hcju/isabelleps huggingface dataset:
case_1: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
numeral_eq_Suc: fixes k :: "num" shows "numeral k = Suc (pred_numeral k)"
\<T>_is_surjection_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "yoneda_lemma C S \<phi> F a"   and "natural_transformation (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F \<tau>" shows "transformation_by_components.map (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) (yoneda_lemma.\<T>o C S \<phi> F a (yoneda_lemma.\<E> S \<phi> a \<tau>)) = \<tau>"
assms_2: shows "A \<inter> space M \<in> sets M"
sublinear_intros_8: fixes S :: "'a \<Rightarrow> 'b set"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> sublinear (wp (p a))"   and "\<And>s a. a \<in> S s \<Longrightarrow> healthy (wp (p a))"   and "\<And>s. S s \<noteq> {}" shows "sublinear (wp (SetDC p S))"
c: shows "(c, ps) \<Rightarrow>\<^sub>A t \<Down> ps'"
Liminf_compose_continuous_mono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "mono f"   and "F \<noteq> bot" shows "Liminf F (\<lambda>n. f (g n)) = f (Liminf F g)"
f_1: fixes x :: "'a" assumes "x \<in> S" shows "norm (f x) = norm x"
C_def: fixes y :: "int" shows "C y \<equiv> y \<le> 1"
s: shows "s \<in> declassified\<^sub>l\<^sub>s\<^sub>t \<A> \<I>"
first_countable_basis: fixes "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.first_countable_topology open" shows "\<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
SeqAppendP_Mem2: fixes f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and g :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm" shows "{SeqAppendP f1 k1 f2 k2 g, HaddP k1 x x', x IN k2, HPair x y IN f2} \<turnstile> HPair x' y IN g"
has_real_derivative_polynomial_function: fixes p :: "real \<Rightarrow> real" assumes "real_polynomial_function p" shows "\<exists>p'. real_polynomial_function p' \<and> (\<forall>x. (p has_real_derivative p' x) (at x))"
chain_boundary_boundary_alt: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain (Suc p) X c" shows "chain_boundary p (chain_boundary (Suc p) c) = 0"
degree_minus: fixes p :: "'c poly" shows "degree (- p) = degree p"
trace_eq_callee_of_restI: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf" assumes "trace_rest_eq rest1 rest2 IA IU p q" shows "trace_callee_eq (callee_of_rest rest1) (callee_of_rest rest2) (IA <+> IU) p q"
gFreshGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshGOp MOD \<equiv> \<forall>ys y delta inp' inp binp' binp. wlsInp delta inp' \<and> gWlsInp MOD delta inp \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> freshInp ys y inp' \<and> gFreshInp MOD ys y inp' inp \<and> freshBinp ys y binp' \<and> gFreshBinp MOD ys y binp' binp \<longrightarrow> gFresh MOD ys y (Op delta inp' binp') (gOp MOD delta inp' inp binp' binp)"
app_hrestrict: fixes x :: "hf"   and A :: "hf"   and f :: "hf" assumes "x \<^bold>\<in> A" shows "app (hrestrict f A) x = app f x"
eval_mono: fixes s :: "state"   and e :: "expr"   and v :: "val"   and n :: "nat"   and t :: "state"   and m :: "nat" assumes "s -e\<succ>v-n\<rightarrow> t"   and "n \<le> m" shows "s -e\<succ>v-m\<rightarrow> t"
wf_subst: fixes \<Delta> :: "binding list"   and B :: "binding"   and \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and U :: "POPLmark.type" assumes "\<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> U]\<^sub>e @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f T[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau>"
insort_greater: fixes ls :: "'a list"   and y :: "'a" assumes "\<forall>x\<in>set ls. x < y" shows "insort y ls = ls @ [y]"
InvariantUniqQAfterApplyBackjump: fixes state :: "State" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = applyBackjump state in InvariantUniqQ (getQ state')"
normalize_length_eq: fixes p :: "'a list" assumes "lead_coeff p \<noteq> \<zero>" shows "length (local.normalize p) = length p"
word_and_or_mask_aligned: assumes "is_aligned b n"   and "a \<le> mask n" shows "b + a = b OR a"
JHS_Tr0_2: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "\<forall>i\<le>s * r - Suc 0. \<natural>cmp_rfn G r f s g i \<triangleright> cmp_rfn G r f s g (Suc i)"
lprefix_lsetD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys" shows "lset xs \<subseteq> lset ys"
simps: fixes t\<^sub>1 :: "'b"   and t\<^sub>2 :: "'b" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>1 / \<tau>\<^sub>c\<^sub>s\<^sub>t t\<^sub>2 = \<tau>\<^sub>c\<^sub>s\<^sub>t (t\<^sub>1 / t\<^sub>2)"
store_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (store_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
zero_less_iff_neq_zero: fixes n :: "'b multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
indicator_def: fixes S :: "'b set"   and x :: "'b" shows "indicator S x = (if x \<in> S then 1::'a else (0::'a))"
perm_pi_simp_2: fixes pi :: "(name \<times> name) list"   and x :: "'d" shows "rev pi \<bullet> pi \<bullet> x = x"
comp_red_monic_basis_GB: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "is_Groebner_basis (set (comp_red_monic_basis xs))"
trans1: shows "\<Union> (get_offending_flows (get_ACS M) \<lparr>nodes = V, edges = E \<union> backflows stateful \<union> backflows {x}\<rparr>) \<subseteq> backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {x}\<rparr>)"
P_accD_2: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" assumes "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (Kp x))\<^sup>*" shows "(x2, y2) \<in> K2.acc"
simps: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" shows "((a1, a2) \<in> g3mod1) = ((\<exists>A. a1 = [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] \<and> a2 = ( \<Empt> \<Rightarrow>* \<LM> \<box> A  \<RM>)) \<or> (\<exists>A. a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] \<and> a2 = ( \<LM> \<diamond> A  \<RM> \<Rightarrow>* \<Empt>)))"
finterE: fixes c :: "'e"   and A :: "'e fset"   and B :: "'e fset" assumes "c |\<in>| A |\<inter>| B" obtains "c |\<in>| A" and "c |\<in>| B"
lossless_sample_coprime: fixes N :: "nat" assumes "\<not> prime N"   and "2 < N" shows "lossless_spmf (sample_coprime N)"
hyp_edge: fixes c :: "'form"   and "is" :: "nat list"   and n :: "nat"   and s :: "'subst"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" assumes "c \<in> set conclusions"   and "is \<in> it_paths (it' c)"   and "tree_at (it' c) is = HNode n s ants" shows "hyp_edge_at c is n s \<in> edges"
assms_17: shows "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"
gbm_empty_impl: shows "graph_empty gbm_\<alpha> gbm_invar gbm_empty"
set_keys_to_list: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "set (pp_pm.extended_ord.punit.keys_to_list p) = keys p"
divide_nonneg_nonneg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq zero y" shows "less_eq zero (divide x y)"
neg_0_less_iff_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (uminus a) = less a zero"
filter_preserves_multiset: fixes M :: "'k \<Rightarrow> nat"   and P :: "'k \<Rightarrow> bool" assumes "M \<in> multiset" shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
assms: fixes x :: "'a" assumes "x \<in> set (monom_vars_list xs)" shows "\<alpha> x = \<beta> x"
joint_d_gchains: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a set" assumes "d_gchain G n f"   and "d_gchain G m g"   and "g 0 \<subseteq> f n" shows "d_gchain G (Suc (n + m)) (jointfun n f m g)"
lookup_def: fixes r :: "'b ref" shows "!r = tap (\<lambda>h. Ref.get h r)"
norm_prod_def: fixes x :: "'d \<times> 'e" shows "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
AbstMakeForm_def: fixes k :: "hf"   and y :: "hf"   and y' :: "hf"   and i :: "hf"   and u :: "hf"   and u' :: "hf"   and j :: "hf"   and w :: "hf"   and w' :: "hf" shows "AbstMakeForm k y y' i u u' j w w' \<equiv> Ord k \<and> (k = i \<and> k = j \<and> y = q_Disj u w \<and> y' = q_Disj u' w' \<or> k = i \<and> y = q_Neg u \<and> y' = q_Neg u' \<or> succ k = i \<and> y = q_Ex u \<and> y' = q_Ex u')"
in_mset_imp_in_array: fixes x :: "'a"   and arr :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int" shows "(x \<in># arr_mset arr l r) = (\<exists>i. bounded l r i \<and> arr i = x)"
glb_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" assumes "class.po below" shows "po.glb below S = (THE x. po.is_glb below S x)"
card_inter_lt_single: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "card (A \<inter> B) \<le> card A"
le_iff_meet: fixes x :: "int"   and y :: "int" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(x \<le> y) = (x \<squnion>\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> y = y)"
forests_bot_3: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and e :: "'b"   and f :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.pd_kleene_allegory star one times sup less_eq less bot inf top conv uminus"   and "(less_eq one e \<and> less_eq (times e e) e) \<and> conv e = e"   and "less_eq (times f (conv f)) one \<and> less_eq (times f (star f)) (uminus one)" shows "inf (times x (inf (uminus e) f)) (times y (inf e f)) = bot"
simps_6: fixes c :: "'c"   and uy :: "'d" shows "collect_clks (GT c uy) = {c}"
inv_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.in_hom f a b" shows "VV.in_hom (VV.inv f) b a"
sup_ge1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<preceq> x \<oplus> y"
reachable_ss_V: fixes s :: "nat"   and c :: "nat \<times> nat \<Rightarrow> 'c" assumes "s \<in> Graph.V c" shows "Graph.reachableNodes c s \<subseteq> Graph.V c"
converse_rtrancl_induct: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P b"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>*; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
empty_subseqs: fixes xs :: "'c list" shows "[] \<in> set (subseqs xs)"
simple_function_completion: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "simple_function M f" shows "simple_function (completion M) f"
hyps_3: fixes x :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "stl (smap abss xa__) = smap abss x"   and "pred_stream (\<lambda>s. \<phi> (reps (abss s)) \<longrightarrow> \<phi> s) x"   and "pred_stream (\<lambda>s. \<psi> (reps (abss s)) \<longrightarrow> \<psi> s) x" shows "(holds \<phi> suntil holds \<psi>) x"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "hindered \<Gamma>" obtains f :: "'a \<times> 'a \<Rightarrow> ennreal" where "hindrance \<Gamma> f" and "current \<Gamma> f" and "wave \<Gamma> f"
simps_2: fixes ts :: "ct list"   and ts' :: "t list" shows "c_types_agree (TopType ts) ts' = ct_suffix ts (to_ct_list ts')"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
reachable_extend_f: fixes p :: "'c"   and F :: "'a program" assumes "p \<in> reachable (extend h F)" shows "f p \<in> reachable F"
listset_ConsD: fixes x :: "'a"   and xs :: "'a list"   and A :: "'a set"   and As :: "'a set list" assumes "x # xs \<in> listset (A # As)" shows "x \<in> A \<and> xs \<in> listset As"
mat_extension_def: fixes dims :: "nat list"   and vars :: "nat set"   and m :: "'a mat" shows "mat_extension dims vars m = tensor_mat dims vars m (1\<^sub>m (partial_state.d2 dims vars))"
considersContains: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "considers f iA n" shows "contains f iA n"
scomp_apply: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "'b" shows "scomp f g x = (case f x of (x, xa) \<Rightarrow> g x xa)"
division_disjoint_union: fixes p1 :: "'a set set"   and s1 :: "'a set"   and p2 :: "'a set set"   and s2 :: "'a set" assumes "p1 division_of s1"   and "p2 division_of s2"   and "interior s1 \<inter> interior s2 = {}" shows "p1 \<union> p2 division_of s1 \<union> s2"
scale_minus_left: fixes a :: "complex"   and x :: "'b" shows "- a *\<^sub>C x = - (a *\<^sub>C x)"
le_any_nat: fixes v :: "val"   and n :: "nat" assumes "v \<sqsubseteq> VNat n" shows "v = VNat n"
\<rho>_wary: shows "wary_subst \<rho>"
induct: fixes P :: "'c list \<Rightarrow> bool"   and a0 :: "'c list" assumes "\<And>xs. \<lbrakk>\<And>x. \<lbrakk>x = length xs; \<not> x \<le> 1\<rbrakk> \<Longrightarrow> P (take (x div 2) xs); \<And>x. \<lbrakk>x = length xs; \<not> x \<le> 1\<rbrakk> \<Longrightarrow> P (drop (x div 2) xs)\<rbrakk> \<Longrightarrow> P xs" shows "P a0"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = a \<squnion> b" shows "b \<le> a"
simps_1: shows "cospan_shape.Cod cospan_shape.AA = cospan_shape.AA"
M: shows "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r / sqrt 2"
assms_3: shows "W B\<^sub>1 + w u \<le> real c"
ReachableSplit: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, t) \<in> Reachable"   and "1 \<le> k" shows "(\<exists>n m r ll. (M, l, s, n, ll, r) \<in> Step \<and> (M, ll, r, m, t) \<in> Reachable \<and> Suc m + n = k) \<or> (\<exists>n ops S h c m par R ops1 code l0. s = (ops, S, h) \<and> get_ins M l = Some (invokeS c m) \<and> mbody_is c m (par, code, l0) \<and> (ops, par, R, ops1) \<in> Frame \<and> ((par, code, l0), l0, ([], R, h), n, t) \<in> Reachable \<and> Suc n = k)"
dist_cblinfun_def: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "dist a b = norm (a - b)"
ie4: shows "\<gamma>2 \<delta>S \<le> \<delta>"
mopup_correct: fixes n :: "nat"   and am :: "nat list"   and rs :: "nat"   and ires :: "cell list"   and k :: "nat" assumes "n < length am"   and "am ! n = rs" shows "\<exists>stp i j. steps0 (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) (mopup_a n @ shift mopup_b (2 * n)) stp = (0, Bk \<up> i @ Bk # Bk # ires, Oc # Oc \<up> rs @ Bk \<up> j)"
awalk_appendI: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and q :: "('a \<times> 'a) list"   and w :: "'a" assumes "awalk u p v"   and "awalk v q w" shows "awalk u (p @ q) w"
rtrancl_eta_AppL: fixes s :: "dB"   and s' :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* s'" shows "s \<degree> t \<rightarrow>\<^sub>\<eta>\<^sup>* s' \<degree> t"
mem_ball_0: fixes x :: "'a"   and e :: "real" shows "(x \<in> ball (0::'a) e) = (norm x < e)"
Group: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "Group V"
hom_char: fixes f :: "'c"   and b :: "'c"   and a :: "'c" shows "Cop.in_hom f b a = \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"
float_round_up_ge: fixes x :: "real"   and xa :: "float"   and prec :: "nat" assumes "x \<le> real_of_float xa" shows "x \<le> real_of_float (float_round_up prec xa)"
finite_W: fixes y :: "'a" assumes "y \<in> carrier" shows "finite {j \<in> N. y \<in> W j}"
invertible_is_bijective: fixes \<alpha> :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "\<alpha> \<in> S \<rightarrow>\<^sub>E S" shows "monoid.invertible (S \<rightarrow>\<^sub>E S) (compose S) (identity S) \<alpha> = bij_betw \<alpha> S S"
causal_program_history_Read: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list"   and v :: "nat" assumes "causal_program_history (Read volatile a t # is\<^sub>s\<^sub>b) sb" shows "causal_program_history is\<^sub>s\<^sub>b (sb @ [Read\<^sub>s\<^sub>b volatile a t v])"
Field_card_of: fixes A :: "'a set" shows "Field |A| = A"
swap_commute: fixes arr :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" shows "swap arr x y = swap arr y x"
ab: shows "a $ 1 \<le> b $ 1 \<and> a $ 2 \<le> b $ 2"
SC_LSC: fixes \<Gamma> :: "'b formula multiset" shows "\<Gamma> \<Rightarrow> {#} = \<Gamma> \<Rightarrow>\<^sub>n"
iFROM_not_empty: fixes n :: "nat" shows "[n\<dots>] \<noteq> {}"
iso_adj_int_im: fixes x :: "'a set"   and y :: "'a set" assumes "maxsimp x"   and "maxsimp y"   and "x \<sim> y"   and "x \<noteq> y" shows "f ` x \<inter> f ` y \<lhd> f ` x"
simps_2: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_WbisT tval aval indis (c1 ;; c2) = (PL_Indis.SC_WbisT tval aval indis c1 \<and> PL_Indis.SC_WbisT tval aval indis c2 \<or> noWhile (c1 ;; c2) \<and> PL_Indis.SC_discr aval indis (c1 ;; c2) \<or> PL_Indis.SC_siso tval aval indis (c1 ;; c2))"
is_character_data_ptr_kind_obtains: fixes node_ptr :: "(_) node_ptr" assumes "is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" obtains character_data_ptr :: "(_) character_data_ptr" where "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr = node_ptr"
Call_result_5: fixes rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t"
sylvester_mat_def: fixes p :: "'a poly"   and q :: "'a poly" shows "sylvester_mat p q \<equiv> sylvester_mat_sub (degree p) (degree q) p q"
Par_strict_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "A B ParStrict C D \<and> B A ParStrict C D \<and> A B ParStrict D C \<and> B A ParStrict D C \<and> C D ParStrict A B \<and> C D ParStrict B A \<and> D C ParStrict A B \<and> D C ParStrict B A"
lookup_pair_tl: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (list.tl xs) k = (if \<forall>k'\<in>fst ` set xs. le x k k' then 0::'c else lookup_pair x xs k)"
lcos_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier (G\<lparr>carrier := H\<rparr>)" shows "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> <#\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> M = M"
computable_complement_3: fixes A :: "nat set" shows "computable A = computable (- A)"
Since: fixes \<phi> :: "formula"   and \<psi> :: "formula"   and I :: "\<I>" assumes "fv \<phi> \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (Since \<phi> I \<psi>)"
per: fixes r :: "'rule" assumes "r \<in> R" shows "per r"
signed_take_bit_eq_iff_take_bit_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "(ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n b) = (take_bit (Suc n) a = take_bit (Suc n) b)"
set_restriction_fun_def: fixes P :: "'a \<Rightarrow> bool" shows "set_restriction_fun P \<equiv> \<lambda>A. {x \<in> A. P x}"
simps_3: fixes I :: "\<I>"   and v :: "'a option list set"   and va :: "'a option list set list"   and t :: "nat" shows "mprev_next I (v # va) [t] = ([], v # va, [t])"
primitive_def: fixes u :: "'b list" shows "primitive u = (\<forall>r k. r \<^sup>@ k = u \<longrightarrow> k = 1)"
grab_eqD: fixes N :: "nat set"   and n :: "nat"   and A :: "nat set"   and M :: "nat set" assumes "grab N n = (A, M)"   and "infinite N" shows "A \<lless> M \<and> finite A \<and> card A = n \<and> infinite M \<and> A \<subseteq> N \<and> M \<subseteq> N"