hcju/isabelleps huggingface dataset:
prodB1_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'e) Ring_scheme" shows "prodB1 R S = (\<lambda>k. if k = 0 then R else if k = Suc 0 then S else undefined)"
map_1: fixes f :: "'h \<Rightarrow> 'g" shows "map f [] = []"
subtract_simps_1: fixes n :: "nat"   and I :: "\<I>" shows "left (subtract n I) = left I - n"
algo_impl_correct: shows "(all_left_impl, SPEC \<circ> all_left_spec) \<in> (array_assn int_assn, array_assn int_assn) \<rightarrow>\<^sub>a array_assn (option_assn nat_assn)"
smult_eq_0_iff: fixes a :: "'b"   and p :: "'b poly" shows "(Polynomial.smult a p = 0) = (a = (0::'b) \<or> p = 0)"
reduce_clause_in_Q: fixes C :: "'a literal list"   and N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and j :: "nat"   and D :: "'a literal list"   and D' :: "'a literal list"   and Q :: "('a literal list \<times> nat) list"   and Q' :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N"   and "\<forall>(E, k)\<in>set P. j < k \<longrightarrow> is_irreducible [C] E"   and "reduce [C] D D' \<noteq> D'" shows "wstate_of_dstate (N, P, Q @ (D @ D', j) # Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, (D @ reduce [C] D D', j) # P, Q @ Q', n)"
transpose_empty: fixes xs :: "'g list list" shows "(transpose xs = []) = (\<forall>x\<in>set xs. x = [])"
simulation2_rtrancl: fixes s2 :: "'s2"   and tls2 :: "'tl2 list"   and s2' :: "'s2"   and s1 :: "'s1" assumes "s2 -2-tls2\<rightarrow>* s2'"   and "s1 \<approx> s2" shows "\<exists>s1' tls1. s1 -1-tls1\<rightarrow>* s1' \<and> s1' \<approx> s2' \<and> tls1 [\<sim>] tls2"
Lang_def: fixes G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" shows "Lang G S = {w. [Inl S] -G\<rightarrow>\<^sup>+ map Inr w}"
simps_2: fixes l :: "'b btree"   and a :: "'b"   and r :: "'b btree" shows "inorder_up\<^sub>i (Up\<^sub>i l a r) = inorder l @ [a] @ inorder r"
and_mask_0_iff_le_mask: fixes w :: "'b word"   and n :: "nat" shows "(w AND NOT (mask n) = 0) = (w \<le> mask n)"
ex_un_null: shows "\<exists>!n. \<forall>f. C' n f = n \<and> C' f n = n"
cong: fixes zero :: "'b" shows "sep_algebra.sep_empty zero \<equiv> sep_algebra.sep_empty zero"
rep_eq: fixes x :: "'a dense_filter_type"   and xa :: "'a dense_filter_type" shows "Rep_dense_filter_type (x \<sqinter> xa) = Rep_dense_filter_type x \<sqinter> Rep_dense_filter_type xa"
mult_left_le: fixes one :: "'h"   and times :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and plus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero :: "'h"   and c :: "'h"   and a :: "'h" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
f_shrink_def: fixes xs :: "'d message_af list"   and k :: "nat" shows "xs \<div> k \<equiv> f_aggregate xs k last_message"
assms_2: shows "(a, b) \<in> set (ordered_fmap S)"
ide_char': fixes \<mu> :: "'a arrow_of_spans_data" shows "V.ide \<mu> = identity_arrow_of_spans (\<cdot>) \<mu>"
distinct_verts_imp_distinct: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v"   and "distinct (pawalk_verts u p)" shows "distinct p"
prems_2: shows "v0_v1_Digraph G v0 v1"
simps_1: fixes hsx :: "'b tree"   and x :: "'b"   and hsy :: "'b tree"   and y :: "'b"   and hs :: "'b tree" shows "link \<langle>hsx, x, \<langle>hsy, y, hs\<rangle>\<rangle> = (if x < y then \<langle>\<langle>hsy, y, hsx\<rangle>, x, hs\<rangle> else \<langle>\<langle>hsx, x, hsy\<rangle>, y, hs\<rangle>)"
lcs_tool1: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "\<rho> a \<cdot> b \<in> H" shows "\<rho> b \<cdot> a \<in> H"
comm_ring_axioms: shows "comm_ring local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel local.one_rel"
poly_from_roots: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "(\<Prod>x\<in>A. [:- f x, 1::'d:]) = (\<Sum>X\<in>Pow A. monom ((- (1::'d)) ^ card X * prod f X) (card (A - X)))"
repeat_Bk: fixes rn :: "nat" shows "(\<exists>rna. Bk \<up> rn = Bk # Bk \<up> rna) \<or> rn = 0"
monoid_axioms: shows "Group.monoid (add_monoid (R Quot I))"
spmf_fast_uniform: fixes n :: "nat"   and x :: "nat" shows "spmf (Fast_Dice_Roll.fast_uniform n) x = (if x < n then 1 / real n else 0)"
lfp_le_gfp: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "lfp f \<le> gfp f"
SFAssInitReds: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields"   and sh'' :: "sheap" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>Val v',(h', l', sh'),b'\<rangle>"   and "sh' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs' = sfs(F \<mapsto> v)"   and "sh'' = sh'(D \<mapsto> (sfs', i))" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,(h', l', sh''),False\<rangle>"
disjoint_family_on_insert: fixes i :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "i \<notin> I" shows "disjoint_family_on A (insert i I) = (A i \<inter> \<Union> (A ` I) = {} \<and> disjoint_family_on A I)"
is_le_relI_2: shows "ord_term_lin.is_le_rel (\<preceq>\<^sub>t)"
MkArr_in_hom: fixes A :: "'obj"   and B :: "'obj"   and f :: "'obj" assumes "A \<in> Obj"   and "B \<in> Obj"   and "f \<in> (if A = B then {A} else {})" shows "\<guillemotleft>MkArr A B f : MkIde A \<rightarrow> MkIde B\<guillemotright>"
simps: fixes a1 :: "prog"   and a2 :: "fin_string \<Rightarrow> real"   and a3 :: "fin_string \<Rightarrow> real" shows "([a1]a2 \<downharpoonleft> a3) = ((\<exists>\<phi> \<nu> \<omega>. a1 = ? \<phi> \<and> a2 = \<nu> \<and> a3 = \<omega> \<and> ([\<phi>]\<nu> \<downharpoonright> True) \<and> \<nu> = \<omega>) \<or> (\<exists>\<alpha> \<nu> \<mu> \<beta> \<omega>. a1 = (\<alpha> ;; \<beta>) \<and> a2 = \<nu> \<and> a3 = \<omega> \<and> ([\<alpha>]\<nu> \<downharpoonleft> \<mu>) \<and> ([\<beta>]\<mu> \<downharpoonleft> \<omega>)) \<or> (\<exists>\<theta> \<nu> r \<omega> x. a1 = (x := \<theta>) \<and> a2 = \<nu> \<and> a3 = \<omega> \<and> ([\<theta>]\<nu> \<down> r) \<and> \<omega> = \<nu>(x := r)) \<or> (\<exists>\<alpha> \<nu> \<omega> \<beta>. a1 = (\<alpha> \<union>\<union> \<beta>) \<and> a2 = \<nu> \<and> a3 = \<omega> \<and> ([\<alpha>]\<nu> \<downharpoonleft> \<omega>)) \<or> (\<exists>\<beta> \<nu> \<omega> \<alpha>. a1 = (\<alpha> \<union>\<union> \<beta>) \<and> a2 = \<nu> \<and> a3 = \<omega> \<and> ([\<beta>]\<nu> \<downharpoonleft> \<omega>)))"
subst_dom_comp_eq: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "subst_domain \<theta> \<inter> range_vars \<sigma> = {}" shows "subst_domain (\<theta> \<circ>\<^sub>s \<sigma>) = subst_domain \<theta> \<union> subst_domain \<sigma>"
order_trans_rules_20: fixes a :: "'a"   and b :: "'a"   and P :: "bool" assumes "a < b"   and "b < a" shows "P"
simps_1: fixes r :: "mregex" shows "LPDi 0 r = {r}"
owns_app: fixes r :: "room"   and s\<^sub>2 :: "event list"   and s\<^sub>1 :: "event list" assumes "no_Check_in s\<^sub>2 r" shows "owns (s\<^sub>2 @ s\<^sub>1) r = owns s\<^sub>1 r"
lhs_step_SPEC: fixes \<Phi> :: "'a \<Rightarrow> bool"   and m :: "'a nres" assumes "\<And>x. \<Phi> x \<Longrightarrow> RETURN x \<le> m" shows "SPEC \<Phi> \<le> m"
lunit_hcomp_4: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<l>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<a>\<^sub>D[trg\<^sub>D f, f, g] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[f] \<star>\<^sub>D g)"
in_prefix: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "prefix xs ys"   and "x \<in> set xs" shows "x \<in> set ys"
cop_per2__col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and X :: "'a"   and Y :: "'a"   and Z :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A X Y Z"   and "A \<noteq> Z"   and "Tarski_neutral_dimensionless.Per Bet Cong X Z A"   and "Tarski_neutral_dimensionless.Per Bet Cong Y Z A" shows "Tarski_neutral_dimensionless.Col Bet X Y Z"
intros_1: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and qs :: "('b, 'c) fm list"   and i :: "'c"   and v' :: "nat"   and k :: "'c" assumes "branch !. v = Some (qs, i)"   and "qs !. v' = Some (\<^bold>\<diamond> Nom k)" shows "descendants k i branch {(v, v')}"
iwlsFSbSw_iterAll_termFSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Iteration.model" assumes "iwlsFSbSw MOD" shows "termFSwSbImorph (iter MOD) (iterAbs MOD) MOD"
eval_Runit_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "\<lbrace>\<^bold>\<r>\<^bold>[a\<^bold>]\<rbrace> = \<r>[\<lbrace>a\<rbrace>]"
tfr_stp_all_wt_subst_apply: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "bvars\<^sub>s\<^sub>t S \<inter> range_vars \<theta> = {}" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S \<cdot>\<^sub>s\<^sub>t \<theta>)"
lm_ball_eq_ran: fixes v :: "('b, 'c) assoc_list"   and P :: "'c \<Rightarrow> bool" shows "lm.ball v (\<lambda>(k, v). P v) = (ran (lm.\<alpha> v) \<subseteq> Collect P)"
inject: fixes x1 :: "'a atom formula list"   and x2 :: "'a atom formula list"   and y1 :: "'a atom formula list"   and y2 :: "'a atom formula list" shows "(Effect x1 x2 = Effect y1 y2) = (x1 = y1 \<and> x2 = y2)"
xk: shows "(x, k) \<in> p1 \<union> p2"
simps_4: fixes a :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "\<langle>a: t \<in> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = \<langle>a: (t \<cdot> \<theta>) \<in> s \<cdot> \<theta>\<rangle>"
relpow_Suc_D2': fixes n :: "nat"   and R :: "('a \<times> 'a) set" shows "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
lcf_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "f (deg R f) \<in> carrier R"
less_eq_int_code_1: shows "(0 \<le> 0) = True"
arr_dom_iff_arr: fixes f :: "'a" shows "B.arr (B.dom f) = B.arr f"
ine_def: fixes sP :: "specID"   and E :: "Expression" shows "ine sP E \<equiv> \<exists>ch. ch \<in> ins sP \<and> exprChannel ch E"
assms_3: shows "f \<squnion> f\<^sup>T \<le> w \<sqinter> - d \<sqinter> - d\<^sup>T \<squnion> w\<^sup>T \<sqinter> - d \<sqinter> - d\<^sup>T"
d_IN_mono: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and y :: "'k"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "\<And>x. f (x, y) \<le> g (x, y)" shows "d_IN f y \<le> d_IN g y"
map\<^sub>1_simps_5: fixes f :: "'a" assumes "C.ide f" shows "D.cod (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) = F.map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f"
assms: shows "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
simps_2: fixes x :: "'b"   and xs :: "['b]" shows "compare\<cdot>(x : xs)\<cdot>[] = GT"
le_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq a (divide b c) = (if less zero c then less_eq (times a c) b else if less c zero then less_eq b (times a c) else less_eq a zero)"
constrains_imp_subset: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set" assumes "F \<in> A co A'" shows "A \<subseteq> A'"
nat_pow_closed: fixes x :: "'c"   and n :: "nat" assumes "x \<in> carrier A" shows "x [^]\<^bsub>A\<^esub> n \<in> carrier A"
"0_3": shows "list_all is_Update (unlabel (transaction_updates T))"
StatEq': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel'"   and "\<Psi>' \<simeq> \<Psi>''" shows "(\<Psi>'', S, T) \<in> Rel'"
f7: shows "4 \<le> rk {a, b, c, R, A}"
is_discrete: fixes f :: "(bool, unit) arr" shows "ide f = arr f"
list_emb_Cons: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "list_emb P xs ys" shows "list_emb P xs (y # ys)"
complex_sin_eq: fixes w :: "complex"   and z :: "complex" shows "(sin w = sin z) = (\<exists>n\<in>\<int>. w = z + complex_of_real (2 * n * pi) \<or> w = - z + complex_of_real ((2 * n + 1) * pi))"
load_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "load_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; flagi = get_operand_flag (op_list ! 0); rd = if instr_name \<in> {load_store_type LDUBA, load_store_type LDA, load_store_type LDSBA, load_store_type LDSHA, load_store_type LDSHA, load_store_type LDDA} then get_operand_w5 (op_list ! 4) else get_operand_w5 (op_list ! 3) in do psr_val \<leftarrow> gets (cpu_reg_val PSR); s_val \<leftarrow> gets (\<lambda>s. get_S psr_val); if instr_name \<in> {load_store_type LDA, load_store_type LDUBA, load_store_type LDSBA, load_store_type LDSHA, load_store_type LDUHA, load_store_type LDDA} \<and> s_val = 0 then do raise_trap privileged_instruction; return () od else if instr_name \<in> {load_store_type LDA, load_store_type LDUBA, load_store_type LDSBA, load_store_type LDSHA, load_store_type LDUHA, load_store_type LDDA} \<and> flagi = 1 then do raise_trap illegal_instruction; return () od else load_sub1 instr rd s_val od"
fbox_export3: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows " |x \<cdot> ad p] q =  |x] (d p + d q)"
inducts: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" assumes "not_final_thread s t" obtains x :: "'x" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, ln__)\<rfloor>" and "\<not> final xa__"   | x :: "'x" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(xb__, lna__)\<rfloor>" and "lna__ \<noteq> no_wait_locks"   | x :: "'x" and w :: "'h wait_set_status" and ln :: "'e \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "wset s t = \<lfloor>w\<rfloor>"
induct: fixes P :: "real list \<Rightarrow> real list \<Rightarrow> bool"   and a0 :: "real list"   and a1 :: "real list" assumes "P [] []"   and "\<And>a as b bs. (\<lbrakk>\<not> a < b; \<not> b < a\<rbrakk> \<Longrightarrow> P as bs) \<Longrightarrow> P (a # as) (b # bs)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
simps_16: fixes v :: "atom fm" shows "push_forall (AllQ (AllQ v)) = (if freeIn 0 (AllQ v) then lowerFm 0 1 (AllQ v) else AllQ (AllQ v))"
domain_complex: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex X"
prems_2: shows "wfrestrictedvars\<^sub>s\<^sub>t S' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t (\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S) \<union> V"
ext_irrefl_trans_strong_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_irrefl_trans_strong ext \<equiv> ext_irrefl ext \<and> ext_irrefl_trans_strong_axioms ext"
subst_add: fixes x :: "char list"   and y :: "char list"   and e :: "'b environment"   and a :: "'b"   and b :: "'b" assumes "x \<noteq> y" shows "e\<lparr>x:a\<rparr>\<lparr>y:b\<rparr> = e\<lparr>y:b\<rparr>\<lparr>x:a\<rparr>"
that_3: shows "D = {} \<longrightarrow> B = {}"
\<K>_finite: fixes st :: "'s \<times> ('c \<Rightarrow> real) set" shows "finite (\<K> st)"
r_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier \<G>"   and "a \<in> carrier \<G>" shows "(x = a \<otimes> x) = (a = \<one>)"
ds2_components1: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme"   and a :: "'e" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "a \<in> carrier M" shows "\<exists>a1\<in>carrier M1. a \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> a1 \<in> carrier M2"
restrict_to_halfspaces_def: fixes sctns :: "'d sctn set"   and X :: "'d set" shows "restrict_to_halfspaces sctns X = SPEC (\<lambda>R. R = X \<inter> below_halfspaces sctns)"
vimage_UNIV: fixes f :: "'a \<Rightarrow> 'b" shows "f -` UNIV = UNIV"
add_idem: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.join_semilattice plus less_eq less" shows "plus x x = x"
equal_pair_count_le: fixes xs :: "('b \<times> 'b) list" shows "equal_pair_count xs \<le> length xs"
permute_diff: fixes p :: "perm"   and q :: "perm"   and x :: "'a" shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
assms_2: shows "b \<rightarrow>\<^sup>* c"
free_generator_sub: fixes H :: "'a set" assumes "free_generator R M H" shows "H \<subseteq> carrier M"
wlog_linorder_le: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "\<And>a b. a \<le> b \<Longrightarrow> P a b"   and "P b a \<Longrightarrow> P a b" shows "P a b"
continuous_at_ball: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = (\<forall>e>0. \<exists>d>0. f ` ball x d \<subseteq> ball (f x) e)"
disjoint: fixes ts :: "traffic"   and c :: "cars" shows "res ts c \<sqinter> clm ts c = bot"
Timess_eq_image: fixes r :: "'b rexp"   and rs :: "'b rexp set" shows "Timess rs r = (\<lambda>r'. Times r' r) ` rs"
tagged_division_ofD_3: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "K \<subseteq> i"
stack_consistent_foo_2: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap" shows "stack_consistent (a # as) (Alts e1 e2 # S) = (Astack S \<sqsubseteq> a \<and> stack_consistent as S)"
mod_0_div_mult_cancel: fixes n :: "nat"   and m :: "nat" shows "(n mod m = 0) = (n div m * m = n)"
subset_entailment: fixes P' :: "'BExp set"   and P :: "'BExp set" assumes "P' \<subseteq> P" shows "P \<turnstile> P'"
simps_14: fixes f1 :: "'a"   and f2 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a"   and x3 :: "BitList" shows "rec_BitList f1 f2 f3 (Bit1 x3) = f3 x3 (rec_BitList f1 f2 f3 x3)"
set_take_disj_set_drop_if_distinct: fixes vs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "distinct vs"   and "i \<le> j" shows "set (take i vs) \<inter> set (drop j vs) = {}"
wordinterval_CIDR_split1_def: fixes r :: "'b wordinterval" shows "wordinterval_CIDR_split1 r \<equiv> let ma = wordinterval_lowest_element r in case ma of None \<Rightarrow> (None, r) | Some a \<Rightarrow> case largest_contained_prefix a r of None \<Rightarrow> (None, r) | Some m \<Rightarrow> (Some m, wordinterval_setminus r (prefix_to_wordinterval m))"
sub2_wit_gram_schmidt_norm_mus: fixes fs :: "'a vec list"   and mus :: "'a list list"   and gs :: "'a vec list" assumes "sub2_wit [] fs = (mus, gs)"   and "set fs \<subseteq> carrier_vec n" shows "norms_mus fs = (map sq_norm gs, mus)"
assms_5: shows "recfn (length xs) (Pr n f g)"
ip_cidr_set_def: fixes i :: "'b word"   and r :: "nat" shows "ip_cidr_set i r \<equiv> {j. i && ~~ mask (LENGTH('b) - r) = j && ~~ mask (LENGTH('b) - r)}"
filter_sup_right_isotone_var: fixes F :: "'a set"   and G :: "'a set"   and H :: "'a set" shows "filter_sup F (G \<inter> H) \<subseteq> filter_sup F H"
run_asset_i_unused: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "\<exists>A. PriKey A \<notin> used (fst (run_asset_i n s))"
simps: fixes z :: "name"   and t :: "tm"   and u :: "tm" assumes "atom z \<sharp> (t, u)" shows "Disjoint t u = All2 z t (Neg (Var z IN u))"
intros_4: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> equality_axioms" shows "H \<turnstile> A"
that: fixes e :: "real" assumes "0 < e"   and "cball x e \<subseteq> S" shows "thesis"
assms: fixes s :: "char list"   and x :: "'a"   and r :: "char list" assumes "p s = Inr (x, r)" shows "length r \<le> length s"
inv_closed: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
fds_converges_remainder: fixes f :: "'b fds"   and s :: "'b"   and m :: "nat" assumes "fds_converges f s" shows "fds_converges (fds_remainder m f) s"
boundedE: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b \<inter># c" obtains "a \<subseteq># b" and "a \<subseteq># c"
monom_mult_smult: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
ord_isom_le: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = f a \<preceq>\<^bsub>E\<^esub> f b"
prems: shows "\<forall>w\<in>set (w # ws). dim_vec w = n"
\<Delta>\<Phi>_insert: fixes x :: "'b"   and h :: "'b hp option" shows "\<Phi> (Pairing_Heap_List2.insert x h) - \<Phi> h \<le> log 2 (real (size_heap h + 1))"
llist_all2_conv_lzip: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>(x, y)\<in>lset (lzip xs ys). P x y))"