hcju/isabelleps huggingface dataset:
rHom_add: fixes f :: "'e \<Rightarrow> 'f"   and A :: "('e, 'g) Ring_scheme"   and R :: "('f, 'h) Ring_scheme"   and x :: "'e"   and y :: "'e" assumes "f \<in> rHom A R"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "f (x \<plusminus>\<^bsub>A\<^esub> y) = f x \<plusminus>\<^bsub>R\<^esub> f y"
redex_not_Seq: fixes c :: "('a, 'b, 'c) com"   and c1 :: "('a, 'b, 'c) com"   and c2 :: "('a, 'b, 'c) com"   and P :: "bool" assumes "redex c = Seq c1 c2" shows "P"
recognition_prefix_closure: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and t' :: "(String.literal \<times> value list) list" assumes "recognises_execution e s r (t @ t')" shows "recognises_execution e s r t"
commute: fixes a :: "'e"   and b :: "'e" shows "max a b = max b a"
coeff_int_sum: fixes p :: "'c \<Rightarrow> 'b poly"   and A :: "'c set"   and i :: "int" shows "coeff_int (sum p A) i = (\<Sum>x\<in>A. coeff_int (p x) i)"
select_convs_3: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "sys_reg \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = sys_reg"
any_moves_remain_winning_Even: fixes A :: "'c llist set"   and N :: "enat"   and p :: "'c list"   and \<sigma> :: "'c list \<Rightarrow> 'c"   and m :: "'c" assumes "GSgame A N"   and "odd (length p)"   and "GSgame.strategy_winning_by_Even A N \<sigma> p" shows "GSgame.strategy_winning_by_Even A N \<sigma> (p @ [m])"
measurable_top: fixes M :: "'a measure" shows "\<top> \<in> M \<rightarrow>\<^sub>M count_space UNIV"
exhaust: fixes y :: "vote" obtains "y = A"   | "y = B"
strict_sublist_def: fixes xs :: "'c list"   and ys :: "'c list" shows "strict_sublist xs ys = (sublist xs ys \<and> xs \<noteq> ys)"
node_ptr_kinds_eq2_h: shows "|h \<turnstile> node_ptr_kinds_M|\<^sub>r = |h2 \<turnstile> node_ptr_kinds_M|\<^sub>r"
Cycle_def: fixes L :: "'d list set" shows "Cycle L = {u @ w |u w. w @ u \<in> L}"
P2: shows "Bet C B Q \<and> Cong B Q A P"
range_tms_def: fixes e :: "nat \<Rightarrow> real"   and xs :: "taylor_model list" shows "range_tms e xs = map (range_tm e) xs"
rHom_def: fixes A :: "('d, 'f) Ring_scheme"   and R :: "('e, 'g) Ring_scheme" shows "rHom A R = {f \<in> aHom A R. (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y) \<and> f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>}"
affine_bounded_eq_trivial: fixes S :: "'a set" assumes "affine S" shows "bounded S = (S = {} \<or> (\<exists>a. S = {a}))"
lex_ext_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and n :: "nat"   and ss :: "'a list"   and ts :: "'a list" shows "lex_ext f n ss ts = (let lts = length ts in if length ss = lts \<or> lts \<le> n then lex_ext_unbounded f ss ts else (False, False))"
preserves_assoc_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "F \<a>\<^sub>B[f, g, h] = \<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>C \<a>\<^sub>C[F f, F g, F h] \<cdot>\<^sub>C (C.inv (\<Phi>\<^sub>F (f, g)) \<star>\<^sub>C F h) \<cdot>\<^sub>C C.inv (\<Phi>\<^sub>F (f \<star>\<^sub>B g, h))"
impl_spec: fixes m :: "('a, 'b) TopoS_packed"   and sinvar_spec :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" assumes "TopoS_modelLibrary m sinvar_spec" shows "TopoS_List_Impl (nm_default m) sinvar_spec (nm_sinvar m) (nm_receiver_violation m) (nm_offending_flows m) (nm_node_props m) (nm_eval m)"
mult_of_simps_1: fixes R :: "('b, 'c) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
trueI: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com" shows "G|-{P}. c .{\<lambda>Z s. True}"
eventually_const: fixes F :: "'a filter"   and P :: "bool" assumes "F \<noteq> bot" shows "(\<forall>\<^sub>F x in F. P) = P"
Step_determ: 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 n :: "nat"   and l1 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and m :: "nat"   and l2 :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, n, l1, t) \<in> Step"   and "(M, l, s, m, l2, r) \<in> Step" shows "n = m \<and> t = r \<and> l1 = l2"
sum_list_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "A (0::'a) (0::'b)"   and "rel_fun A (rel_fun A A) (+) (+)" shows "rel_fun (list_all2 A) A sum_list sum_list"
Hi: shows "Hi o\<^sub>L H' (x, y) = 1\<^sub>L"
defs_restrict: fixes c :: "com"   and t :: "char list \<Rightarrow> int option" shows "defs c t |` (- lvars c) = t |` (- lvars c)"
crit_pair_cbelow_sym: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and q :: "'t \<Rightarrow>\<^sub>0 'e" assumes "crit_pair_cbelow_on d m F p q" shows "crit_pair_cbelow_on d m F q p"
jkbpT_traces_of_length: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview" shows "{t \<in> PreEnvironmentJView.jkbpT jkbp envInit envAction envTrans envVal jview T. tLength t = n} = PreEnvironmentJView.jkbpTn jkbp envInit envAction envTrans envVal jview n T"
filter_partition_by_median_2: fixes l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a"   and ps :: "(real, 'a) vec list" assumes "(l, m, r) = partition_by_median k ps" shows "\<forall>p\<in>set r. \<not> p $ k \<le> m"
BlockRed: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and V :: "String.literal"   and vo :: "'addr val option"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and h' :: "'heap"   and l' :: "String.literal \<Rightarrow> 'addr val option"   and T :: "ty" assumes "extTA,P,t \<turnstile> \<langle>e,(h, l(V := vo))\<rangle> -ta\<rightarrow> \<langle>e',(h', l')\<rangle>" shows "extTA,P,t \<turnstile> \<langle>{V:T=vo; e},(h, l)\<rangle> -ta\<rightarrow> \<langle>{V:T=l' V; e'},(h', l'(V := l V))\<rangle>"
res: shows "dmu_array (IArray fs_init) m (IArray []) 0 = res"
new_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
sf_aux2: fixes z :: "nat"   and x :: "nat"   and y :: "nat" assumes "sf z \<le> sf (x + y) + x" shows "z \<le> x + y"
unstar_star_n: fixes P :: "nat \<Rightarrow> bool" shows "unstar (star_n P) = eventually P \<U>"
centre_in_cball: fixes x :: "'c"   and e :: "real" shows "(x \<in> cball x e) = (0 \<le> e)"
wellformed_complete_item_\<beta>: fixes x :: "item" assumes "wellformed_item x"   and "is_complete x" shows "item_\<beta> x = []"
fin_PQ: shows "finite {s + t |s t. lookup p s \<noteq> (0::'b) \<and> lookup q t \<noteq> (0::'b)}"
l_zero: fixes x :: "'a set" assumes "x \<in> carrier (R Quot J)" shows "\<zero>\<^bsub>R Quot J\<^esub> \<oplus>\<^bsub>R Quot J\<^esub> x = x"
rep_eq: fixes x :: "'c \<Rightarrow> bool" shows "set_bset (bCollect x) = Collect x"
transpos_ij_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "(\<tau>\<^bsub>i j\<^esub>) i = j"
Py_def: fixes x :: "'c" shows "Py x = (if x \<in> Y ` space M then prob (Y -` {x} \<inter> space M) else 0)"
not_call_def: fixes p :: "('e, 'f, 'g, 'h) seqp" shows "not_call p \<equiv> \<forall>pn. p \<noteq> call(pn)"
cert_okD2: fixes c :: "'c list"   and n :: "nat"   and T :: "'c"   and B :: "'c"   and A :: "'c set" assumes "cert_ok c n T B A" shows "c ! n = B"
ANDConcEnd: fixes xs :: "('a set set, 'b) Combinators list"   and xa :: "('a set set, 'b) Combinators" assumes "allNetsDistinct (xs @ [xa])"   and "singleCombinators xs" shows "allNetsDistinct xs"
filtration_measurable: fixes f :: "bool stream \<Rightarrow> 'b"   and t :: "nat"   and N :: "'b measure" assumes "f \<in> F t \<rightarrow>\<^sub>M N" shows "random_variable N f"
node\<^sub>i_no_split: fixes ts :: "('a btree \<times> 'a) list"   and k :: "nat"   and t :: "'a btree" assumes "length ts \<le> 2 * k" shows "abs_split.node\<^sub>i k ts t = abs_split.T\<^sub>i (Node ts t)"
assms_3: shows "x \<in>\<^sub>\<circ> B"
bijective_alt: fixes R :: "('a \<times> 'b) set" shows "bijective R = (single_valued R \<and> single_valued (R\<inverse>))"
not_definable: fixes R :: "('a \<Rightarrow> ValD) set"   and args :: "('a \<Rightarrow> ValD) list"   and result :: "'a \<Rightarrow> ValD" assumes "PCF_lr R"   and "set args \<subseteq> R"   and "result \<notin> R" shows "\<nexists>f. definable f \<and> appFLv f args = result"
true_cls_singleton: fixes I :: "'a set"   and L :: "'a literal" shows "(I \<Turnstile> {#L#}) = (I \<Turnstile>l L)"
rem_UnPath_parity_v': fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "is_trail v ps v'" shows "(v \<noteq> v') = (odd (degree v' (rem_unPath ps G)) = even (degree v' G))"
length_le_max_insert_rec_ci: fixes gs :: "'a list"   and ffp :: "nat"   and x1 :: "nat"   and x2 :: "'b \<Rightarrow> nat"   and x3 :: "'a \<Rightarrow> 'b" assumes "length gs \<le> ffp" shows "length gs \<le> max x1 (Max (insert ffp (x2 ` x3 ` set gs)))"
gh: shows "group_hom (reduced_homology_group p X) (relative_homology_group p X S) (hom_induced p X {} X S id)"
ring_times_0_x: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<cdot>\<^sub>r x = \<zero>"
representsE: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'a)"
asym_cases: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "less_eq x y" obtains "asympartp less_eq x y"   | "sympartp less_eq x y"
assms: shows "\<exists>q. q\<^sup>2 = n"
residual_is_design: assumes "multiplicity bl = 1" shows "design (bl\<^sup>c) \<B>\<^sup>R"
delete_node_def: fixes v :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "delete_node v g \<equiv> \<lparr>nodes = nodes g - {v}, edges = edges g \<inter> (- {v}) \<times> UNIV \<times> - {v}\<rparr>"
chain_lSup_lprefix: fixes A :: "'c llist set"   and zs :: "'c llist" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "\<And>xs. xs \<in> A \<Longrightarrow> xs \<sqsubseteq> zs" shows "lSup A \<sqsubseteq> zs"
inject_2: fixes x2 :: "'h"   and y2 :: "'h" shows "(Inr x2 = Inr y2) = (x2 = y2)"
fimage_vdiff: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<bullet> A -\<^sub>\<circ> r `\<^sub>\<bullet> B \<subseteq>\<^sub>\<circ> r `\<^sub>\<bullet> (A -\<^sub>\<circ> B)"
filterlim_inf: fixes f :: "'b \<Rightarrow> 'c"   and F2 :: "'c filter"   and F3 :: "'c filter"   and F1 :: "'b filter" shows "filterlim f (inf F2 F3) F1 = (filterlim f F2 F1 \<and> filterlim f F3 F1)"
W_All: fixes P :: "W \<Rightarrow> bool" shows "(\<forall>w. P w) = (\<forall>w\<in>set [W1, W2, W3]. P w)"
F2_antimono: fixes ds :: "'b set" shows "antimono (F2 ds)"
assms_2: shows "\<Inter> \<A> \<noteq> {}"
LAss1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and V :: "nat" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>V:=e,s\<rangle> -ta\<rightarrow> \<langle>V:=e',s'\<rangle>"
quot_of_ratfps_0: shows "quot_of_ratfps 0 = (0, 1)"
norm_ACI_NFOR: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "\<phi> = NFOR (norm_ACI ` disjuncts \<phi>)"
select_convs_2: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "doctype \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = doctype"
Awhile_10: shows "(\<Union>a\<in>set upds. case a of (a, ba, c) \<Rightarrow> c) \<subseteq> LVE"
ereal_SUP: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "\<bar>SUP a\<in>A. ereal (f a)\<bar> \<noteq> \<infinity>" shows "ereal (\<Squnion> (f ` A)) = (SUP a\<in>A. ereal (f a))"
intro: fixes A :: "('e, 'f) aGroup_scheme" assumes "(\<plusminus>\<^bsub>A\<^esub>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A"   and "\<And>a b c. \<lbrakk>a \<in> carrier A; b \<in> carrier A; c \<in> carrier A\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)"   and "\<And>a b. \<lbrakk>a \<in> carrier A; b \<in> carrier A\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b = b \<plusminus>\<^bsub>A\<^esub> a"   and "mop A \<in> carrier A \<rightarrow> carrier A"   and "\<And>a. a \<in> carrier A \<Longrightarrow> -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>"   and "\<zero>\<^bsub>A\<^esub> \<in> carrier A"   and "\<And>a. a \<in> carrier A \<Longrightarrow> \<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a" shows "aGroup A"
Lcm_int_greater_eq_0: fixes K :: "int set" shows "0 \<le> Lcm K"
possible_single: fixes x :: "'b"   and x' :: "'b" shows "possible (single x) x' = (x = x')"
path_connected_complement_bounded_convex: fixes S :: "'a set" assumes "bounded S"   and "convex S"   and "2 \<le> DIM('a)" shows "path_connected (- S)"
Cod_hcomp: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "Cod (\<mu> \<star> \<nu>) = Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>"
bin_rest_strunc: fixes n :: "nat"   and bin :: "int" shows "signed_take_bit (Suc n) bin div 2 = signed_take_bit n (bin div 2)"
schur_upper_triangular_1: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])" shows "schur_upper_triangular A es \<in> carrier_mat n n"
lessThan_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "'c" shows "ord.lessThan less u \<equiv> {x. less x u}"
iTILL_Suc: fixes x :: "nat"   and n :: "nat" assumes "x < n" shows "Suc x \<in> [\<dots>n]"
swapBinp_def2: fixes xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" shows "binp %%[x1 \<and> x2]_xs = asBinp (qSwapBinp xs x1 x2 (pickBinp binp))"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b option \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option \<Rightarrow> 'a"   and f4 :: "bool \<Rightarrow> 'a"   and f5 :: "bool \<Rightarrow> 'a"   and f6 :: "gc_phase \<Rightarrow> 'a"   and mem_store_action :: "('c, 'd, 'b) mem_store_action" shows "P (case mem_store_action of mw_Mark x xa \<Rightarrow> f1 x xa | mw_Mutate x xa xb \<Rightarrow> f2 x xa xb | mw_Mutate_Payload x xa xb \<Rightarrow> f3 x xa xb | mw_fA x \<Rightarrow> f4 x | mw_fM x \<Rightarrow> f5 x | mw_Phase x \<Rightarrow> f6 x) = (\<not> ((\<exists>x11 x12. mem_store_action = mw_Mark x11 x12 \<and> \<not> P (f1 x11 x12)) \<or> (\<exists>x21 x22 x23. mem_store_action = mw_Mutate x21 x22 x23 \<and> \<not> P (f2 x21 x22 x23)) \<or> (\<exists>x31 x32 x33. mem_store_action = mw_Mutate_Payload x31 x32 x33 \<and> \<not> P (f3 x31 x32 x33)) \<or> (\<exists>x4. mem_store_action = mw_fA x4 \<and> \<not> P (f4 x4)) \<or> (\<exists>x5. mem_store_action = mw_fM x5 \<and> \<not> P (f5 x5)) \<or> (\<exists>x6. mem_store_action = mw_Phase x6 \<and> \<not> P (f6 x6))))"
simps_6: fixes t :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
Sup_Inf_var: 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 bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a"   and X :: "'a set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf (\<Inter> (ord.upset less_eq ` X)) = Sup X"
generate_valid_stateful_policy_IFSACS_select_simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "hosts (generate_valid_stateful_policy_IFSACS G M edgesList) = nodes G"
push_values: fixes k :: "'c"   and p :: "'d"   and q :: "('c, 'd) pq" shows "set |PQ.push k p q| = set |q| \<union> {k}"
float_zero2: fixes x :: "('a, 'b) IEEE.float" shows "is_zero (- x) = is_zero x"
distinct_remove_nth: fixes xs :: "'b list"   and i :: "nat" assumes "distinct xs" shows "distinct (remove_nth i xs)"
exists_lin_indpt_sublist: fixes X :: "'a vec set" assumes "X \<subseteq> carrier_vec n" shows "\<exists>Ls. lin_indpt_list Ls \<and> local.span (set Ls) = local.span X \<and> set Ls \<subseteq> X"
b_InfM: fixes k :: "nat" shows "\<Union> (range (b k)) \<subseteq> {\<Sqinter> (M k)..}"
imult_is_infinity: fixes a :: "enat"   and b :: "enat" shows "(a * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
sup_relative_same_increasing: fixes y :: "'a"   and x :: "'a"   and z :: "'a"   and w :: "'a" assumes "y \<le> x"   and "x \<sqinter> z = x \<sqinter> w" shows "y \<sqinter> z = y \<sqinter> w"
pole_def: fixes l :: "proj2_line" shows "pole l \<equiv> proj2_abs (M *v proj2_line_rep l)"
r2f_f2r_galois: fixes f :: "'b \<Rightarrow> 'c set"   and R :: "('b \<times> 'c) set" shows "(\<R> f = R) = (\<F> R = f)"
int_msb_xor: fixes x :: "int"   and y :: "int" shows "msb (x XOR y) = (msb x \<noteq> msb y)"
elims: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and xb :: "nat"   and y :: "('b \<times> nat) option" assumes "acquire_locks x xa xb = y" obtains L :: "('b \<times> nat) option" and t :: "'b" where "x = La__" and "xa = tb__" and "xb = 0" and "y = La__"   | L :: "('b \<times> nat) option" and t :: "'b" and m :: "nat" where "x = L" and "xa = t" and "xb = Suc m" and "y = acquire_locks (lock_lock L t) t m"
valid_eqvt: fixes P :: "'state"   and x :: "('c, 'pred, 'act) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
add_neg_numeral_special_7: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus one) = zero"
saturatedD: fixes N :: "'a literal multiset set"   and CC :: "'a literal multiset multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset" assumes "saturated N"   and "Infer CC D E \<in> \<Gamma>"   and "set_mset CC \<subseteq> N"   and "D \<in> N" shows "E \<in> N"
assms_1: shows "set (xs @ [x]) = set (ys @ [x])"
changes_snoc: fixes xs :: "'b list"   and b :: "'b"   and a :: "'b" shows "changes (xs @ [b, a]) = (if a * b < (0::'b) then 1 + changes (xs @ [b]) else if b = (0::'b) then changes (xs @ [a]) else changes (xs @ [b]))"
integrable_iff_bounded: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>)"
subfrmlsn_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subfrmlsn \<psi> \<subseteq> subfrmlsn \<phi>"
jumpF_poly_noroot_2: fixes p :: "real poly"   and a :: "real"   and q :: "real poly" assumes "poly p a \<noteq> 0" shows "jumpF_polyR q p a = 0"
stuck_sim: fixes a :: "'b"   and b :: "'a" assumes "(a, b) \<in> R\<inverse>"   and "b \<notin> Domain a.E" shows "a \<notin> Domain b.E"
sifum_types_axioms: shows "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"
oconf_upd_obj: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option" assumes "P,h \<turnstile> obj \<surd>"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "P,h(a \<mapsto> (C, fs')) \<turnstile> obj \<surd>"
conc_fun_fail_iff_1: fixes R :: "('c \<times> 'd) set"   and S :: "'d nres" shows "(\<Down> R S = FAIL) = (S = FAIL)"
subtop_topology: fixes S :: "'b set set"   and T :: "'b set set" assumes "subtopology S T"   and "topology T" shows "topology S"
InterI: fixes C :: "'a set set"   and A :: "'a" assumes "\<And>X. X \<in> C \<Longrightarrow> A \<in> X" shows "A \<in> \<Inter> C"
vec1I_def: fixes ze :: "'c"   and on :: "'c"   and n :: "nat"   and i :: "nat" shows "vec1I ze on n i \<equiv> replicate i ze @ on # replicate (n - 1 - i) ze"