hcju/isabelleps huggingface dataset:
local: fixes p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and l :: "'d"   and \<sigma> :: "nat \<Rightarrow> 'b"   and a :: "'c seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> ctermsl (\<Gamma> pn)"   and "not_call p"   and "l \<in> labels \<Gamma> p"   and "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, pp) \<in> oreachable A S U"   and "p \<in> sterms \<Gamma> pp"   and "(\<sigma>', q) \<in> oreachable A S U"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, l), a, \<sigma>', l')"
union: shows "GEN_OP un (\<union>) (\<langle>B\<rangle>Rs2 \<rightarrow> \<langle>B\<rangle>Rs3 \<rightarrow> \<langle>B\<rangle>Rs3)"
filter_set_fold: fixes A :: "'a set"   and m :: "('a, 'b) mapping"   and t :: "'b" assumes "finite A" shows "filter_set m A t = Finite_Set.fold (\<lambda>a m. case Mapping.lookup m a of None \<Rightarrow> m | Some u \<Rightarrow> if t = u then Mapping.delete a m else m) m A"
inject_2: fixes x21 :: "'d rexp"   and x22 :: "'d rexp"   and y21 :: "'d rexp"   and y22 :: "'d rexp" shows "(Alt x21 x22 = Alt y21 y22) = (x21 = y21 \<and> x22 = y22)"
l2_inv3_def: shows "l2_inv3 \<equiv> {s. \<forall>Rb A B gnx. Auth A B \<langle>Number 1, gnx, Exp Gen (NonceF (Rb $ ny))\<rangle> \<in> chan s \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xgnx = Some gnx \<and> guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))))}"
coef_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and s :: "('b, 'a) tm"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "0 < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef coef_sym s i))"
e_lam_elim2: fixes v1 :: "val"   and v2 :: "val"   and f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "VFun (finsert (v1, v2) f) \<in> E (ELam x e) \<rho>" obtains "v2 \<in> E e ((x, v1) # \<rho>)"
simps_2: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and p :: "int"   and xs :: "((int \<times> int) set set, int) Combinators list" shows "wellformed_policy3 (AllowPortFromTo a b p # xs) = ((\<forall>r. r \<in> set xs \<longrightarrow> dom (C r) \<inter> dom (C (AllowPortFromTo a b p)) = {}) \<and> wellformed_policy3 xs)"
augFlow_resFlow: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "Flow cf s t (augmentingFlow p)"
continuous_map_id_subt: fixes X :: "'b topology"   and S :: "'b set" shows "continuous_map (subtopology X S) X id"
path_join_last: fixes p :: "nat list"   and q :: "nat list" assumes "joinable p q"   and "q \<noteq> []" shows "last (path_join p q) = last q"
RTranCl_elim: fixes h :: "'a"   and h' :: "'a"   and succs :: "'a \<Rightarrow> 'a list" assumes "h [succs]\<rightarrow>* h'" obtains "h' = h"   | g' :: "'a" where "g' \<in> set (succs h)" and "g' [succs]\<rightarrow>* h'"
finite_representation: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d" assumes "module scale" shows "finite {b. module.representation scale basis v b \<noteq> (0::'c)}"
oid_of_option_def: fixes x :: "\<langle>'d\<rangle>\<^sub>\<bottom>" shows "oid_of x = oid_of (the x)"
it_paths_RNodeE: fixes "is" :: "nat list"   and r :: "'a"   and ants :: "'a rose_tree list" assumes "is \<in> it_paths (RNode r ants)" obtains "is = []"   | i :: "nat" and isa :: "nat list" where "is = i # isa" and "i < length (children (RNode r ants))" and "isa \<in> it_paths (children (RNode r ants) ! i)"
prSwap_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prSwap g SEM \<equiv> \<forall>zs z1 z2 s X val. wls s X \<and> sWlsVal SEM val \<longrightarrow> g (X #[z1 \<and> z2]_zs) val = g X (val ^[z1 \<and> z2]_zs)"
lex_prod_def: fixes ra :: "('c \<times> 'c) set"   and rb :: "('d \<times> 'd) set" shows "ra <*lex*> rb = {((a, b), a', b'). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
ipresCons_imp_ipresFreshAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igFreshCls MOD"   and "igConsIPresIGWls MOD" shows "ipresFreshAll h hA MOD"
wfI_pf: fixes R :: "('a \<times> 'a) set" assumes "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}" shows "wf R"
leftderives_implies_derives: fixes u :: "symbol list"   and v :: "symbol list" assumes "leftderives u v" shows "derives u v"
cong__le3412: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "C D Le A B"
senior_states_cases_subseteq: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and p :: "'d"   and n :: "nat"   and q :: "'d" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" obtains "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n \<subseteq> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n"   | "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n \<subseteq> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n"
ord_spmf_None: fixes ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d spmf" shows "ord_spmf ord (return_pmf None) x"
admissible_all: fixes lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>y. ccpo.admissible lub ord (\<lambda>x. P x y)" shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y. P x y)"
cp_name_name1: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'a" shows "pi1 \<bullet> pi2 \<bullet> x = (pi1 \<bullet> pi2) \<bullet> pi1 \<bullet> x"
al_redu: fixes M :: "trm"   and M' :: "trm" assumes "M \<longrightarrow>\<^sub>l M'" shows "M \<longrightarrow>\<^sub>a M'"
prefixclosed_def: fixes tr :: "'b list set" shows "prefixclosed tr \<equiv> \<forall>l1\<in>tr. \<forall>l2. l2 \<preceq> l1 \<longrightarrow> l2 \<in> tr"
negHorbit_HomSet_indVfbasis_W: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "set (concat (negHorbit_homVfbasis T)) \<subseteq> W"
iso_same_card: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme" assumes "G \<cong> H" shows "card (carrier G) = card (carrier H)"
agent_abs_tObsC_abs_subset: fixes t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace"   and a :: "'a" assumes "tObsC t' = tObsC t" shows "agent_abs a t \<subseteq> tObsC_abs t'"
add_top: fixes x :: "'d" assumes "(0::'d) \<le> x" shows "x + \<top> = \<top>"
stream_space_single_preimage: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and r :: "'a stream"   and n :: "nat" assumes "borel_adapt_stoch_proc F X"   and "r \<in> range (proj_stoch_proc X n)" shows "stream_space_single (proj_stoch_proc X n) r \<inter> range (proj_stoch_proc X n) = {r}"
Overriding: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and C :: "qtname"   and new :: "mdecl"   and old :: "qtname \<times> mdecl"   and supr :: "qtname"   and accclass :: "qtname" assumes "G \<turnstile> membr member_in class"   and "membr = (C, mdecl new)"   and "G \<turnstile> (C, new) overrides old"   and "G\<turnstile>class\<prec>\<^sub>C supr"   and "G \<turnstile>Method old in supr dyn_accessible_from accclass" shows "G \<turnstile> membr in class dyn_accessible_from accclass"
internal_case_prod_conv: fixes c :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and a :: "'g"   and b :: "'h" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
is_run_def: fixes s :: "nat \<Rightarrow> 'a" shows "A.is_run s \<equiv> \<exists>l. A.is_lrun l s"
compl_less_compl_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less (uminus x) (uminus y) = less y x"
func_le_dom: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<preceq> g" shows "dom f \<subseteq> dom g"
ennreal_le_minus_iff: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a \<le> b - c) = (a + c \<le> b \<or> a = 0 \<and> b \<le> c)"
sgn_quat_def: fixes x :: "quat" shows "sgn x = x /\<^sub>R norm x"
size1_imbal: fixes t :: "'b tree" assumes "\<not> bal_log t"   and "bal_log (hchild t)"   and "t \<noteq> \<langle>\<rangle>" shows "(2 powr (1 - 1 / c) - 1) * real (size1 t) - 1 < real (imbal t)"
set_sorted_list_of_set: fixes A :: "'t set" assumes "finite A" shows "set (ord_term_lin.sorted_list_of_set A) = A"
NF_anti_mono: fixes A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" assumes "A \<subseteq> B" shows "NF B \<subseteq> NF A"
assms_1: fixes i :: "nat" shows "u i \<in> borel_measurable M"
list_ex1_simps_1: fixes P :: "'b \<Rightarrow> bool" shows "list_ex1 P [] = False"
field_abs_le_zero_epsilon: fixes x :: "'a" assumes "\<And>e. (0::'a) < e \<Longrightarrow> \<bar>x\<bar> \<le> e" shows "\<bar>x\<bar> = (0::'a)"
const: shows "\<forall>y\<in>C. y \<sim> x"
y: shows "y = (s \<cdot> r) \<^sup>@ j"
alpha_rp_eqE: fixes r :: "('c, 'd) alpha_rp_scheme"   and r' :: "('c, 'd) alpha_rp_scheme" assumes "r = r'" obtains "ok r = ok r'" and "tr r = tr r'" and "wait r = wait r'" and "ref r = ref r'" and "alpha_rp.more r = alpha_rp.more r'"
finite_Inf_in: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> inf x y \<in> A" shows "Inf A \<in> A"
simps_1: fixes runz :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat" shows "ir_runs2sigs runz (Running [A, B] (Kab, Ts, Ta)) = card (ir_running runz A B Kab Ts Ta)"
z_def: shows "z \<equiv> getFr [out, inp, zz, x] [] [\<sigma>, \<tau>]"
ND0aux4: fixes x :: "'a"   and y :: "'a"   and p :: "'b"   and b :: "('a, 'b) Combinators list" assumes "AllowPortFromTo x y p \<in> set b" shows "y \<in> set (net_list_aux b)"
faccs_correct2: fixes \<delta> :: "('d, 'e) ta_rule set"   and n :: "'e tree"   and q :: "'d" assumes "accs \<delta> n q" shows "q \<in> faccs \<delta> n"
mat_of_rows_index: fixes i :: "nat"   and rs :: "'b Matrix.vec list"   and j :: "nat"   and n :: "nat" assumes "i < length rs"   and "j < n" shows "mat_of_rows n rs $$ (i, j) = rs ! i $v j"
Restr_Field: fixes r :: "('b \<times> 'b) set" shows "Restr r (Field r) = r"
two_comparisons_into_case_order_11: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and R :: "'b"   and Q :: "'b" assumes "comparator comp" shows "(if x = y then P else if le_of_comp comp y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
cod: fixes i :: "'b" assumes "i \<in> S" shows "\<psi> (cod (\<phi> i)) \<in> C'.codomains i"
case_converter_conv_run_converter: fixes f :: "('l \<Rightarrow> ('m \<times> ('l, 'm, 'n, 'o) converter, 'n, 'o) gpv) \<Rightarrow> 'k"   and conv :: "('l, 'm, 'n, 'o) converter" shows "(case conv of Converter x \<Rightarrow> f x) = f (run_converter conv)"
ball_simps_10: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(\<not> (\<forall>x\<in>A. P x)) = (\<exists>x\<in>A. \<not> P x)"
proper_iff: fixes f :: "'b \<Rightarrow> ereal" shows "proper f = ((\<forall>x. f x \<noteq> - \<infinity>) \<and> (\<exists>x. f x \<noteq> \<infinity>))"
IH: assumes "no_Emptys hs" shows "is_root (pass\<^sub>2 hs)"
wf_gt: shows "wfP (\<lambda>x y. gt y x)"
decision: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Decision X\<rangle> \<Rightarrow> Decision X"
card_partition_on: fixes A :: "'a set"   and k :: "nat" assumes "finite A" shows "card {P. partition_on A P \<and> card P = k} = Stirling (card A) k"
equivp_imp_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivp R" shows "transp R"
mat_of_row_index: fixes x :: "nat"   and y :: "'a vec" assumes "x < dim_vec y" shows "mat_of_row y $$ (0, x) = y $ x"
take_bit_Suc: 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 n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit (Suc n) a = plus (times (take_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))) (numeral.numeral one plus (num.Bit0 num.One))) (modulo a (numeral.numeral one plus (num.Bit0 num.One)))"
is_full: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and a' :: "'a"   and g :: "'b" assumes "full_functor A B F"   and "partial_magma.ide A a"   and "partial_magma.ide A a'"   and "partial_magma.in_hom B g (F a') (F a)" shows "\<exists>f. partial_magma.in_hom A f a' a \<and> F f = g"
Diagonalize_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t\<^bold>\<rfloor> = Cod \<^bold>\<lfloor>t\<^bold>\<rfloor>"
extendEmpty: fixes C :: "'a sequent" shows "extend ( \<Empt> \<Rightarrow>* Em) C = C"
assms: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i \<le> b \<bullet> i"
timpl_closure_set_timpls_trancl_eq: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and c :: "('c set \<times> 'c set) set" shows "timpl_closure_set M (c\<^sup>+) = timpl_closure_set M c"
hcomp_simps_4: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "B.cod (\<nu> \<star>\<^sub>B \<mu>) = B.cod \<nu> \<star>\<^sub>B B.cod \<mu>"
e_length_append: fixes xs :: "nat"   and ys :: "nat" shows "e_length (e_append xs ys) = e_length xs + e_length ys"
collapse_4: fixes event :: "('a, 'b, 'c) event" assumes "isSnapshot event" shows "Snapshot (occurs_on event) = event"
zero_le_one: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
pls_cong_ax_holds: fixes e :: "(name, nat) finfun" shows "eval_fmla e pls_cong_ax"
assms_2: assumes "exception_\<I> (\<I>_real' \<oplus>\<^sub>\<I> \<I>_common') \<turnstile>g \<A> \<surd>" shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) adv"
assms_2: shows "Broadcast m1 \<in> set (history i)"
derives1_implies_derives: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.derives1 \<NN> \<TT> \<RR> a b" shows "CFG.derives \<NN> \<TT> \<RR> a b"
cfun_eqI: fixes f :: "'r \<rightarrow> 's"   and g :: "'r \<rightarrow> 's" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
dist_self: fixes dist :: "'g \<Rightarrow> 'g \<Rightarrow> real"   and uniformity :: "('g \<times> 'g) filter"   and "open" :: "'g set \<Rightarrow> bool"   and x :: "'g" assumes "class.metric_space dist uniformity open" shows "dist x x = 0"
zz_def: shows "zz \<equiv> Variable (Suc (Suc 0))"
W_var_ge: fixes e :: "expr"   and A :: "type_scheme list"   and n :: "nat"   and S :: "nat \<Rightarrow> typ"   and t :: "typ"   and m :: "nat" assumes "W e A n = Some (S, t, m)" shows "n \<le> m"
order_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
equiv_communicating: fixes K :: "'a \<Rightarrow> 'a pmf" shows "equiv UNIV (MC_syntax.communicating K)"
ide_MkIde: fixes A :: "'a" assumes "A \<in> Collect B.obj" shows "ide (MkIde A)"
wt_App_ge_arg: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "wt (App s t) \<ge>\<^sub>p wt t"
A_def: fixes ws :: "nat \<Rightarrow> real" shows "A ws = tensors_from_net (insert_weights shared_weights (deep_model_l rs) ws) $ y"
simps_2: fixes h :: "'b tree" shows "exec Del_min [h] = del_min h"
strict_subset_subst_strictly_subsumes: fixes C :: "'a literal multiset"   and \<eta> :: "'s"   and D :: "'a literal multiset" assumes "C \<cdot> \<eta> \<subset># D" shows "strictly_subsumes C D"
non_empty_multiset_induct: fixes M :: "'b multiset"   and P :: "'b multiset \<Rightarrow> bool" assumes "M \<noteq> {#}"   and "\<And>x. P {#x#}"   and "\<And>x M. P M \<Longrightarrow> P (add_mset x M)" shows "P M"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom f)"
inject_2: fixes x21 :: "Ty"   and x22 :: "Ty"   and y21 :: "Ty"   and y22 :: "Ty" shows "(x21 -> x22 = y21 -> y22) = (x21 = y21 \<and> x22 = y22)"
reflFresh_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'c, 'e, 'f) Recursion.model_scheme" shows "reflFresh h MOD \<equiv> \<forall>ys y s X. wls s X \<longrightarrow> gFresh MOD ys y X (h X) \<longrightarrow> fresh ys y X"
abs_indicator: fixes A :: "'c set"   and x :: "'c" shows "\<bar>indicator A x\<bar> = indicator A x"
closure_of_closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of X closure_of S = X closure_of S"
share_other_untouched: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "sharing_consistent \<S> \<O> sb"   and "a \<notin> \<O> \<union> all_acquired sb" shows "share sb \<S> a = \<S> a"
ex_subset_image_inj: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<exists>T\<subseteq>f ` S. P T) = (\<exists>T\<subseteq>S. inj_on f T \<and> P (f ` T))"
Pxyz: shows "distributed M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) (\<lambda>x. ennreal (Pxyz x))"
Int_empty_right: fixes A :: "'f set" shows "A \<inter> {} = {}"
cont_Rep_cfun2: fixes f :: "'c \<rightarrow> 'd" shows "cont (Rep_cfun f)"
iFROM_iprev_if: fixes t :: "nat"   and n :: "nat" shows "iprev t [n\<dots>] = (if n < t then t - Suc 0 else t)"
empty_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "local.empty \<equiv> \<lambda>i. Const 0"
simps_3: fixes a :: "value"   and b :: "value" shows "value_eq (Some a) (Some b) = (if a = b then true else false)"
eexp_special_values_iff_2: fixes x :: "ereal" shows "(eexp x = 1) = (x = 0)"
big_to_small: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int" assumes "cs \<Rightarrow> t" shows "cs \<rightarrow>* (SKIP, t)"
pre_3: fixes \<sigma> :: "'\<sigma>" assumes "I S0 \<sigma>" shows "P \<sigma>"
that: fixes q :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "\<And>b. q b \<in> P[X]"   and "p = (\<Sum>b\<in>B. q b * b)" shows "thesis"
fv_Let: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "fv (Let \<Gamma> e) = fv \<Gamma> \<union> fv e - domA \<Gamma>"
compose_def: fixes A :: "'c set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "FuncSet.compose A g f = (\<lambda>x\<in>A. g (f x))"
inv_func: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "f\<inverse>\<^bsub>B,A\<^esub> \<in> B \<rightarrow> A"
to_pair_unit_fst: fixes x :: "'a \<times> unit" shows "to_pair_unit (fst x) = x"
simps_1: fixes X :: "'b set set" assumes "SimplicialComplex X" shows "SimplicialComplex.min_maxsimpchain X [] = True"
PDG': shows "PDG src trg knd lve NewEntry lDef lUse state_val NewExit (StrongPostdomination.weak_control_dependence src trg lve NewExit)"
is_syz_sig_adds: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t"   and v :: "'t" assumes "dickson_grading d"   and "is_syz_sig d u"   and "u adds\<^sub>t v"   and "d (pp_of_term v) \<le> dgrad_max d" shows "is_syz_sig d v"
divide_nat_def: fixes m :: "nat"   and n :: "nat" shows "m div n = (if n = 0 then 0 else Max {k. k * n \<le> m})"
label_wrt_special_coset2': fixes \<phi> :: "'a \<Rightarrow> 'a"   and w0 :: "'a permutation"   and v :: "'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "v \<in> w0 `\<rightarrow> C0" shows "{w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v} = w0 +o \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {the_inv_into (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) fundantivertex (\<phi> v)}\<rangle>"
qle_refl: fixes x :: "'a" shows "x \<preceq> x"
butlast_snoc: fixes xs :: "'a list"   and x :: "'a" shows "butlast (xs ## x) = xs"
less_prefix_def: fixes x :: "'b prefix"   and y :: "'b prefix" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
subalgebra_filtration: fixes N :: "'a measure" assumes "subalgebra N M"   and "Filtration.filtration M F" shows "Filtration.filtration N F"
TBtheorem1a: fixes PQ :: "specID"   and E :: "Expression"   and P :: "specID"   and Q :: "specID" assumes "ine PQ E"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ" shows "ine P E \<or> ine Q E"