hcju/isabelleps huggingface dataset:
iMODb_cut_greater: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and t :: "nat" shows "[ r, mod m, c ] \<down>> t = (if t < r then [ r, mod m, c ] else if r + m * c \<le> t then {} else [ t + m - (t - r) mod m, mod m, c - Suc ((t - r) div m) ])"
1: shows "arr f \<and> arr (B.\<ll> (B.dom f))"
hs_dfs_step_def: fixes post :: "'a \<Rightarrow> 'a dlist"   and S :: "('a, unit) HashMap.hashmap \<times> 'a list" shows "hs_dfs_step post S \<equiv> let (Q, W) = S; \<sigma> = hd W in ls.iterate (post \<sigma>) (\<lambda>x (Q, W). if hs.memb x Q then (Q, W) else (hs.ins x Q, x # W)) (Q, tl W)"
elims: fixes x :: "rat vec"   and xa :: "rat vec"   and y :: "constraint" assumes "x_times_c_geq_y_times_b x xa = y" obtains c :: "rat vec" and b :: "rat vec" where "x = c" and "xa = b" and "y = GEQPP (vec_to_lpoly (c @\<^sub>v 0\<^sub>v (dim_vec b))) (vec_to_lpoly (0\<^sub>v (dim_vec c) @\<^sub>v b))"
stabilizer_is_subgroup: fixes m :: "'c" assumes "m \<in> M" shows "subgroup (stabilizer m) G"
InvariantUniqAfterApplyDecide: fixes Vbl :: "nat set"   and state :: "State" assumes "\<not> Vbl \<subseteq> vars (elements (getM state))"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)" shows "let state' = applyDecide state Vbl in InvariantUniq (getM state')"
a_subgroup_in_rcosets: shows "H \<in> a_rcosets H"
Inf_nat_def: fixes X :: "nat set" shows "Inf X = least X"
less_cSup_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and y :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_above less_eq X" shows "less y (Sup X) = (\<exists>x\<in>X. less y x)"
inext_mono2_infin_fin: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "n \<noteq> Max I \<or> infinite I" shows "n < inext n I"
SYM: fixes X :: "freemsg"   and Y :: "freemsg" assumes "X \<sim> Y" shows "Y \<sim> X"
Un_absorb1: fixes A :: "'b set"   and B :: "'b set" assumes "(A::'b set) \<subseteq> (B::'b set)" shows "(A::'b set) \<union> (B::'b set) = B"
inf_sup_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y + z) \<cdot> x = y \<cdot> x + z \<cdot> x"
assoc'_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "arr \<a>\<^sup>-\<^sup>1[f, g, h]"
word_and_less': fixes b :: "'b word"   and c :: "'b word"   and a :: "'b word" assumes "b < c" shows "a AND b < c"
that: fixes a :: "'a"   and b :: "'a" assumes "S = closed_segment a b" shows "thesis"
has_contour_integral_linepath: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and b :: "complex" shows "(f has_contour_integral i) (linepath a b) = ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral i) {0..1}"
left_edge_explicit: shows "(\<lambda>y. twoC (y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
ordLeq_iff_ordLess_or_ordIso: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "(r \<le>o r') = (r <o r' \<or> r =o r')"
r_into_rtranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
a: shows "(S, Branch A B) \<in> set G \<and> w = l \<cdot> r \<and> l \<in> Lang G A \<and> r \<in> Lang G B"
det_non_zero_imp_unit: fixes A :: "'b mat"   and n :: "nat"   and b :: "'c" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'b)" shows "A \<in> Units (ring_mat TYPE('b) n b)"
OclIsMaintained_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "X.oclIsMaintained() \<equiv> \<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>oid_of (X \<tau>) \<in> dom (heap (fst \<tau>)) \<and> oid_of (X \<tau>) \<in> dom (heap (snd \<tau>))\<rfloor>\<rfloor> else invalid \<tau>"
induct: fixes x1 :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and x2 :: "IMP"   and x3 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and P :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set \<Rightarrow> IMP \<Rightarrow> ((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> Deriv"   and "\<And>G c. \<lbrakk>(G, c, twiddle) \<in> Deriv; P G c twiddle\<rbrakk> \<Longrightarrow> P G c (Sec (\<lambda>(s, t).  s \<approx> t ))"   and "\<And>G. P G Skip (Sec (\<lambda>(s, t).  s \<approx> t ))"   and "\<And>e G x. \<forall>s ss.  s \<approx> ss  \<longrightarrow> evalE e s = evalE e ss \<Longrightarrow> P G (Assign x e) (Sec (\<lambda>(s, t).  s \<approx> update t x (evalE e t) ))"   and "\<And>G c1 \<Phi> c2 \<Psi>. \<lbrakk>(G, c1, Sec \<Phi>) \<in> Deriv; P G c1 (Sec \<Phi>); (G, c2, Sec \<Psi>) \<in> Deriv; P G c2 (Sec \<Psi>)\<rbrakk> \<Longrightarrow> P G (Comp c1 c2) (Sec (\<lambda>(s, t). \<exists>r. \<Phi> (r, t) \<and> (\<forall>w.  r \<approx> w  \<longrightarrow> \<Psi> (s, w))))"   and "\<And>b G c1 \<Phi> c2 \<Psi>. \<lbrakk>\<forall>s ss.  s \<approx> ss  \<longrightarrow> evalB b s = evalB b ss; (G, c1, Sec \<Phi>) \<in> Deriv; P G c1 (Sec \<Phi>); (G, c2, Sec \<Psi>) \<in> Deriv; P G c2 (Sec \<Psi>)\<rbrakk> \<Longrightarrow> P G (Iff b c1 c2) (Sec (\<lambda>(s, t). (evalB b t \<longrightarrow> \<Phi> (s, t)) \<and> (\<not> evalB b t \<longrightarrow> \<Psi> (s, t))))"   and "\<And>b G c \<Phi>. \<lbrakk>\<forall>s ss.  s \<approx> ss  \<longrightarrow> evalB b s = evalB b ss; (G, c, Sec \<Phi>) \<in> Deriv; P G c (Sec \<Phi>)\<rbrakk> \<Longrightarrow> P G (While b c) (Sec (PhiWhile b \<Phi>))"   and "\<And>\<Phi> G. \<lbrakk>({Sec (FIX \<Phi>)} \<union> G, body, Sec (\<Phi> (FIX \<Phi>))) \<in> Deriv; P ({Sec (FIX \<Phi>)} \<union> G) body (Sec (\<Phi> (FIX \<Phi>))); Monotone \<Phi>\<rbrakk> \<Longrightarrow> P G Call (Sec (FIX \<Phi>))"   and "\<And>G. P G Skip twiddle"   and "\<And>x G e. CONTEXT x = high \<Longrightarrow> P G (Assign x e) twiddle"   and "\<And>G c1 c2. \<lbrakk>(G, c1, twiddle) \<in> Deriv; P G c1 twiddle; (G, c2, twiddle) \<in> Deriv; P G c2 twiddle\<rbrakk> \<Longrightarrow> P G (Comp c1 c2) twiddle"   and "\<And>G c1 c2 b. \<lbrakk>(G, c1, twiddle) \<in> Deriv; P G c1 twiddle; (G, c2, twiddle) \<in> Deriv; P G c2 twiddle\<rbrakk> \<Longrightarrow> P G (Iff b c1 c2) twiddle"   and "\<And>G c b. \<lbrakk>(G, c, twiddle) \<in> Deriv; P G c twiddle\<rbrakk> \<Longrightarrow> P G (While b c) twiddle"   and "\<And>G. \<lbrakk>({twiddle} \<union> G, body, twiddle) \<in> Deriv; P ({twiddle} \<union> G) body twiddle\<rbrakk> \<Longrightarrow> P G Call twiddle" shows "P x1 x2 x3"
standard_control_dependences_eq: fixes n :: "'node"   and n' :: "'node" shows "n controls\<^sub>s n' \<equiv> \<exists>as. n controls\<^sub>s n' via as"
value_mI_gen: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "j \<le> n" shows "(\<nu>\<^bsub>K P j\<^esub>) (\<Sigma>\<^sub>e K (\<lambda>k. Zl_mI K P I k \<cdot>\<^sub>r mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m_zmax_pdsI K n P I\<^esup>) n) = LI K (\<nu>\<^bsub>K P j\<^esub>) I"
assms_5: fixes i :: "'a" assumes "i \<in> I" shows "c \<le>\<^sub>o\<^sub>w f i"
synthetic_div_unique: fixes p :: "'a poly"   and c :: "'a"   and q :: "'a poly"   and r :: "'a" assumes "p + Polynomial.smult c q = pCons r q" shows "r = poly p c \<and> q = synthetic_div p c"
assms_2: fixes x :: "'d"   and xs :: "('d, 'a) tllist" shows "monotone orda ordb (\<lambda>f. tcons f x xs)"
that: fixes v :: "'t" assumes "lookup p v = (0::'b)"   and "lookup q v \<noteq> (0::'b)"   and "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = lookup q u" shows "thesis"
indRelR_cond_respects_predA_and_reflects_predB: fixes PredA :: "('procS, 'procT) Proc \<Rightarrow> bool"   and PredB :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "((\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_pred Rel PredA) \<and> (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_reflects_pred Rel PredB)) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_pred Rel PredA \<and> rel_reflects_pred Rel PredB)"
nullable_satisfies_bounded: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" assumes "Length \<AA> = 0" shows "nullable \<phi> = (\<AA> \<Turnstile>\<^sub>b \<phi>)"
update_convs_7: fixes back_edges' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and counter :: "nat"   and discovered :: "'a \<Rightarrow> nat option"   and finished :: "'a \<Rightarrow> nat option"   and pending :: "('a \<times> 'a) set"   and stack :: "'a list"   and tree_edges :: "('a \<times> 'a) set"   and back_edges :: "('a \<times> 'a) set"   and cross_edges :: "('a \<times> 'a) set"   and more :: "'b" shows "back_edges_update back_edges' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges' back_edges, cross_edges = cross_edges, \<dots> = more\<rparr>"
Diagonalize_rep_preserves_seq: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "seq g f" shows "Seq \<^bold>\<lfloor>rep g\<^bold>\<rfloor> \<^bold>\<lfloor>rep f\<^bold>\<rfloor>"
cofactor_list_syz_nth: fixes i :: "nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "i < n" shows "cofactor_list_syz n p ! i = proj_poly i p"
idiff_infinity_right: fixes a :: "int" shows "eint a - \<infinity> = \<infinity>"
ennreal_SUP_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> nat \<Rightarrow> ennreal" assumes "\<And>i. i \<in> I \<Longrightarrow> incseq (f i)" shows "(SUP n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. Sup (range (f i)))"
High_target_Entry_edge: shows "\<exists>a. valid_edge a \<and> sourcenode a = (_Entry_) \<and> targetnode a = (_High_) \<and> kind a = (\<lambda>s. True)\<^sub>\<surd>"
deterministic_RP_complete: fixes N0 :: "(('a, nat) Term.term literal list \<times> nat) list"   and n0 :: "nat" assumes "deterministic_RP (RP.St0 N0 n0) = None" shows "satisfiable (RP.grounded_N0 N0)"
rtrancl_while_Some: fixes p :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a list"   and x :: "'a"   and ws :: "'a list"   and Z :: "'a set" assumes "rtrancl_while p f x = Some (ws, Z)" shows "if ws = [] then Z = {(x, y). y \<in> set (f x)}\<^sup>* `` {x} \<and> (\<forall>z\<in>Z. p z) else \<not> p (hd ws) \<and> hd ws \<in> {(x, y). y \<in> set (f x)}\<^sup>* `` {x}"
hull_subset: fixes s :: "'d set"   and S :: "'d set \<Rightarrow> bool" shows "s \<subseteq> S hull s"
Cons_2: shows "convert_atom_list var (At # L) (xs' @ x # xs) = Some L'"
integral_dominated_convergence_at_top: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and s :: "real \<Rightarrow> 'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>t. s t \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"   and "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x" shows "((\<lambda>t. integral\<^sup>L M (s t)) \<longlongrightarrow> integral\<^sup>L M f) at_top"
canI_associator_hcomp_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "\<a>[f \<star> g, h, k] = can ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>) (((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>)"
smap_def: shows "smap \<equiv> \<mu> smap. \<Lambda> f v. Fixrec.run\<cdot>(match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>[::]) +++ match_scons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(f\<cdot>x :# smap\<cdot>f\<cdot>xs)))"
assms: shows "SubRel.ces v1 es v2 subs1"
plus_fun_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f + g) x = f x + g x"
prems_4: shows "\<forall>c\<in>set (c' # cs). v c \<le> n"
simps_5: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Cos a) vs = isDERIV x a vs"
simple_path_eq_arc: fixes g :: "real \<Rightarrow> 'a" assumes "pathfinish g \<noteq> pathstart g" shows "simple_path g = arc g"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelRT Enc TRel"
simps_2: fixes f :: "'b \<Rightarrow> 'a Heap"   and x :: "'b"   and xs :: "'b list" shows "fold_map f (x # xs) = f x \<bind> (\<lambda>y. fold_map f xs \<bind> (\<lambda>ys. return (y # ys)))"
Rep_formula: fixes x :: "('e, 'f, 'g) formula" shows "Rep_formula x \<in> {t\<^sub>\<alpha>. hereditarily_fs t\<^sub>\<alpha>}"
label_trap: fixes n :: "nat"   and es :: "e list" shows "\<lparr>[Label n es [Trap]]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
trg_cod: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "trg (cod \<mu>) = trg \<mu>"
proper_map_imp_subset_topspace: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "proper_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
wf_ruleset_normalize_rules_dnf: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list" assumes "wf_ruleset \<gamma> p rs" shows "wf_ruleset \<gamma> p (normalize_rules_dnf rs)"
Max_by_conds_1: fixes Q :: "process set" shows "finite (vote_set (votes sa) Q)"
eq: shows "(\<lambda>t. exp (- t)) ` {0<..} = {0<..<1}"
dgrad_sig_set_closed_sig_red: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_red :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "F \<subseteq> dgrad_sig_set' (length fs) d"   and "p \<in> dgrad_sig_set' (length fs) d"   and "sig_red sing_red top_tail F p q" shows "q \<in> dgrad_sig_set' (length fs) d"
simps: fixes a1 :: "'rule \<times> 'form"   and a2 :: "'form"   and a3 :: "('form, 'var) antecedent fset" shows "natEff_Inst a1 a2 a3 = (\<exists>c r. a1 = (r, c) \<and> a2 = c \<and> a3 = f_antecedent r \<and> c \<in> set (consequent r))"
le_less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y < z" shows "x < z"
is_tiny_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
codomains_comp: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.seq g f" shows "A_BxA.codomains (g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]\<^sub>x\<^sub>A f) = A_BxA.codomains g"
space_trivial_measure: fixes X :: "'a set" shows "space (trivial_measure X) = X"
Polys_closed_map_scale: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and c :: "'c" assumes "p \<in> P[X]" shows "c \<cdot> p \<in> P[X]"
from_list_correct: fixes xs :: "('fun \<times> ('b, 'c, 'fun, 'd, 'e) Inca.instr fundef) list"   and k :: "'fun" shows "map_of xs k = Finca_get (Sinca.Fenv.from_list xs) k"
trm_simp_8: fixes P :: "'b trm"   and Q :: "'b trm" assumes "Snd P = Snd Q" shows "P = Q"
decomp_rm\<^sub>e\<^sub>s\<^sub>t_ik_subset: fixes S :: "('fun, 'var) extstrand_step list" shows "ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t S) \<subseteq> ik\<^sub>e\<^sub>s\<^sub>t S"
unit_mult_div_div: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "times b (divide one a) = divide b a"
mult_2_right: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" assumes "class.semiring_numeral one times plus" shows "times z (numeral.numeral one plus (num.Bit0 num.One)) = plus z z"
graph_restrict_right_aimpl: fixes Vr :: "'a set"   and E :: "('a \<times> 'a) set" shows "op_graph_restrict_right Vr E = E_of_succ (\<lambda>v. {x \<in> E `` {v}. x \<in> Vr})"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) atm" assumes "\<not> Ik.satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "Ik.int \<xi>'__ T1 \<noteq> Ik.int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "\<not> intP p (map (Ik.int \<xi>) Tl)"
ACI_norm_eqI: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "ACI r s" shows "\<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>"
simpleDefs_phiDefs_var_disjoint: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> phiDefs g n"   and "n \<in> set (\<alpha>n g)" shows "var g v \<notin> oldDefs g n"
lt_param: fixes q :: "real" assumes "q = (1 + r - d) / (u - d)"   and "viable_market Mkt" shows "q < 1"
of_nat_mono_maybe_le: fixes x :: "nat"   and y :: "nat" assumes "x < 2 ^ LENGTH('b)"   and "y < 2 ^ LENGTH('b)" shows "(y \<le> x) = (word_of_nat y \<le> word_of_nat x)"
comm_mult_left_mono: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "(0::'c) \<le> c" shows "c * a \<le> c * b"
Re_Ln_pos_lt: fixes z :: "complex" assumes "z \<noteq> 0" shows "(\<bar>Im (Ln z)\<bar> < pi / 2) = (0 < Re z)"
ATOM_EQ_axioms: shows "ATOM_EQ aneg anormal I\<^sub>a depends\<^sub>0 decr solvable\<^sub>0 trivial subst\<^sub>0"
len_not_eq_0: fixes len_of :: "'b itself \<Rightarrow> nat" assumes "class.len len_of" shows "len_of TYPE('b) \<noteq> 0"
simps: fixes t1 :: "'b option list"   and t2 :: "'b option list" shows "New_max.merge t1 t2 = map New_max.merge_option (zip t1 t2)"
Healthy_carrier_image: fixes A :: "'d upred set"   and \<H> :: "'d upred \<Rightarrow> 'd upred" assumes "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H" shows "\<H> ` A = A"
event_set: fixes e :: "'a event" shows "e \<in> insert tick (range ev)"
mpoly_map_vars_prod: fixes f :: "nat \<Rightarrow> nat"   and g :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "mpoly_map_vars f (prod g A) = (\<Prod>x\<in>A. mpoly_map_vars f (g x))"
eq: fixes ko :: "'b key_order"   and x :: "'b"   and y :: "'b" shows "(key_compare ko x y = Eq) = (x = y)"
proj_not_active: fixes n :: "nat"   and t :: "cnf llist"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "enat n < llength t"   and "\<not> active c (lnth t n)" shows "dynamic_component.proj tCMP active c (ltake (enat (Suc n)) t) = dynamic_component.proj tCMP active c (ltake (enat n) t)"
simps_2: fixes n :: "nat" shows "gener_bool_list (Suc n) = {True # w |w. w \<in> gener_bool_list n} \<union> {False # w |w. w \<in> gener_bool_list n}"
assms_1: shows "Bisimulation_Invariants A B sim PA QA PB QB"
xor_commute: fixes a :: "'a"   and b :: "'a" shows "a XOR b = b XOR a"
bij_list_succ: fixes xs :: "'b list" assumes "distinct xs" shows "bij (list_succ xs)"
distinct_33: fixes x2 :: "sethi_type"   and x8 :: "shift_type" shows "sethi_type x2 \<noteq> shift_type x8"
All_mapping_filter_iff: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('b, 'c) mapping"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "All_mapping (Mapping.filter P m) Q = All_mapping m (\<lambda>k v. P k v \<longrightarrow> Q k v)"
row_rank_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "row_rank A = vec.dim (row_space A)"
simps_20: fixes x5 :: "char list" shows "NT \<noteq> Class x5"
dual_complete_lattice: shows "class.complete_lattice Sup Inf (\<squnion>) (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x) (\<sqinter>) \<top> \<bottom>"
"1_3": shows "Ind {a} (set v\<^sub>1)"
insert_is_Un: fixes a :: "'x"   and A :: "'x set" shows "insert a A = {a} \<union> A"
assms_2: shows "rec_inseparable (f -` A) (f -` B)"
WP1n_DA_notinSet: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "DenyAll \<notin> set (tl p)"
prems: shows "\<Gamma>\<turnstile>merge_guards (Catch c1 c2) \<down> Normal s"
prems_3: shows "a \<in> dom (share (x # sb) \<S>)"
mono_Seq: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(P `;` S) \<sqsubseteq> (Q `;` S)"
simps_1: fixes r :: "mregex" shows "RPDi 0 r = {r}"
cases: fixes a1 :: "'d rexp"   and a2 :: "'d rexp" assumes "a1 ~ a2" obtains r :: "'d rexp" and s :: "'d rexp" and t :: "'d rexp" where "a1 = Alt (Alt r__ s__) t__" and "a2 = Alt r__ (Alt s__ t__)"   | r :: "'d rexp" and s :: "'d rexp" and t :: "'d rexp" where "a1 = Alt ra__ (Alt sa__ t)" and "a2 = Alt (Alt ra__ sa__) t"   | r :: "'d rexp" and s :: "'d rexp" where "a1 = Alt rb__ sb__" and "a2 = Alt sb__ rb__"   | r :: "'d rexp" where "a1 = rc__" and "a2 = Alt rc__ rc__"   | r :: "'d rexp" where "a1 = rd__" and "a2 = rd__"   | r :: "'d rexp" and r' :: "'d rexp" and s :: "'d rexp" and s' :: "'d rexp" where "a1 = Alt re__ sc__" and "a2 = Alt r'__ s'__" and "re__ ~ r'__" and "sc__ ~ s'__"   | r :: "'d rexp" and r' :: "'d rexp" and s :: "'d rexp" and s' :: "'d rexp" where "a1 = Conc rf__ s" and "a2 = Conc r'a__ s'" and "rf__ ~ r'a__" and "s ~ s'"   | r :: "'d rexp" and r' :: "'d rexp" where "a1 = Star r" and "a2 = Star r'" and "r ~ r'"
RFNoMemVal: shows "RPCFailure \<notin> MemVal"
ord_spmf_bindI: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'b spmf"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "ord_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> ord_spmf P (f x) (g y)" shows "ord_spmf P (p \<bind> f) (q \<bind> g)"
brw_inner_step_abs: fixes \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<Sigma>' :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and r :: "('d, 'e) ta_rule" assumes "(\<Sigma>, \<Sigma>') \<in> brw_inner_step r" shows "br'_inner_step r (brw_\<alpha> \<Sigma>) = brw_\<alpha> \<Sigma>'"
T: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "\<phi> c d"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" shows "\<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t'"
Collect_mem_eq: fixes A :: "'b set" shows "{x. s2p A x} = A"
inj_on_chamber_system: shows "inj_on ((`) f) domain.\<C>"