hcju/isabelleps huggingface dataset:
converse_rtrancl_induct_red: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and R :: "char list exp \<Rightarrow> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> char list exp \<Rightarrow> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "\<And>e h l. R e h l e h l"   and "\<And>e\<^sub>0 h\<^sub>0 l\<^sub>0 e\<^sub>1 h\<^sub>1 l\<^sub>1 e' h' l'. \<lbrakk>P \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1)\<rangle>; R e\<^sub>1 h\<^sub>1 l\<^sub>1 e' h' l'\<rbrakk> \<Longrightarrow> R e\<^sub>0 h\<^sub>0 l\<^sub>0 e' h' l'" shows "R e h l e' h' l'"
mono_trans_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "mono_trans t \<equiv> \<forall>P Q. sound P \<and> sound Q \<and> P \<tturnstile> Q \<longrightarrow> t P \<tturnstile> t Q"
simps_1: fixes r :: "real"   and w :: "'b" shows "disc_rfr_proc r 0 w = 1"
merge_mono: fixes ss\<^sub>2 :: "(nat \<times> 'a) list"   and ss\<^sub>1 :: "(nat \<times> 'a) list"   and x :: "'a"   and c :: "'a list"   and pc :: "nat" assumes "set ss\<^sub>2 {\<sqsubseteq>\<^bsub>r\<^esub>} set ss\<^sub>1"   and "x \<in> A"   and "snd ` set ss\<^sub>1 \<subseteq> A"   and "snd ` set ss\<^sub>2 \<subseteq> A" shows "merge c pc ss\<^sub>2 x \<sqsubseteq>\<^bsub>r\<^esub> merge c pc ss\<^sub>1 x"
concurrent_direct_steps_simulates_store_buffer_history_step: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "valid_ownership \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_reads m\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "valid_history program_step ts\<^sub>s\<^sub>b"   and "valid_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"   and "tmps_distinct ts\<^sub>s\<^sub>b"   and "valid_sops ts\<^sub>s\<^sub>b"   and "valid_data_dependency ts\<^sub>s\<^sub>b"   and "load_tmps_fresh ts\<^sub>s\<^sub>b"   and "enough_flushs ts\<^sub>s\<^sub>b"   and "valid_program_history ts\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "safe_reach_direct safe_delayed (ts, m, \<S>)" shows "valid_ownership \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_reads m\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> valid_history program_step ts\<^sub>s\<^sub>b' \<and> valid_sharing \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b' \<and> tmps_distinct ts\<^sub>s\<^sub>b' \<and> valid_data_dependency ts\<^sub>s\<^sub>b' \<and> valid_sops ts\<^sub>s\<^sub>b' \<and> load_tmps_fresh ts\<^sub>s\<^sub>b' \<and> enough_flushs ts\<^sub>s\<^sub>b' \<and> valid_program_history ts\<^sub>s\<^sub>b' \<and> valid ts\<^sub>s\<^sub>b' \<and> (\<exists>ts' \<S>' m'. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>'))"
add_neg_numeral_special_1: 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"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus (numeral.numeral one plus m)) = neg_numeral.sub minus plus one num.One m"
simps_3: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "option_le P (Some x) (Some y) = P x y"
finsum_empty: fixes G :: "('c, 'd) ring_scheme"   and f :: "'e \<Rightarrow> 'c" assumes "abelian_monoid G" shows "finsum G f {} = \<zero>\<^bsub>G\<^esub>"
assms_2: shows "j < dim_col (Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)))"
case_u_less_v: assumes "u < v" shows "\<exists>u. p @ [a] \<in> \<P> (charslength (p @ [a])) u"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "C.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f)"
borel_measurable_add: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m_refines_LPHIL0: shows "LPHIL0 \<sqsubseteq> LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m 0"
CConcStartA: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C a)" shows "x \<in> dom (C (list2FWpolicy (a # b)))"
eval_trm_fv_cong: fixes t :: "Formula.trm"   and v :: "event_data list"   and v' :: "event_data list" assumes "\<forall>x\<in>fv_trm t. v ! x = v' ! x" shows "Formula.eval_trm v t = Formula.eval_trm v' t"
natural_transformation_axioms: shows "natural_transformation A C (K \<circ> F) (K \<circ> H) (K \<circ> \<tau>'\<tau>.map)"
pairwise_finiteI: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "finite {b. \<exists>a. P a b}"   and "finite {a. \<exists>b. P a b}" shows "finite {(a, b). P a b}"
PsiInv_Init: shows "\<turnstile> InitPsi \<longrightarrow> PsiInv"
len_view_hchop_right: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> len v2 ts c = ext v2"
nop_vdomain: shows "\<D>\<^sub>\<circ> vint_mult = \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
SS_def: shows "SS \<equiv> \<lambda>t1 t2. psubst S [(t1, xx), (t2, yy)]"
surj_card_le: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A"   and "B \<subseteq> f ` A" shows "card B \<le> card A"
connected_refl: fixes v :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>" shows "T.connected v v"
gallery_least_length: fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<noteq> D" shows "domain.gallery (C # (ARG_MIN length Cs. domain.gallery (C # Cs @ [D])) @ [D])"
variation_0_1: fixes y :: "real" shows "variation 0 y = 0"
image_vimage_eq: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'b set" shows "f ` f -` A = A \<inter> range f"
approachable_sequenceE: fixes f :: "real \<Rightarrow> 'b"   and p :: "'b" assumes "\<And>t e. \<lbrakk>0 \<le> t; 0 < e\<rbrakk> \<Longrightarrow> \<exists>tt\<ge>t. dist (f tt) p < e" obtains s :: "nat \<Rightarrow> real" where "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" and "(f \<circ> s) \<longlonglongrightarrow> p"
induct: fixes P :: "(('b, 'a) Term.term \<times> ('b, 'c) Term.term) list \<Rightarrow> ('a \<Rightarrow> ('b, 'c) Term.term option) \<Rightarrow> bool"   and a0 :: "(('b, 'a) Term.term \<times> ('b, 'c) Term.term) list"   and a1 :: "'a \<Rightarrow> ('b, 'c) Term.term option" assumes "\<And>\<sigma>. P [] \<sigma>"   and "\<And>x t Pa \<sigma>. (\<sigma> x = None \<or> \<sigma> x = Some t \<Longrightarrow> P Pa (\<sigma>(x \<mapsto> t))) \<Longrightarrow> P ((Var x, t) # Pa) \<sigma>"   and "\<And>f ss g ts Pa \<sigma>. (\<And>x2. decompose (Fun f ss) (Fun g ts) = Some x2 \<Longrightarrow> P (x2 @ Pa) \<sigma>) \<Longrightarrow> P ((Fun f ss, Fun g ts) # Pa) \<sigma>"   and "\<And>f ss x Pa \<sigma>. P ((Fun f ss, Var x) # Pa) \<sigma>" shows "P a0 a1"
le_step_down_word: fixes i :: "'a word"   and n :: "'a word"   and P :: "bool" assumes "i \<le> n"   and "i = n \<longrightarrow> P"   and "i \<le> n - 1 \<longrightarrow> P" shows "P"
dom_comp: fixes g :: "J.Arr"   and f :: "J.Arr" assumes "J.seq g f" shows "J.dom (g \<cdot>\<^sub>J f) = J.dom f"
p: shows "Ifm (real_of_int x # bs) (NDvd j (CN 0 c e))"
cycle_free_diag_intro: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)" shows "linordered_ab_monoid_add.cycle_free plus zero less_eq m n"
append_Cons_nth_not_middle: fixes i :: "nat"   and xs :: "'a list"   and u :: "'a"   and ys :: "'a list"   and z :: "'a" assumes "i \<noteq> length xs" shows "(xs @ u # ys) ! i = (xs @ z # ys) ! i"
table_of_remap_SomeD: fixes t :: "(('c \<times> 'a) \<times> 'b) list"   and k :: "'c"   and k' :: "'a"   and x :: "'b" assumes "table_of (map (\<lambda>((k, k'), x). (k, k', x)) t) k = Some (k', x)" shows "table_of t (k, k') = Some x"
R26_bc_1: shows "pmf (sds R26) b = 0"
torder_onE: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "torder_on A r" obtains "porder_on A r" and "total_onP A r"
coprime_dvd_mult_right_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "dvd.dvd times a (times c b) = dvd.dvd times a b"
Says_imp_knows: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> knows A evs"
inf_right_idem: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) y = inf x y"
mem_grow_impl_correct: fixes m :: "mem"   and n :: "nat"   and m' :: "mem" assumes "mem_grow_impl m n = Some m'" shows "mem_grow m n = m'"
inclusion_is_functor: shows "functor (\<cdot>\<^sub>C) (\<cdot>) \<F>C.inclusion_of_generators"
removelE: fixes s :: "'queue" assumes "queue_invar s"   and "queue_\<alpha> s \<noteq> []" obtains s' :: "'queue" where "queue_dequeue s = (hd (queue_\<alpha> s), s')" and "queue_invar s'" and "queue_\<alpha> s' = tl (queue_\<alpha> s)"
surjective: fixes r :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) NatTrans_scheme" shows "r = \<lparr>NTDom = NTDom r, NTCod = NTCod r, NatTransMap = NatTransMap r, \<dots> = NatTrans.more r\<rparr>"
inf_le2: fixes x :: "'a"   and y :: "'a" shows "inf x y \<le> y"
fps_nonzero_nth: fixes f :: "'a fps" shows "(f \<noteq> 0) = (\<exists>n. fps_nth f n \<noteq> (0::'a))"
is_natural_2: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "\<alpha>' (VVV.cod f) \<cdot> HoVH f = \<alpha>' f"
vmc_path_lset_induction_closed_subset: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and T :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "\<And>v. \<lbrakk>v \<in> S; \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w); v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)\<rbrakk> \<Longrightarrow> \<sigma> v \<in> S \<union> T"   and "\<And>v w. \<lbrakk>v \<in> S; \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w); v \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>); v \<rightarrow>\<^bsub>G\<^esub> w\<rbrakk> \<Longrightarrow> w \<in> S \<union> T"   and "v0 \<in> S"   and "lset P \<inter> T = {}" shows "lset P \<subseteq> S"
stocks_def: fixes Mkt :: "('d, 'c) discrete_market" shows "stocks Mkt = fst (Rep_discrete_market Mkt)"
strict_mono_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less"   and "order.strict_mono less f" shows "order.mono less_eq f"
c: shows "R1 \<subseteq> upRules \<and> R = Ax \<union> R1"
row_add_iterate_PA_preserves_pivot_row: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and P :: "(('a, 'c) vec, 'c) vec"   and j :: "'b"   and b :: "'c" assumes "n < nrows A"   and "mod_type_class.to_nat i \<le> n" shows "fst (row_add_iterate_PA (P, A) n i j) $ i $ b = P $ i $ b"
simps_2: fixes c :: "'a"   and f :: "'a poly"   and m :: "nat" shows "graeffe_poly_impl_main c f (Suc m) = graeffe_one_step c (graeffe_poly_impl_main c f m)"
cont2val_def: shows "cont2val \<equiv> \<Lambda> f. f\<cdot>ID"
assms_6: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
common_first_summand: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM" shows "P x (x \<oplus> y) \<and> P x (x \<oplus> z)"
prod_filter_principal_singleton2: fixes F :: "'d filter"   and x :: "'e" shows "F \<times>\<^sub>F principal {x} = filtermap (\<lambda>a. (a, x)) F"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'b list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
GreenThm_typeII_divisible_region_boundary_gen: fixes \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "only_horizontal_division \<gamma> two_chain_typeII" shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
tgt_VX: shows "io_targets PM (initial PM) (vs @ xs) = {target (vs @ xs || trVX) (initial PM)}"
wf_Un: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "wf r"   and "wf s"   and "Domain r \<inter> Range s = {}" shows "wf (r \<union> s)"
interp_rga_distinct: fixes xs :: "('a \<times> 'a option) list" assumes "rga_ops xs" shows "distinct (interp_rga xs)"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'g list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
seqE_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, s\<rangle> \<Rightarrow> t" obtains ti :: "state" where "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], s\<rangle> \<Rightarrow> ti" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, ti\<rangle> \<Rightarrow> t" and "no_matching_Goto \<gamma> p [r]"   | "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], s\<rangle> \<Rightarrow> t" and "\<not> no_matching_Goto \<gamma> p [r]"
assms_1: shows "x \<in> set (independent_pdevs xs)"
add: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
RP_state_repetition_distribution_productF: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and m :: "nat"   and xs :: "('e \<times> 'f) list"   and vs :: "('e \<times> 'f) list"   and V :: "'e list set"   and V'' :: "('e \<times> 'f) list set" assumes "OFSM M2"   and "OFSM M1"   and "|M2| * m \<le> length xs"   and "|M1| \<le> m"   and "vs @ xs \<in> L M2 \<inter> L M1"   and "is_det_state_cover M2 V"   and "V'' \<in> Perm V M1" shows "\<exists>q\<in>nodes M2. m < card (RP M2 q vs xs V'')"
that_1: shows "\<forall>x\<in>set_spmf q. trace_eq_simcl S p (f x)"
NatTransExt: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans \<eta>1"   and "NatTrans \<eta>2"   and "NTDom \<eta>1 = NTDom \<eta>2"   and "NTCod \<eta>1 = NTCod \<eta>2"   and "\<And>X. X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub> \<Longrightarrow> \<eta>1 $$ X = \<eta>2 $$ X" shows "\<eta>1 = \<eta>2"
omit_branch_def: fixes xs :: "(nat \<times> nat) set"   and branch :: "(('b, 'c) fm list \<times> 'c) list" shows "omit_branch xs branch \<equiv> mapi (\<lambda>v. omit_block (proj xs v)) branch"
zero_aux: fixes f :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f" shows "f 0 = (0::'b)"
owner_ii_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s" shows "PriKey A \<notin> spied s"
sup: fixes f :: "'b \<Rightarrow> 'c"   and F1 :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and F2 :: "'b filter" assumes "f \<in> O[F1](g)"   and "f \<in> O[F2](g)" shows "f \<in> O[sup F1 F2](g)"
big_small_asymmetric: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> o[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'c)"
incrIndexList_help8: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "butlast ls \<noteq> []"
safe_max_Var: fixes x :: "'c" assumes "x \<in> Csafe" shows "max_Var_floatariths ode_e \<le> DIM('c)"
ord_le_group_order: fixes a :: "nat" assumes "finite (carrier (units_of R))"   and "a \<in> carrier (units_of R)" shows "units.ord a \<le> order (units_of R)"
that_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) \<le> L * dist x y"
assms_2: shows "P (\<lambda>while. return_pmf None)"
fixp_induct_uc: fixes U :: "'h \<Rightarrow> 'f \<Rightarrow> 'g"   and F :: "'h \<Rightarrow> 'h"   and C :: "('f \<Rightarrow> 'g) \<Rightarrow> 'h"   and f :: "'h"   and P :: "('f \<Rightarrow> 'g) \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (lfp.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "lfp.admissible P"   and "P (\<lambda>_. \<Squnion> {})"   and "\<And>f. P (U f) \<Longrightarrow> P (U (F f))" shows "P (U f)"
get_shadow_root_is_component_unsafe: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr" shows "set |h \<turnstile> get_dom_component (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^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 host)|\<^sub>r \<inter> set |h \<turnstile> get_dom_component (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 shadow_root_ptr)|\<^sub>r = {}"
pw_SUP_inres: fixes f :: "'c \<Rightarrow> 'b nres"   and X :: "'c set"   and r :: "'b" shows "inres (Sup (f ` X)) r = (\<exists>M\<in>X. inres (f M) r)"
semigroup_ow_def: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semigroup_ow U f \<equiv> (\<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> f a b \<in> U) \<and> (\<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> f (f a b) c = f a (f b c))"
dfs_invar_def: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "dfs_invar \<Sigma>i R \<equiv> dfs_invar_add \<inter> {s. dfs_\<alpha> s \<in> sse_invar \<Sigma>i R}"
unprot_invt: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP"   and "ProblemIkTpart.unprot prot protFw \<sigma> \<or> protFw \<sigma>" shows "ProblemIkTpart_TEModel.invt prot protFw eintT eintF \<sigma> a = a"
ends_in_segment_2: fixes b :: "'a"   and a :: "'a" shows "b \<in> {a--b}"
not_zeroring: fixes Zero_ring :: "('a, 'c) Ring_scheme \<Rightarrow> bool" shows "\<not> Zero_ring S"
rel_G_pos_distr: fixes Co1 :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Co1' :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Co2 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Co2' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Contra1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Contra2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Contra2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and uu :: "('v \<times> 'w \<times> 'x \<times> 'y \<times> 'z \<times> 'aa \<times> 'ab) itself"   and L1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and L2 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and L1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and L2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2')"
ncoplanar_perm_21: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet D B C A"
listsum_cong: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "xs = ys"   and "\<And>y. y \<in> set ys \<Longrightarrow> f y = g y" shows "ListSum xs f = ListSum ys g"