s (ts ! j))) then if prs then (True, True) else let ss' = map ((!) ss) (status \<sigma>\<sigma> (f, length ss)); ts' = map ((!) ts) (status \<sigma>\<sigma> (g, length ts)); cf = c (f, length ss); cg = c (g, length ts) in if cf = Lex \<and> cg = Lex then lex_ext (wpo.wpo n S NS prc prl \<sigma>\<sigma> c ssimple large) n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext (wpo.wpo n S NS prc prl \<sigma>\<sigma> c ssimple large) ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
sub: fixes a :: "'a" assumes "a \<in> K" shows "a \<in> G"
conc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> \<zero>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(b \<otimes> a = c \<otimes> a) = (b = c)"
add_eq: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "residues_nat n" shows "x \<oplus>\<^bsub>Residues_nat n\<^esub> y = (x + y) mod n"
insert_spec_nth_oid: fixes xs :: "'c list"   and n :: "nat"   and oid :: "'c" assumes "distinct xs"   and "n < length xs" shows "insert_spec xs (oid, Some (xs ! n)) ! Suc n = oid"
ys_def: shows "ys = sorted_wrt_list_of_set R' A"
F2_3: shows "\<forall>t\<in>fst ` set (linearize data_prev). t \<le> ot \<and> ot - t < left I"
superI: fixes C :: "char list"   and D :: "char list"   and P :: "'a prog" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "super P C = D"
acyclicI_order: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b \<subset># f a" shows "acyclic r"
defs_3: fixes r :: "Monitor.args"   and more :: "'a" shows "args.extend r more \<equiv> \<lparr>args_ivl = args_ivl r, args_n = args_n r, args_L = args_L r, args_R = args_R r, args_pos = args_pos r, \<dots> = more\<rparr>"
Gram_Schmidt_upt_k_suc: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gram_Schmidt_upt_k A (Suc k) = Gram_Schmidt_column_k (Gram_Schmidt_upt_k A k) (Suc k)"
card_partition: fixes C :: "'a set set"   and k :: "nat" assumes "finite C"   and "\<And>c. c \<in> C \<Longrightarrow> card c = k"   and "\<And>c1 c2. \<lbrakk>c1 \<in> C; c2 \<in> C; c1 \<noteq> c2\<rbrakk> \<Longrightarrow> c1 \<inter> c2 = {}" shows "card (\<Union> C) = k * card C"
l7_22_aux_R1: fixes A1 :: "'p"   and C :: "'p"   and B1 :: "'p"   and B2 :: "'p"   and M1 :: "'p"   and M2 :: "'p"   and A2 :: "'p" assumes "Bet A1 C C"   and "Bet B1 C B2"   and "Cong C A1 C B1"   and "Cong C C C B2"   and "M1 Midpoint A1 B1"   and "M2 Midpoint A2 B2"   and "C A1 Le C C" shows "Bet M1 C M2"
comp_cod_arr: fixes f :: "(bool, unit) arr"   and b :: "(bool, unit) arr" assumes "arr f"   and "cod f = b" shows "COMP b f = f"
echelon_form_of_det_def: fixes A :: "(('c, 'd) vec, 'd) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" shows "echelon_form_of_det A bezout = echelon_form_of_upt_k_det bezout (1::'c, A) (ncols A - 1)"
9: shows "\<forall>m\<ge>s. \<exists>n\<ge>m. d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j))) \<le> real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1)))"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
complex_diff: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "Complex a b - Complex c d = Complex (a - c) (b - d)"
iMin_atLeast: fixes n :: "'a" shows "iMin {n..} = n"
member_conv_keys: fixes t :: "'c set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.member t = (\<lambda>x. x \<in> set (RBT_Set2.keys t))"
compute_ord_p: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "ord_p p q = (let aux = comp_opt_p p q in aux = Some Lt \<or> aux = Some Eq)"
simps_20: fixes vb :: "('b, 'c, 'd) Tree set['b]"   and v :: "'d"   and va :: "('b, 'c, 'd) Tree" shows "(tConj vb =\<^sub>\<alpha> tAct v va) = False"
token_succeeds_def: fixes x :: "nat" shows "token_succeeds x = (\<exists>n. token_run x n \<in> F)"
Cons_1: fixes eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list" assumes "separateAtoms L = (eq, les, leq, neq)" shows "set L = set (map EqUni eq @ map LessUni les @ map LeqUni leq @ map NeqUni neq)"
nat_to_set_upper_bound1: fixes x :: "nat"   and u :: "nat" assumes "x \<in> nat_to_set u" shows "x < u"
dBox_skip: fixes \<Phi> :: "'b upred" shows "\<^bold>[II\<^bold>]\<Phi> = \<Phi>"
bisim1ALengthNull: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list" shows "P,a\<bullet>length,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 a), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
simps_2: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmarkRecord.type" shows "\<down>\<^sub>\<tau> (Suc n) k T = \<down>\<^sub>\<tau> n k (T[k \<mapsto>\<^sub>\<tau> Top]\<^sub>\<tau>)"
simps_4: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and p :: "'b fm"   and q :: "'b fm" shows "(M, s \<Turnstile> p \<^bold>\<and> q) = (M, s \<Turnstile> p \<and> M, s \<Turnstile> q)"
uses'_codem_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and nodes_of_uses :: "('val, 'node set) mapping" shows "uninst_code.uses'_codem u g next next' nodes_of_uses = fold (\<lambda>n. Mapping.update n (insert next' (Set.remove (snd next) (the (Mapping.lookup (u g) n))))) (sorted_list_of_set (case Mapping.lookup nodes_of_uses (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> id x)) (u g)"
cr_fract_def: shows "fract_as_normalized_quot.cr_fract \<equiv> \<lambda>x y. x = quot_of_fract y"
DisjointPaths_axioms: shows "DisjointPaths (remove_vertex v) v0 v1 paths"
l3_inv1_trans: shows "{l3_inv1} TS.trans l3 {> l3_inv1}"
\<Phi>_is_functor: shows "functor (\<cdot>\<acute>) (\<cdot>) \<Psi>"
lift_ms_def: fixes C :: "'a" shows "lift_ms C = MS (MSLCons (C, 0) MSLNil) (eval C)"
coeff_0_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "constant_term (p * q) = constant_term p * constant_term q"
map_cong0: fixes x :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>z. z \<in> range x \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
interval_bij_bij_cart: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and u :: "(real, 'a) vec"   and v :: "(real, 'a) vec"   and x :: "(real, 'a) vec" assumes "\<forall>i. a $ i < b $ i \<and> u $ i < v $ i" shows "interval_bij (a, b) (u, v) (interval_bij (u, v) (a, b) x) = x"
star_sup_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "(mone \<oplus> x)\<^sup>\<odot> = x\<^sup>\<odot>"
qe_interior_def: shows "qe_interior = DLO.lift_nnf_qe qe_interior\<^sub>1"
ccINF_lower: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
iFROM_UN_set_def: shows "iFROM_UN_set \<equiv> \<Union>n. {[n\<dots>]}"
assms_1: shows "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
assms: fixes kv :: "'b \<times> 'c" assumes "kv \<in> set (fst xs)" shows "f kv = g kv"
clearjunk_map: fixes f :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'e"   and ps :: "('c \<times> 'd) list" assumes "\<And>kv. fst (f kv) = fst kv" shows "clearjunk (map f ps) = map f (clearjunk ps)"
uniq_opp: fixes a :: "'a"   and b :: "'a"   and p\<^sub>2 :: "'a point"   and p\<^sub>1 :: "'a point" assumes "on_curve a b p\<^sub>2"   and "local.add a p\<^sub>1 p\<^sub>2 = Infinity" shows "p\<^sub>2 = opp p\<^sub>1"
segments_subset_convex_hull_2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "closed_segment a c \<subseteq> convex hull {a, b, c}"
simps: fixes v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "right (T v l r) = r"
mset_children: shows "queue_to_multiset (children (getMinTree q)) = tree_to_multiset (getMinTree q) - {#(val (getMinTree q), prio (getMinTree q))#}"
cbox_def: fixes a :: "'b"   and b :: "'b" shows "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
in_idealE_rep_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and B :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "hom_grading d"   and "B \<subseteq> punit.dgrad_p_set d m"   and "p \<in> punit.dgrad_p_set d m"   and "p \<in> ideal B" obtains r :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" where "keys r \<subseteq> B" and "Poly_Mapping.range r \<subseteq> punit.dgrad_p_set d m" and "p = ideal.rep r"
Rep_formula_inf: fixes x :: "'a formula"   and y :: "'a formula" shows "Rep_formula (x \<sqinter> y) = Rep_formula x \<inter> Rep_formula y"
hyps_2: fixes c :: "'a"   and s :: "'a FOL_Fitting.term" shows "subcs c s z \<turnstile> subc c s (p[App d []/0])"
R_seq: fixes p :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_R p r ; rel_R r q \<subseteq> rel_R p q"
IH_1: shows "h_minus i e \<le> h_plus i e"
ABVS: fixes b :: "'a" assumes "b \<in> B" shows "a \<noteq> b \<and> (b, a) \<in> r"
r_consist_phi_recfn: shows "recfn 2 r_consist_phi"
funpow_dom: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "f ` A \<subseteq> A" shows "f^n ` A \<subseteq> A"
permutations_of_multiset_not_empty: fixes A :: "'a multiset" shows "permutations_of_multiset A \<noteq> {}"
iter_unguarded: fixes s :: "'a"   and n :: "nat" assumes "\<not> guard s" shows "iter n s = return_spmf s"
obsv_init_def: shows "obsv_init = {\<lparr>next_round = 0, votes = \<lambda>r. Map.empty, decisions = Map.empty, obsv_state.obs = \<lambda>r. Map.empty\<rparr>}"
redundancy_criterion_standard_extension_effective: fixes \<Gamma> :: "'b inference set"   and \<Gamma>' :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "\<Gamma> \<subseteq> \<Gamma>'"   and "effective_redundancy_criterion \<Gamma> Rf Ri" shows "effective_redundancy_criterion \<Gamma>' Rf (\<lambda>N. Ri N \<union> (\<Gamma>' - \<Gamma>))"
assms_4: shows "h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h'"
t: fixes y :: "V"   and xs :: "V list"   and x :: "V" assumes "y \<notin> list.set xs"   and "y \<noteq> x" shows "y \<notin> list.set (x # xs)"
subset: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "\<Sqinter> B \<sqinter> \<Sqinter> A = \<Sqinter> A"