hcju/isabelleps huggingface dataset:
cross_ratio: fixes z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> u \<and> v \<noteq> w \<or> z \<noteq> w \<and> u \<noteq> v"   and "z \<noteq> \<infinity>\<^sub>h"   and "u \<noteq> \<infinity>\<^sub>h"   and "v \<noteq> \<infinity>\<^sub>h"   and "w \<noteq> \<infinity>\<^sub>h" shows "cross_ratio z u v w = (z -\<^sub>h u) *\<^sub>h (v -\<^sub>h w) :\<^sub>h ((z -\<^sub>h w) *\<^sub>h (v -\<^sub>h u))"
prj_emb_prj: fixes x :: "udom" assumes "DEFL('e) \<sqsubseteq> DEFL('f)" shows "PRJ('e)\<cdot>(EMB('f)\<cdot>(PRJ('f)\<cdot>x)) = PRJ('e)\<cdot>x"
cod_simps_3: shows "H\<mu>H\<nu>\<pi>.cod.leg1 = H\<mu>H\<nu>\<pi>.leg1"
effect_nominal_ts_axioms: shows "effect_nominal_ts (\<turnstile>) (\<rightarrow>) effect_apply L"
rref_Gauss_Jordan_upt_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "reduced_row_echelon_form_upt_k (Gauss_Jordan_upt_k A k) (Suc k)"
dimpoly_all_zero_limit: fixes d :: "nat"   and x :: "linear_poly" assumes "\<And>i. d \<le> i \<Longrightarrow> Abstract_Linear_Poly.coeff x i = 0" shows "dim_poly x \<le> d"
map_sel_3: fixes a :: "('g, 'h, 'i) generat"   and f1 :: "'g \<Rightarrow> 'k"   and f2 :: "'h \<Rightarrow> 'l"   and f3 :: "'i \<Rightarrow> 'j" assumes "\<not> is_Pure a" shows "continuation (map_generat f1 f2 f3 a) = f3 (continuation a)"
embedded_serial_sas_plus_plan_operator_structure: fixes ops :: "('a, 'b) sas_plus_operator list"   and \<psi> :: "('a, 'b) sas_plus_operator list"   and \<Psi> :: "('a, 'b) sas_plus_problem" assumes "ops \<in> set (List_Supplement.embed \<psi>)" obtains op :: "('a, 'b) sas_plus_operator" where "op \<in> set \<psi>" and "map (sasp_op_to_strips \<Psi>) ops = [\<phi>\<^sub>O \<Psi> op]"
trivial_reduced_homology_group_contractible_space: fixes X :: "'b topology"   and p :: "int" assumes "contractible_space X" shows "trivial_group (reduced_homology_group p X)"
larger_is_join2: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.cpo below"   and "below x y" shows "cpo.join below x y = y"
compatAbsVsubst_imp_compatAbsSubst: fixes xs :: "'varSort"   and s :: "'b"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c" assumes "compatAbsVsubst xs s f" shows "compatAbsSubst xs s f"
finestPart: fixes X :: "'c set" shows "finestpart X = {{x} |x. x \<in> X}"
ltl_Until_WeakUntil_con_1: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> U\<^sub>c \<psi> = \<xi> \<Turnstile>\<^sub>c \<phi> W\<^sub>c \<psi> and\<^sub>c F\<^sub>c \<psi>"
supportE_single2: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "supportE (\<lambda>l. P) = {}"
likelihood_backward: fixes s :: "'b"   and os :: "'a list" shows "likelihood s os = backward s os"
Min_eq_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Min less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
prems_2: shows "are_operators_interfering op\<^sub>1 op\<^sub>2"
abss_\<S>D': fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "abss s \<in> \<S>"   and "abss s = (l, R)" obtains u :: "'c \<Rightarrow> real" where "s = (l, u)" and "u \<in> [u]\<^sub>\<R>" and "[u]\<^sub>\<R> \<in> \<R>" and "R = [u]\<^sub>\<R>"
nn_integral_multc: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * c \<partial>M = integral\<^sup>N M f * c"
pedersen_bind: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "ped_commit.bind_advantage \<eta> \<A> = discrete_log.advantage \<eta> (dis_log_\<A> \<eta> \<A>)"
simps_2: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and P :: "'e"   and x :: "'f"   and \<Pi> :: "('e, 'f) chain" shows "chainmap f g (\<lbrace> P \<rbrace> \<cdot> x \<cdot> \<Pi>) = \<lbrace> f P \<rbrace> \<cdot> g x \<cdot> chainmap f g \<Pi>"
integrable_bound: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "integrable M f"   and "g \<in> borel_measurable M"   and "AE x in M. norm (g x) \<le> norm (f x)" shows "integrable M g"
wf\<^sub>s\<^sub>t\<^sub>s'_update\<^sub>s\<^sub>t_ineq: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list" assumes "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "wf\<^sub>s\<^sub>t\<^sub>s' (update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)) (\<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
W3a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a l\<rightarrow> b) l\<rightarrow> (b l\<rightarrow> c) r\<rightarrow> a l\<rightarrow> c = (1::'a)"
bijective: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" assumes "bijective \<alpha> S T" shows "bij_betw \<alpha> S T"
strong_sum_fusion: fixes x :: "i"   and y :: "i" shows "x \<oplus> y = (\<sigma>z. z = x \<or> z = y)"
LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: shows "LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m = (\<lambda>s. Mprefix (Tr\<^sub>l\<^sub>p s) (\<lambda>e. LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m (Up\<^sub>l\<^sub>p s e)))"
integrable_const: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c" shows "integrable (\<lambda>_. c) a b"
tconcat'_def: fixes b :: "unit \<Rightarrow> 'b" shows "TLList.tconcat' b = tconcat (b ())"
projs_5: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>6 (dsn, dsk, flag, hops, nhip, pre) = nhip"
seqE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C (C g f)" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.dom C g = partial_magma.cod C f"
dd: shows "dist_perm (swaps ss s1) (swaps ss s1)"
nasty_arithmetic2: fixes T :: "real"   and t :: "real" assumes "0 < T"   and "t \<le> 0" obtains q :: "nat" and r :: "real" where "t = real q * - T + r" and "0 \<le> r" and "r < T"
assms: shows "has_ticket (map_ticket (\<lambda>t. t mod N) (ts t)) tki"
LetNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and t :: "'a OCL_Types.type option"   and init :: "'a expr"   and body :: "'a expr"   and b :: "'a expr" assumes "\<Gamma> \<turnstile> expr.Let v t init body \<Rrightarrow> b" obtains init\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and body\<^sub>2 :: "'a expr" where "t = Some \<tau>__" and "b = expr.Let v (Some \<tau>__) init\<^sub>2__ body\<^sub>2__" and "\<Gamma> \<turnstile> init \<Rrightarrow> init\<^sub>2__" and "\<Gamma>(v \<mapsto>\<^sub>f \<tau>__) \<turnstile> body \<Rrightarrow> body\<^sub>2__"   | init\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and body\<^sub>2 :: "'a expr" where "t = None" and "b = expr.Let v (Some \<tau>) init\<^sub>2 body\<^sub>2" and "\<Gamma> \<turnstile> init \<Rrightarrow> init\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>E init\<^sub>2 : \<tau>" and "\<Gamma>(v \<mapsto>\<^sub>f \<tau>) \<turnstile> body \<Rrightarrow> body\<^sub>2"
sets_empty_iffI: fixes A :: "'d set"   and B :: "'e set" assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a. a \<in> A" shows "(A = {}) = (B = {})"
cp_valid: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and \<tau> :: "'d state \<times> 'd state" shows "(\<upsilon> X) \<tau> = (\<upsilon> (\<lambda>_. X \<tau>)) \<tau>"
termMOD_igFreshCls: shows "igFreshCls termMOD"
Bs_2: shows "B2 \<in> carrier_mat k (n - k)"
gen_binomial_real: fixes z :: "real"   and a :: "real" assumes "\<bar>z\<bar> < 1" shows "(\<lambda>n. (a gchoose n) * z ^ n) sums (1 + z) powr a"
assms: shows "falsifies\<^sub>c G (C \<cdot>\<^sub>l\<^sub>s \<tau>)"
simps: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "real"   and h :: "real"   and x :: "'b"   and j :: "nat" shows "rk_eval A c f t h x j = f (t + h * c j) (x + h *\<^sub>R (\<Sum>l = 1..<j. A j l *\<^sub>R rk_eval A c f t h x l))"
ind: assumes "abc_final (abc_steps_l (0, lm) p n) p"   and "p \<noteq> []" shows "\<exists>n'. abc_notfinal (abc_steps_l (0, lm) p n') p \<and> abc_final (abc_steps_l (0, lm) p (Suc n')) p"
simps_14: fixes f :: "'b"   and v :: "'b local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<l>\<^bold>[v\<^bold>] = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> \<^bold>\<l>\<^bold>[v\<^bold>]"
mult_pos_neg: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) < a"   and "b < (0::'b)" shows "a * b < (0::'b)"
rt_fresher_refl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> rt"
index_add_mat_1: fixes i :: "nat"   and B :: "'b mat"   and j :: "nat"   and A :: "'b mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A + B) $$ (i, j) = A $$ (i, j) + B $$ (i, j)"
eff_Seq: fixes c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state" shows "eff (c1 ;; c2) s = eff c1 s"
int_not_posinf: fixes ra :: "32 Word.word" assumes "real_of_int (sint ra) < real_of_int (sint POS_INF)"   and "real_of_int (sint NEG_INF) < real_of_int (sint ra)" shows "ra \<noteq> POS_INF"
that_1: assumes "of_nat m / (2::'a) ^ k \<in> \<nat>" shows "thesis"
closest_supchamberD_1: fixes F :: "'a set"   and D :: "'a set" assumes "F \<in> X"   and "chamber D" shows "chamber (closest_supchamber F D)"
dvd_div_mult_self: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times (divide b a) a = b"
mapping_of: fixes x :: "'a mpoly" shows "mapping_of x \<in> UNIV"
zcf_poly_shift: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "zcf (poly_shift f) = f 1"
R_def: shows "R = (\<lambda>x. \<Sum>i\<in>{0<..nat \<lfloor>sqrt x\<rfloor>}. frac (x / real i))"
right_neutral: fixes a :: "'c list" shows "a \<cdot> [] = a"
es_inputs_are_events_def: fixes ES :: "'c ES_rec" shows "es_inputs_are_events ES \<equiv> I\<^bsub>ES\<^esub> \<subseteq> E\<^bsub>ES\<^esub>"
baEANow: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<phi> :: "'b \<Rightarrow> bool" assumes "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)"   and "active c (t n)" shows "\<phi> (tCMP c (t n))"
app_def: fixes t1 :: "('a, 'b) FingerTreeStruc"   and t2 :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.app t1 t2 = FingerTreeStruc.app3 t1 [] t2"
ennreal_top_eq_mult_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(top = a * b) = (a = top \<and> b \<noteq> 0 \<or> b = top \<and> a \<noteq> 0)"
IDE_implies_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.IDE C a" shows "partial_magma.ide (free_monoidal_category.comp C) a"
ordertype_image_plus: fixes \<alpha> :: "V"   and u :: "V" assumes "Ord \<alpha>" shows "ordertype ((+) u ` elts \<alpha>) VWF = \<alpha>"
add_mask_lower_bits: fixes x :: "'b word"   and n :: "nat"   and p :: "'b word" assumes "is_aligned x n"   and "\<forall>n'\<ge>n. n' < LENGTH('b) \<longrightarrow> \<not> p !! n'" shows "x + p AND NOT (mask n) = x"
subcls1D: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "\<exists>fs ms Bs. class P C = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> D \<in> baseClasses Bs"
\<sigma>_subterm_inj: shows "subterm_inj_on \<sigma> (subst_domain \<sigma>)"
varInClauseVars: fixes variable :: "nat"   and clause :: "Literal list" shows "(variable \<in> vars clause) = (\<exists>literal. literal el clause \<and> var literal = variable)"
Un_singleton_iff: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" shows "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) (K\<langle>N\<rangle> \<prec> P) [] (\<one>)"
ivl_disj_un_singleton_1: fixes l :: "'c" shows "{l} \<union> {l<..} = {l..}"
cd_sub: shows "set CAs \<union> {DA} \<subseteq> N \<union> {C}"
mset_eq_permutation: fixes xs :: "'c list"   and ys :: "'c list" assumes "mset xs = mset ys" obtains p :: "nat \<Rightarrow> nat" where "p permutes {..<length ys}" and "permute_list p ys = xs"
adjacent_reusables: fixes w :: "nat \<times> nat"   and t :: "'b tree" assumes "0 < fst w"   and "well_shaped t"   and "l t \<le> fst w"   and "r t \<le> snd w" shows "adjacent (fst w, r t) (reusables t w)"
Diagonalize_red: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a" shows "monoidal_language.Diagonalize C (monoidal_language.red C a) = monoidal_language.Diagonalize C a"
r_delay_def: shows "r_delay \<equiv> Cn 2 r_delay_aux [Cn 2 r_length [recf.Id 2 1], recf.Id 2 0, recf.Id 2 1]"
isChoice_def: fixes process :: "'a process" shows "isChoice process = (\<exists>x21 x22. process = Choice x21 x22)"
UniqueReceiverOf: fixes q :: "'a"   and msg :: "('a, 'b) message"   and p :: "'a" assumes "isReceiverOf q msg"   and "p \<noteq> q" shows "\<not> isReceiverOf p msg"
sum_def: shows "sum \<equiv> comm_monoid_set.F (+) (0::'a)"
LCons_lprefix_conv: fixes x :: "'c"   and xs :: "'c llist"   and ys :: "'c llist" shows "lprefix (LCons x xs) ys = (\<exists>ys'. ys = LCons x ys' \<and> lprefix xs ys')"
N_N'_in_mult1: shows "(N, N') \<in> mult1 {(x, y). x \<in> A \<and> y \<in> A \<and> R x y}"
transition_def: fixes Y :: "'b list set"   and c :: "'b"   and X :: "'b list set" shows "Y \<Turnstile>c\<Rightarrow>X \<equiv> Y \<cdot> {[c]} \<subseteq> X"
inj_V_of_list: fixes V_of :: "'a \<Rightarrow> V" assumes "inj V_of" shows "inj (V_of_list V_of)"
up: fixes C :: "('a, 'b) sequent"   and A :: "('a, 'b) form"   and Ps :: "('a, 'b) sequent list"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>)"   and "(Ps, C) \<in> R" shows "rightPrincipal (Ps, C) A R"
substitute_remove_anyways: fixes t :: "'a ttree"   and f :: "'a \<Rightarrow> 'a ttree"   and x :: "'a"   and T :: "'a set"   and t' :: "'a ttree" assumes "repeatable t"   and "f x = t" shows "substitute f T (t \<otimes>\<otimes> t') = substitute (f(x := TTree.empty)) T (t \<otimes>\<otimes> t')"
add_mono: fixes a :: "'h multiset"   and b :: "'h multiset"   and c :: "'h multiset"   and d :: "'h multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
mult_right_less_imp_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times a c) (times b c)"   and "less_eq zero c" shows "less a b"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int" assumes "proper_design point_set block_collection"   and "t_wise_balance_axioms point_set block_collection grouping index" shows "t_wise_balance point_set block_collection grouping index"
Ide_implies_Arr: fixes t :: "'c local.term" assumes "Ide t" shows "Arr t"
invertible_right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "additive.invertible x"   and "x \<in> R"   and "y \<in> R"   and "z \<in> R" shows "(y + x = z + x) = (y = z)"
timpls_transformable_to_inv'_4: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to TI (Fun f T) t"   and "f \<noteq> the_Fun t" shows "is_Abs f \<and> is_Abs (the_Fun t) \<and> (the_Abs f, the_Abs (the_Fun t)) \<in> set TI"
simps_2: fixes L :: "'c \<Rightarrow> ('a \<times> 'a) set"   and R :: "'c \<Rightarrow> ('a \<times> 'a) set"   and E :: "'c \<Rightarrow> ('a \<times> 'a) set"   and x :: "accent \<times> 'c"   and xs :: "(accent \<times> 'c) list" shows "lconv L R E (x # xs) = lstep L R E x O lconv L R E xs"
simps_5: fixes x :: "'a rexp"   and y :: "'a rexp" shows "lang (x \<cdot>\<^sub>r y) = lang x \<cdot> lang y"
quasi_ordered_set_axioms: shows "quasi_ordered_set A (\<lambda>x y. y \<sqsubseteq> x)"
inv_solve_right: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
simps: fixes t :: "'a tree" shows "is_empty t = (t = \<langle>\<rangle>)"
fst_set_fst_f4_red: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "fst ` set (fst (f4_red gs bs ps sps data)) = set (f4_red_aux (gs @ bs) sps)"
gis_on_stack_impl_def: fixes v :: "'q"   and s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gis_on_stack_impl v s = is_on_stack_impl v (snd s)"
JFcol_minimal: fixes K1 :: "'d JF1 \<Rightarrow> 'd set"   and K2 :: "'d JF2 \<Rightarrow> 'd set"   and n :: "nat" assumes "\<And>a. F1set1 (dtor1 a) \<subseteq> K1 a"   and "\<And>b. F2set1 (dtor2 b) \<subseteq> K2 b"   and "\<And>a a'. a' \<in> F1set2 (dtor1 a) \<Longrightarrow> K1 a' \<subseteq> K1 a"   and "\<And>a b'. b' \<in> F1set3 (dtor1 a) \<Longrightarrow> K2 b' \<subseteq> K1 a"   and "\<And>b a'. a' \<in> F2set2 (dtor2 b) \<Longrightarrow> K1 a' \<subseteq> K2 b"   and "\<And>b b'. b' \<in> F2set3 (dtor2 b) \<Longrightarrow> K2 b' \<subseteq> K2 b" shows "\<forall>a b. JF1col n a \<subseteq> K1 a \<and> JF2col n b \<subseteq> K2 b"
dgrad_p_setD: fixes p :: "'a \<Rightarrow>\<^sub>0 'd"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'a" assumes "p \<in> punit.dgrad_p_set d m"   and "v \<in> keys p" shows "d (local.punit.pp_of_term v) \<le> m"
error_returns_heap: fixes h :: "'x"   and e :: "'y"   and h' :: "'x" shows "\<not> h \<turnstile> error e \<rightarrow>\<^sub>h h'"
conjl1_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<lhd> z \<le> y \<lhd> z"
pre_skip_post: fixes b :: "'b upred" shows "(\<lceil>b\<rceil>\<^sub>< \<and> II) = (II \<and> \<lceil>b\<rceil>\<^sub>>)"
assoc: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" shows "append (append a b) c = append a (append b c)"
Trans': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel'"   and "(\<Psi>', T, U) \<in> Rel'" shows "(\<Psi>', S, U) \<in> Rel'"
sup_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "sup_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall>A\<subseteq>carrier X. f (\<Squnion>\<^bsub>X\<^esub>A) = \<Squnion>\<^bsub>Y\<^esub>f ` A)"
\<Gamma>\<^sub>0_def: fixes x :: "'Var" shows "\<Gamma>\<^sub>0 x = None"
r_pdec2: fixes e :: "nat" shows "eval r_pdec2 [e] \<down>= pdec2 e"
extended_Gromov_product_at_commute: fixes e :: "'b"   and x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_Gromov_product_at e x y = extended_Gromov_product_at e y x"
P_LCons: shows "P' = LCons v0 (ltl P')"
inv: fixes a :: "'a" assumes "A.ide a" shows "B.inverse_arrows (\<sigma> a) (\<sigma>' a)"
ereal_diff_le_self: fixes y :: "ereal"   and x :: "ereal" assumes "0 \<le> y" shows "x - y \<le> x"
subset_ball: fixes d :: "real"   and e :: "real"   and x :: "'c" assumes "d \<le> e" shows "ball x d \<subseteq> ball x e"
B'_def: fixes a :: "'a" shows "B' a = snd ` ({x \<in> Sigma A B. (0::'c) \<noteq> f x} \<inter> {(a, b) |b. True})"
dimiC: shows "\<i> \<cdot>\<^sub>m C \<in> carrier_mat n n"
auxiliary_predicate_def: fixes V :: "('exp, 'id) MWLsCom list" shows "auxiliary_predicate V \<equiv> unique_PPV V \<longrightarrow> WHATWHERE_Secure V"
add_order_equality: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat"   and a :: "'b" assumes "class.monoid_add plus zero"   and "order.greater n 0"   and "power.power zero plus a n = zero"   and "\<And>m. \<lbrakk>order.greater m 0; power.power zero plus a m = zero\<rbrakk> \<Longrightarrow> order.greater_eq m n" shows "monoid_add.add_order plus zero a = n"
LN: fixes vs :: "e list"   and lholed :: "Lholed"   and n :: "nat"   and es' :: "e list"   and l :: "Lholed"   and es'' :: "e list"   and k :: "nat"   and es :: "e list"   and lfilledk :: "e list" assumes "const_list vs"   and "lholed = LRec vs n es' l es''"   and "Lfilled k l es lfilledk" shows "Lfilled (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')"
arev_arev_raw: fixes a :: "'b" assumes "a \<in> arcs G" shows "rev_G (rev_G a) = a"
Basic: fixes a :: "'b set"   and A :: "'b set set"   and sp :: "'b set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
Trie: fixes k :: "'a"   and t :: "('a, 'b) trie" assumes "(k, t) \<in> set kts" shows "finite (dom (lookup_trie t))"
exactly_one_in_maximal: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm" assumes "consistency C"   and "V \<in> C"   and "maximal V C" shows "(p \<in> V) \<noteq> ((\<^bold>\<not> p) \<in> V)"
ltl_ldropn: fixes n :: "nat"   and xs :: "'c llist" shows "ltl (ldropn n xs) = ldropn n (ltl xs)"
prenum: fixes e :: "nat"   and x :: "nat" shows "prenum e x \<down>= (if x < e_length e then e_nth e x else 0)"
bis_imp_3: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>01T c2" shows "c1 \<approx>wT c2"
simps: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" shows "r\<^sup>*\<^sup>* a1 a2 = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>*\<^sup>* a b \<and> r b c))"
\<K>_enum: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat"   and i :: "nat" assumes "K \<in> \<K> j0 j"   and "i < card K" shows "j0 < enum K i"