hcju/isabelleps huggingface dataset:
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "aggregation.sum_0 g (insert x A) = g x + aggregation.sum_0 g (A - {x})"
assms: shows "pmdl (set gs) = pmdl (set (init_syzygy_list bs))"
inject_1: fixes x1 :: "Keys"   and y1 :: "Keys" shows "(kE x1 = kE y1) = (x1 = y1)"
frefD: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and P :: "'e \<Rightarrow> bool"   and R :: "('c \<times> 'e) set"   and S :: "('d \<times> 'f) set"   and y :: "'e"   and x :: "'c" assumes "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"   and "P y"   and "(x, y) \<in> R" shows "(f x, g y) \<in> S"
MClkIPSpec_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkIPSpec send rcv cst p = (Init MClkInit rcv cst p \<and> \<box>[MClkNext send rcv cst p]_(cst!p, rtrner send!p, caller rcv!p) \<and> WF(MClkFwd send rcv cst p)_(cst!p, rtrner send!p, caller rcv!p) \<and> SF(MClkReply send rcv cst p)_(cst!p, rtrner send!p, caller rcv!p))"
Assign: fixes D :: "nat set"   and \<theta> :: "nat \<Rightarrow> nat option"   and volatile :: "bool"   and a :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and t :: "nat" assumes "D \<subseteq> dom \<theta>" shows "\<theta>\<turnstile> (Assign volatile (Tmp (D, a)) e A L R W, t) \<rightarrow>\<^sub>p ((Skip, t + used_tmps e), issue_expr t e @ [Write volatile (a \<theta>) (eval_expr t e) (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"
valid_reach_src_tgt: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and istate :: "'a" assumes "IO_Automaton.valid step tr"   and "IO_Automaton.reach istate step (srcOf (hd tr))" shows "IO_Automaton.reach istate step (tgtOf (last tr))"
simps_1: fixes carrier :: "'e set"   and more :: "'f" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
foundation9': fixes \<tau> :: "'b state \<times> 'b state"   and x :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not x" shows "\<tau> |\<noteq> x"
perm_wfactorsD: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and a :: "'c"   and b :: "'c" assumes "Group.comm_monoid G"   and "as <~~> bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G" shows "a \<sim>\<^bsub>G\<^esub> b"
assms_2: shows "x\<^sub>1 = x\<^sub>2"
eventually_all_finite: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and net :: "'b filter" assumes "\<And>y. \<forall>\<^sub>F x in net. P x y" shows "\<forall>\<^sub>F x in net. \<forall>y. P x y"
simps_11: fixes m :: "nat"   and p :: "fm"   and q :: "fm" shows "bound m (Iff p q) = (bound m p \<and> bound m q)"
memory_write_asi_def: fixes asi :: "word8"   and addr :: "word32"   and byte_mask :: "word4"   and data_w32 :: "word32"   and state :: "'b sparc_state" shows "memory_write_asi asi addr byte_mask data_w32 state \<equiv> let asi_int = uint asi; psr_val = cpu_reg_val PSR state; s_val = get_S psr_val in if asi_int = 1 then store_word_mem state addr data_w32 byte_mask (word_of_int 10) else if asi_int = 2 then if uint addr = 0 then let s1 = sys_reg_mod data_w32 CCR state in Some (ccr_flush s1) else if uint addr = 8 then Some (sys_reg_mod data_w32 ICCR state) else if uint addr = 12 then Some (sys_reg_mod data_w32 DCCR state) else None else if asi_int \<in> {8, 9} then let ns = add_instr_cache state addr data_w32 byte_mask in store_word_mem ns addr data_w32 byte_mask asi else if asi_int \<in> {10, 11} then let ns = add_data_cache state addr data_w32 byte_mask in store_word_mem ns addr data_w32 byte_mask asi else if asi_int = 13 then Some (add_instr_cache state addr data_w32 15) else if asi_int = 15 then Some (add_data_cache state addr data_w32 15) else if asi_int = 16 then Some (flush_instr_cache state) else if asi_int = 17 then Some (flush_data_cache state) else if asi_int \<in> {20, 21} then None else if asi_int = 24 then Some (flush_cache_all state) else if asi_int = 25 then let mmu_state' = mmu_reg_mod (mmu state) addr data_w32 in case mmu_state' of None \<Rightarrow> None | Some mmus \<Rightarrow> Some (state\<lparr>mmu := mmus\<rparr>) else if asi_int = 28 then Some (mem_mod_w32 asi (ucast addr) byte_mask data_w32 state) else if asi_int = 29 then None else None"
induct: fixes x1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x2 :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and x3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and P :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> ('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list \<Rightarrow> ('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> bool" assumes "mthr.\<tau>rtrancl3p x1 x2 x3"   and "\<And>s. P s [] s"   and "\<And>s s' tls s'' tl. \<lbrakk>redT s tl s'; \<not> m\<tau>move s tl s'; mthr.\<tau>rtrancl3p s' tls s''; P s' tls s''\<rbrakk> \<Longrightarrow> P s (tl # tls) s''"   and "\<And>s s' tls s'' tl. \<lbrakk>redT s tl s'; m\<tau>move s tl s'; mthr.\<tau>rtrancl3p s' tls s''; P s' tls s''\<rbrakk> \<Longrightarrow> P s tls s''" shows "P x1 x2 x3"
simps_1: fixes tset :: "('a, 'b, 'c, 'd) FL_Formula.Tree set['a]" shows "L_transform_Tree (FL_Formula.Tree.tConj tset) = Formula.Conj\<^sub>\<alpha> (map_bset L_transform_Tree tset)"
removeShadowRules1_alternative_def: fixes p :: "('a, 'b) Combinators list" shows "removeShadowRules1_alternative p = rev (removeShadowRules1_alternative_rev (rev p))"
Ci_in_nth_wP: fixes t :: "nat" assumes "enat (t_C + t) < llength Sts" shows "\<exists>i. (C, i) \<in># wP_of_wstate (lnth Sts (t_C + t))"
assms_2: shows "Di \<in> set (subseqs (dbproj i D))"
toplevel_summands_PLUS_strong: fixes xs :: "'c rexp list" assumes "xs \<noteq> []"   and "list_all (\<lambda>x. \<nexists>r s. x = Plus r s) xs" shows "toplevel_summands (PLUS xs) = set xs"
unit_div_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "(b div a = c div a) = (b = c)"
simps_2: fixes fa :: "floatarith"   and fas :: "floatarith list"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fas' (fa # fas) M slp = (let (M, slp) = slp_of_fa fa M slp in slp_of_fas' fas M slp)"
simps_1: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and zero :: "'e"   and add :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
Aheap_nonrec_lookup: fixes x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "(Aheap_nonrec x e\<cdot>i) x = ABind_nonrec x e\<cdot>i"
exec_instr_stk_offer: fixes ins :: "'addr instr list"   and pc :: "nat"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and frs :: "'addr frame list"   and ta' :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and xcp' :: "'addr option"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and t :: "'thread_id"   and stk'' :: "'addr val list" assumes "check_instr' (ins ! pc) P h stk loc C M pc frs"   and "(ta', xcp', h', (stk', loc', C, M, pc') # frs) \<in> exec_instr (ins ! pc) P t h stk loc C M pc frs" shows "(ta', xcp', h', (stk' @ stk'', loc', C, M, pc') # frs) \<in> exec_instr (ins ! pc) P t h (stk @ stk'') loc C M pc frs"
phi_vec_length: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "(cmod (\<phi> $$ (0, 0)))\<^sup>2 + (cmod (\<phi> $$ (Suc 0, 0)))\<^sup>2 = 1"
no_msg_change_if_no_event: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "isSend ev \<longrightarrow> getId ev \<noteq> i"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "msgs c i = msgs c' i"
Image_singleton_iff: fixes b :: "'c"   and r :: "('d \<times> 'c) set"   and a :: "'d" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
characteristic_formula_supp_aux: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "supp (Abs_bset {indexed_nominal_ts.distinguishing_formula satisfies transition P Q |Q. \<not> indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q}) \<subseteq> supp P"
inf_Some: fixes x :: "'b"   and y :: "'b" shows "Some x \<sqinter> Some y = Some (x \<sqinter> y)"
steps_alt: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f"   and l' :: "'d"   and u' :: "'e \<Rightarrow> 'f"   and l'' :: "'d"   and u'' :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"   and "A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow> \<langle>l'',u''\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>"
events_RUN: fixes A :: "'a set" shows "events_of (RUN A) = A"
f_join_f_shrink_iMOD: fixes k :: "nat"   and xs :: "'b message_af list"   and n :: "nat" assumes "0 < k" shows "xs \<longmapsto> k \<Join>\<^sub> [ n * k + (k - 1), mod k ] = xs \<div> k \<Join>\<^sub> [n\<dots>]"
support_set_def: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "support_set p = {x. \<exists>n w. p x n w \<noteq> 0}"
Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf_axioms: shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"
simps_5: fixes x1 :: "real" shows "MInfty \<noteq> ereal x1"
type_less_eq_x_Required: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<le> \<sigma>[1]" obtains \<rho> :: "'a basic_type" where "\<tau> = \<rho>[1]" and "\<rho> \<le> \<sigma>"
rule1': shows "filter (\<lambda>v. v \<in> set (removeNones vol)) (snd (splitAt u (verticesFrom f v'))) = removeNones vol"
relates_def: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c stream" shows "relates \<phi> s = \<phi> (shd s) (shd (stl s))"
edom_Aheap: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "edom (Aheap \<Gamma> e\<cdot>a) \<subseteq> domA \<Gamma>"
is_reducible_lit_mset_iff: fixes C :: "'a literal list"   and C' :: "'a literal list"   and Ds :: "'a literal list list"   and L :: "'a literal" assumes "mset C = mset C'" shows "is_reducible_lit Ds C' L = is_reducible_lit Ds C L"
assms_3: shows "subgroup K (G\<lparr>carrier := H\<rparr>)"
mult_pCons_right: fixes p :: "'b poly"   and a :: "'b"   and q :: "'b poly" shows "p * pCons a q = Polynomial.smult a p + pCons (0::'b) (p * q)"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier C"   and "b \<in> carrier C"   and "c \<in> carrier C" shows "(a = b \<otimes>\<^bsub>C\<^esub> inv\<^bsub>C\<^esub> c) = (b = a \<otimes>\<^bsub>C\<^esub> c)"
fs'_def: shows "fs' = fs[i := fs ! (i - 1), i - 1 := fs ! i]"
bddh_ge: fixes n :: "nat"   and m :: "nat"   and bdd :: "'a bdd" assumes "n \<le> m"   and "bddh n bdd" shows "bddh m bdd"
lemma_A: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and K :: "nat"   and Tk :: "nat"   and X :: "msg"   and Ta :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Crypt K \<lbrace>Agent A, Number Ta\<rbrace> \<in> parts (knows Spy evs)" shows "Says A B \<lbrace>X, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"
dnso3: fixes x :: "'a" shows "d x + (1::'a) = (1::'a)"
elims_3: fixes x :: "ty" assumes "\<not> is_Class x" obtains "x = Void"   | "x = Boolean"   | "x = Integer"   | "x = NT"   | v :: "ty" where "x = v\<lfloor>\<rceil>"
mono_update: fixes a1 :: "'a"   and a2 :: "'a"   and S1 :: "'a st"   and S2 :: "'a st"   and x :: "char list" assumes "a1 \<le> a2"   and "S1 \<le> S2" shows "update S1 x a1 \<le> update S2 x a2"
hom_paired2: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and G' :: "('g, 'i) monoid_scheme"   and H' :: "('h, 'j) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "((\<lambda>(x, y). (f x, g y)) \<in> hom (G \<times>\<times> H) (G' \<times>\<times> H')) = (f \<in> hom G G' \<and> g \<in> hom H H')"
semiring_norm_175: fixes n :: "nat" shows "Suc n = n + 1"
code_constn_def: fixes n :: "nat"   and c :: "nat" shows "code_constn n c \<equiv> if n = 1 then code_const1 c else quad_encode 3 n (code_const1 c) (singleton_encode (triple_encode 2 n 0))"
simps_11: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
in_set_simps_5: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool" shows "Ball (set [x]) P = P x"
unrest_bop: fixes x :: "'b \<Longrightarrow> 'c"   and u :: "('d, 'c) uexpr"   and v :: "('e, 'c) uexpr"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "x \<sharp> u"   and "x \<sharp> v" shows "x \<sharp> bop f u v"
hoare_partial_sound: fixes P :: "complex mat"   and S :: "com"   and Q :: "complex mat" assumes "\<turnstile>\<^sub>p {P} S {Q}"   and "well_com S" shows "\<Turnstile>\<^sub>p {P} S {Q}"
sup_union_right1: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" assumes "x \<notin># N" shows "N \<union># add_mset x M = add_mset x (N \<union># M)"
redundancy_and_semantics: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set"   and S :: "'a Literal set set"   and I :: "'a set" assumes "propositional_atoms atom_ordering"   and "redundant C S"   and "propositional_atoms.validate_formula I S" shows "propositional_atoms.validate_clause I C"
start_heap_obs_def: shows "start_heap_obs = map2 (\<lambda>C a. NewHeapElem a (Class_type C)) initialization_list (fst (snd start_heap_data))"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "F.F g A = F.F h B"
proj_in_setD: fixes a :: "'a strand_label \<times> 'b"   and l :: "'a"   and A :: "('a strand_label \<times> 'b) list" assumes "a \<in> set (proj l A)" obtains k :: "'a strand_label" and b :: "'b" where "a = (k, b)" and "k = ln l \<or> k = \<star>"
is_inf_lower: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and i :: "'a" assumes "dlat le"   and "dpo.is_inf le x y i" obtains "le i x" and "le i y"
Id_O_R: fixes R :: "('a \<times> 'b) set" shows "Id O R = R"
obs_intra_elem: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S"   and "n' \<in> S" shows "n' \<in> obs_intra n S"
true_cls_union: fixes I :: "'a set"   and C :: "'a literal multiset"   and D :: "'a literal multiset" shows "(I \<Turnstile> C + D) = (I \<Turnstile> C \<or> I \<Turnstile> D)"
measure_nonneg': fixes M :: "'d measure"   and A :: "'d set" shows "\<not> Sigma_Algebra.measure M A < 0"
"2_4": fixes x :: "'t list"   and xa :: "'t \<Rightarrow>\<^sub>0 'b" assumes "x = new_syz_sigs ss bs p"   and "\<not> sig_crit bs x p"   and "xa = sig_trd bs (poly_of_pair p)"   and "rep_list xa \<noteq> 0" shows "snd (snd (fst (rb_aux ((xa # bs, x, add_spairs ps bs xa), z)))) = []"
split_induct: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and P :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'x set \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) set \<Rightarrow> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<times> 'x set) list \<Rightarrow> bool"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "\<And>t U S. \<lbrakk>U \<subseteq> X; finite S; 0 \<in> S\<rbrakk> \<Longrightarrow> P t U S ([(monomial (1::'c) t, U)], [])"   and "\<And>t U S. \<lbrakk>U \<subseteq> X; finite S; 0 \<notin> S; S \<inter> .[U] = {}\<rbrakk> \<Longrightarrow> P t U S ([], [(monomial (1::'c) t, U)])"   and "\<And>t U S V x ps0 ps1 qs0 qs1. \<lbrakk>U \<subseteq> X; finite S; 0 \<notin> S; S \<inter> .[U] \<noteq> {}; V \<subseteq> U; S \<inter> .[V] = {}; \<And>V'. \<lbrakk>V' \<subseteq> U; S \<inter> .[V'] = {}\<rbrakk> \<Longrightarrow> card V' \<le> card V; x \<in> U; x \<notin> V; V = max_subset U (\<lambda>V'. S \<inter> .[V'] = {}); x = (SOME x'. x' \<in> U - V); (ps0, qs0) = split t (U - {x}) S; (ps1, qs1) = split (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S); split t U S = (ps0 @ ps1, qs0 @ qs1); P t (U - {x}) S (ps0, qs0); P (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S) (ps1, qs1)\<rbrakk> \<Longrightarrow> P t U S (ps0 @ ps1, qs0 @ qs1)" shows "P t U S (split t U S)"
splitAt_distinct_fst_snd: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs" shows "set (fst (splitAt ram vs)) \<inter> set (snd (splitAt ram vs)) = {}"
vec_hom_zero_iff: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b vec"   and n :: "nat" assumes "inj_semiring_hom hom" shows "(map_vec hom x = 0\<^sub>v n) = (x = 0\<^sub>v n)"
prems_1: shows "vs @ xs @ [x] \<in> L M1 - L M2"
FL_bisimilar_simulation_step: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<sim>\<cdot>[F] Q"   and "f \<in>\<^sub>f\<^sub>s F"   and "bn \<alpha> \<sharp>* (\<langle>f\<rangle>Q, F, f)"   and "\<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>,P'\<rangle>" obtains Q' :: "'state" where "\<langle>f\<rangle>Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>" and "P' \<sim>\<cdot>[L (\<alpha>, F, f)] Q'"
between_cong: fixes A :: "'p"   and C :: "'p"   and B :: "'p" assumes "Bet A C B"   and "Cong A C A B" shows "C = B"
1: shows "prim_W w v e * e\<^sup>T * top \<le> v"
id_funpow_id: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "f x = x" shows "(f ^^ n) x = x"
ActualS: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>\<phi> in v] = [\<phi> in dw]"
ifex_minimal_implied: fixes a :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'a ifex" assumes "(a, b) \<in> bf_ifex_rel" shows "ifex_minimal b"
exhaust: fixes y :: "'a strand_label" obtains x1 :: "'a" where "y = ln x1"   | "y = \<star>"
homogeneousD_poly_deg: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p"   and "t \<in> keys p" shows "deg_pm t = poly_deg p"
lc_row_to_poly_pivot_fun: fixes S :: "'t set"   and A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "card S = dim_col A"   and "pivot_fun A f (dim_col A)"   and "i < dim_row A"   and "f i < dim_col A" shows "lc (mat_to_polys (pps_to_list S) A ! i) = (1::'c)"
Sr_zero: fixes S :: "'a set" assumes "sr R S" shows "\<zero> \<in> S"
set_plus_mono2: fixes C :: "'d set"   and D :: "'d set"   and E :: "'d set"   and F :: "'d set" assumes "C \<subseteq> D"   and "E \<subseteq> F" shows "C + E \<subseteq> D + F"
TR1_sub_carr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR1 R M N P \<subseteq> carrier P"
not_eq_complement: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b" 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 "not a = minus (uminus a) one"
opn: fixes C :: "'a set" assumes "C \<in> \<B>" shows "open C"
assms_5: shows "bijective (e * top) \<and> bijective (e[[top]])"
minus_mult_scalar_commute: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "- p \<odot> q = p \<odot> - q"
invalidate_kD_inv: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" shows "kD (invalidate rt dests) = kD rt"
splitAt_distinct_fst: fixes vs :: "'a list"   and ram1 :: "'a" assumes "distinct vs" shows "distinct (fst (splitAt ram1 vs))"
BRPar: fixes R :: "pi"   and T :: "pi"   and Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "(R, T) \<in> Rel"   and "(P, Q) \<in> bangRel Rel" shows "(R \<parallel> P, T \<parallel> Q) \<in> bangRel Rel"
Ax: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> analz H"
coprime_imp_coprime: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and d :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one c d"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e c"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e d" shows "algebraic_semidom.coprime times one a b"
distinct_15: fixes x51 :: "'b rexp"   and x52 :: "'b rexp" shows "One \<noteq> Times x51 x52"