hcju/isabelleps huggingface dataset:
verts_in_translation: fixes X :: "'a allegorical_term" shows "inv_translation (vertices (translation X))"
to_Gromov_completion_lim: fixes u :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" shows "((\<lambda>n. to_Gromov_completion (u n)) \<longlongrightarrow> to_Gromov_completion a) F = (u \<longlongrightarrow> a) F"
univ_poly_mult: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "(\<otimes>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub>) = ring.poly_mult R"
add_less_cancel_left: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" shows "(c + a \<subset># c + b) = (a \<subset># b)"
kop_mono: fixes f :: "'d \<Rightarrow> 'f \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'f \<Rightarrow> 'e set" assumes "op\<^sub>K \<circ> f = op\<^sub>K \<circ> g" shows "f = g"
mono_Un: fixes f :: "'a set \<Rightarrow> 'b set"   and A :: "'a set"   and B :: "'a set" assumes "mono f" shows "f A \<union> f B \<subseteq> f (A \<union> B)"
span_clauses_3: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x + y \<in> module.span scale S"
extended_simps_3: fixes x' :: "'b" shows "\<lbrakk>x'\<rbrakk> < \<stileturn>"
induct: fixes P :: "'b list \<Rightarrow> 'b trie \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b trie" assumes "\<And>b m. P [] (trie.Nd b m)"   and "\<And>x xs b m. (\<And>xa. xa = (case m x of None \<Rightarrow> Trie_Fun.empty | Some t \<Rightarrow> t) \<Longrightarrow> P xs xa) \<Longrightarrow> P (x # xs) (trie.Nd b m)" shows "P a0 a1"
UP_ring_subring_car: fixes S :: "'a set" assumes "subring S R" shows "carrier (UP (R\<lparr>carrier := S\<rparr>)) = {h \<in> carrier (UP R). \<forall>n. h n \<in> S}"
assms_1: shows "Bisimulation_Invariants A B sim1 PA QA PB QB"
join_related: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y" shows "x \<squnion> y = y"
e1e2cond: shows "(e1, e2) \<in> f \<and> e2 = i"
one_closed: shows "\<zero>\<^bsub>poly_ring R\<^esub> \<in> carrier (poly_ring R)"
admissible_leI: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "mcont luba orda Sup (\<le>) f"   and "mcont luba orda Sup (\<le>) g" shows "ccpo.admissible luba orda (\<lambda>x. f x \<le> g x)"
assms: fixes x :: "'a" assumes "x \<in> S" shows "P x x"
ran_def: fixes m :: "'c \<Rightarrow> 'b option" shows "ran m = {b. \<exists>a. m a = Some b}"
comp_by_index_main: fixes n1 :: "nat"   and n2 :: "nat" shows "nat_to_pr (comp_by_index n1 n2) = (\<lambda>x. nat_to_pr n1 (nat_to_pr n2 x))"
r1_bc_def: fixes s :: "nat \<Rightarrow> nat option" shows "r1_bc s \<equiv> s \<in> \<R> \<and> learn_bc \<phi> U\<^sub>0 s"
N2H_eqI: fixes n :: "C"   and b :: "nat"   and m :: "nat" assumes "n \<in> hbase (Suc (Suc b))"   and "H2N (Suc (Suc b)) n = m" shows "N2H (Suc (Suc b)) m = n"
hyp_wrong: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and t :: "nat" assumes "r1_bc s" shows "\<phi> (the (s (e_butlast (r1_bc.prefixes s j (Suc t))))) \<noteq> r1_bc.psi s j"
hyps_4: fixes b :: "Vars.var list" assumes "set b \<subseteq> set L" shows "\<Delta> @ \<Gamma> : body \<Down>\<^bsub>b\<^esub> \<Theta> : z"
weak_lattice_axioms: shows "weak_lattice (fpl \<P> \<H>)"
psubst_sterm: fixes I :: "('sf, 'sc, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sc, 'sz) formula" assumes "is_interp I" shows "sterm_sem I \<theta> = sterm_sem (PFadjoint I \<sigma>) \<theta>"
assms: shows "cone (h, U) \<subseteq> P[X]"
presSwapAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presSwapAll h hA MOD \<equiv> presSwap h MOD \<and> presSwapAbs hA MOD"
that: fixes k :: "nat"   and k' :: "nat" assumes "j \<le> k"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> (Suc k) \<noteq> \<pi>' (Suc k')" shows "thesis"
p: fixes i :: "'a"   and t :: "real" assumes "i \<in> I"   and "t \<in> {0..1}" shows "p i (\<lambda>x. h i (t, x))"
UP_mult_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" obtains "a = sup a b"
has_vector_derivative_within_subset: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and T :: "real set" assumes "(f has_vector_derivative f') (at x within S)"   and "T \<subseteq> S" shows "(f has_vector_derivative f') (at x within T)"
reduced_word_for_0_imp_nil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A zero as" shows "as = []"
ts_tuple_rel_Un: fixes ys :: "(nat \<times> 'a option list set) set"   and zs :: "(nat \<times> 'a option list set) set" shows "ts_tuple_rel (ys \<union> zs) = ts_tuple_rel ys \<union> ts_tuple_rel zs"
assms_3: shows "val_Zp y < val_Zp x"
nearly_healthy_wlp_Abort: shows "nearly_healthy (wlp Abort)"
LatticeDisjunctionIntro: fixes F :: "'a \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> H) \<and> (G \<leadsto> H) \<longrightarrow> (F \<or> G \<leadsto> H)"
2: shows "(inv a) k \<in> carrier (residue_ring (p ^ k))"
CESem_simps_5: fixes scrut :: "exp"   and e\<^sub>1 :: "exp"   and e\<^sub>2 :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> (scrut ? e\<^sub>1 : e\<^sub>2) \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> (C\<cdot>r). CB_project\<cdot>((\<N>\<lbrakk> scrut \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r)\<cdot>((\<N>\<lbrakk> e\<^sub>1 \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r)\<cdot>((\<N>\<lbrakk> e\<^sub>2 \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r))"
spmf_geometric_nonpos: fixes p :: "real" assumes "p \<le> 0" shows "geometric_spmf p = return_pmf None"
s_prod_subdistl: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "sup (x \<cdot> y) (x \<cdot> z) \<le> x \<cdot> sup y z"
assms: shows "2 \<le> n \<or> x \<notin> \<int>"
IH_2: fixes n :: "nat"   and m :: "nat" shows "cut_bound n m F2 \<le> cut_bound' (max n m) F2"
fi: shows "(local.tighten_bound_root ^^ i) (il, ir) = (l', r')"
dom_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.dom (g \<cdot> f) = C.dom f"
abstr_id2: fixes c :: "'c"   and cc :: "('c, 'd) cconstraint"   and v :: "'c \<Rightarrow> nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "c \<notin> collect_clks cc"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "abstr cc M v (v c) 0 = M (v c) 0"
butlast_nth: fixes n :: "nat"   and l :: "'b list" assumes "n + 1 < length l" shows "butlast l ! n = l ! n"
filter_set: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "Set.filter P (set xs) = set (filter P xs)"
countable_PiE: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (F i)" shows "countable (Pi\<^sub>E I F)"
Try: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c1 :: "stmt"   and Q :: "'b assn"   and C :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} .c1. {SXAlloc G Q}"   and "G,A\<turnstile>{Q \<and>. (\<lambda>s. G,s\<turnstile>catch C) ;. new_xcpt_var vn} .c2. {R}"   and "(Q \<and>. (\<lambda>s. \<not> G,s\<turnstile>catch C)) \<Rightarrow> R" shows "G,A\<turnstile>{Normal P} .Try c1 Catch(C vn) c2. {R}"
measure_lborel_Icc: fixes l :: "real"   and u :: "real" assumes "l \<le> u" shows "content {l..u} = u - l"
additive_principal_times_omega: fixes x :: "ordinal" assumes "0 < x" shows "additive_principal (x * \<omega>)"
MGFNormalD: fixes G :: "prog"   and A :: "state triple set"   and t :: "Term.term" assumes "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{Normal \<doteq>} t\<succ> {G\<rightarrow>}"
minGraphProps_facesAt_eq: fixes g :: "graph"   and v :: "nat" assumes "minGraphProps g"   and "v \<in> \<V> g" shows "set (facesAt g v) = {f \<in> \<F> g. v \<in> \<V> f}"
span_Int_subset: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "local.punit.pmdl (A \<inter> B) \<subseteq> local.punit.pmdl A \<inter> local.punit.pmdl B"
scale_eq_0_iff: fixes a :: "'c"   and x :: "('c, 'd) vec" shows "(a *s x = 0) = (a = (0::'c) \<or> x = 0)"
open_map_imp_subset: fixes X1 :: "'b topology"   and X2 :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "open_map X1 X2 f"   and "S \<subseteq> topspace X1" shows "f ` S \<subseteq> topspace X2"
scopeExt: fixes x :: "name"   and P :: "ccs"   and Q :: "ccs" assumes "x \<sharp> P" shows "\<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<sim> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
Casms: shows "mn = MatchAnd (Match (C Casms)) ms"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "ord.rbt_del less x Empty = Empty"
poly_mod_diff_left: fixes x :: "'b poly"   and y :: "'b poly"   and z :: "'b poly" shows "(x - y) mod z = x mod z - y mod z"
len_lt: shows "len m i j xs < len m i j (rem_cycles i j xs)"
mono_neutral_cong_left: fixes T :: "'d set"   and S :: "'d set"   and h :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (0::'e)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
finite_prime_divisors: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "finite {p. normalization_semidom.prime zero times one normalize p \<and> dvd.dvd times p x}"
spr_tFirst: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t = spr_jview a t'" shows "envObs a (tFirst t) = envObs a (tFirst t')"
one_arctic_delta_def: shows "1 = Num_arc_delta (0::'c)"
tf: shows "case_nat t' (case_nat t \<Gamma>) \<turnstile>\<^sub>c f : REAL"
S2_def: fixes v :: "bool"   and m :: "'a" shows "S2 v m = sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z' = (\<^bold>g [^] c\<^sub>v) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1 in sample_uniform (order \<G>) \<bind> (\<lambda>s'. let enc_m = \<^bold>g [^] s'; enc_m' = z' \<otimes> m in return_spmf (v, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c\<^sub>v, w0, enc_m, w1, enc_m'))))))))"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "\<phi>.A.arr f"
lt_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'e" shows "lt p = (if p = 0 then min_term else ord_term_lin.Max (keys p))"
aprimedivisor_def: fixes q :: "'a" shows "aprimedivisor q = (SOME p. prime p \<and> p dvd q)"
fixp_lowerbound: fixes f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b"   and z :: "'a \<times> 'b" assumes "monotone (rel_prod orda ordb) (rel_prod orda ordb) f"   and "rel_prod orda ordb (f z) z" shows "rel_prod orda ordb (ab.fixp f) z"
preserves_cod: fixes f :: "'a" assumes "C.arr f" shows "C.cod (C.map f) = C.map (C.cod f)"
assms_2: shows "\<alpha> \<in> {0<..<1}"
measure_spmf_scale_spmf': fixes r :: "real"   and p :: "'c spmf" assumes "r \<le> 1" shows "measure_spmf (scale_spmf r p) = scale_measure (ennreal r) (measure_spmf p)"
I: shows "infdist w ({x--z} \<union> Gzt) \<le> 4 * deltaG TYPE('a)"
irreducible_dvd_prod: fixes p :: "'b poly"   and f :: "'c \<Rightarrow> 'b poly"   and as :: "'c set" assumes "irreducible p"   and "p dvd prod f as" shows "\<exists>a\<in>as. p dvd f a"
E: shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) xs), inv p ` X) \<in> failures P"
matrix_to_iarray_nth: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e"   and j :: "'d" shows "matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = A $ i $ j"
of_weak_ranking_Nil': fixes x :: "'b"   and y :: "'b" shows "of_weak_ranking [] x y = False"
\<tau>1_eq: shows "lst (s, [\<alpha>_step]) = fst \<tau>1"
le_supE: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and x :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq (sup a b) x" obtains "less_eq a x" and "less_eq b x"
update_simps_2: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r" shows "update rt ip r = rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip))))"
assms_1: shows "e2 = e1 @ e \<or> e2 = e @ e1"
c: shows "(c, d) \<in> set (half_segments_of_aform X)"
3: shows "(f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T)\<^sup>+ = (f \<sqinter> q)\<^sup>T\<^sup>\<star> * (f \<sqinter> - q)\<^sup>+ \<squnion> (f \<sqinter> q)\<^sup>T\<^sup>+"
Map_axioms: shows "Map Q_empty Q_update Q_delete Q_lookup Q_invar"
intersect'_nodes_finite: fixes A :: "('e, 'd) dba"   and B :: "('e, 'f) dba" assumes "finite (DBA.nodes A)"   and "finite (DBA.nodes B)" shows "finite (DGBA.nodes (intersect' A B))"
h_def: shows "h \<equiv> \<lambda>x. \<lambda>i\<in>I. f i (x i)"
symmetric_P: fixes I :: "'b"   and J :: "'b" assumes "\<exists>a b. Q I a b"   and "\<exists>a b. Q J a b"   and "P I J" shows "P J I"
assms: shows "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
ivl_less_eq: fixes x :: "'b ivl"   and y :: "'b ivl" shows "(x \<le> y) = (low x < low y \<or> low x = low y \<and> high x \<le> high y)"
toEnumAlt_def: fixes n :: "nat" shows "toEnumAlt n \<equiv> the (enum_alt n)"
coeff: fixes n :: "nat" shows "coeff P (\<lambda>i. \<ominus> p i) n = \<ominus> coeff P p n"
1: shows "y \<le> x\<^sup>\<star> * (y \<sqinter> z\<^sup>\<star>) \<squnion> - z\<^sup>\<star>"
img_linorder: fixes f :: "'b \<Rightarrow> 'a"   and D :: "'b set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "linorder_on (f ` D) cmp" shows "linorder_on D (cmp_img f cmp)"
subst_fresh: fixes xT :: "char list \<times> Unary_PCF.type"   and u :: "expr"   and t :: "expr" assumes "fresh xT u" shows "subst xT t u = u"
case_optionE: fixes P :: "bool"   and Q :: "'a \<Rightarrow> bool"   and x :: "'a option" assumes "case x of None \<Rightarrow> P | \<lfloor>y\<rfloor> \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'a" where "x = \<lfloor>y\<rfloor>" and "Q y"
one_step_implies_mult: fixes J :: "'k multiset"   and K :: "'k multiset"   and r :: "('k \<times> 'k) set"   and I :: "'k multiset" assumes "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(I + K, I + J) \<in> mult r"
measurable_on_cdivide: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and c :: "real" assumes "f measurable_on S" shows "(\<lambda>x. f x / c) measurable_on S"
discount_factor_def: fixes r :: "real"   and n :: "nat" shows "discount_factor r n = (\<lambda>w. inverse (disc_rfr_proc r n w))"
NX: shows "map_minsky (\<lambda>s. 2 * s + 5) id N \<subseteq> X"
exists_MST: fixes g :: "'b ugraph"   and w :: "'b set \<Rightarrow> nat" assumes "Undirected_Graph.connected g" shows "\<exists>t. is_MST w g t"
msubstlepos_def: fixes c :: "poly"   and t :: "tm"   and a :: "poly"   and b :: "tm" shows "msubstlepos c t a b = Le (tm.Add (tm.Mul a t) (tm.Mul c b))"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<sim>\<^sup>s\<^sub>l \<zero>"
fls_inverse_shift: fixes m :: "int"   and f :: "'a fls" shows "inverse (fls_shift m f) = fls_shift (- m) (inverse f)"
Zr_one: shows "1\<^sub>r\<^bsub>Zr\<^esub> = 1"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "inj_on Xd X"
perp_in_perp_bis: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong X B C D \<or> Tarski_neutral_dimensionless.Perp Bet Cong A X C D"
comm_pow_roots: fixes m :: "nat"   and k :: "nat"   and u :: "'b list"   and v :: "'b list" assumes "m \<noteq> 0"   and "k \<noteq> 0" shows "(u \<^sup>@ m \<cdot> v \<^sup>@ k = v \<^sup>@ k \<cdot> u \<^sup>@ m) = (u \<cdot> v = v \<cdot> u)"
sel_6: fixes x61 :: "'b mrexp2"   and x62 :: "bool" shows "fin (Star2 x61 x62) = x62"
llist_all2_llengthD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" assumes "llist_all2 P xs ys" shows "llength xs = llength ys"
assms_1: fixes i :: "'a" shows "f abs_summable_on S i"
diff_fun_space_def: shows "diff_fun_space = {f. diff_fun k charts f \<and> extensional0 carrier f}"
map_tree_ap_tree_pure_tree: fixes f :: "'b \<Rightarrow> 'a"   and u :: "'b tree" shows "pure f \<diamondop> u = map_tree f u"
fBex_triv_one_point2: fixes A :: "'c fset"   and a :: "'c" shows "fBex A ((=) a) = (a |\<in>| A)"
simps_4: fixes vs :: "'b list"   and bs :: "'b list"   and a :: "tm" shows "Ifm vs bs (Le a) = (Itm vs bs a \<le> (0::'b))"
unit_dvd_iff_1: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times (times a b) c = dvd.dvd times a c"
Domainp_set: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
count_inI: fixes M :: "'a multiset"   and x :: "'a" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
simps_15: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Floor a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<notin> \<int>)"
dvd_prime_power_iff: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "{d. d dvd p ^ n} = (^) p ` {0..n}"
if_type: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> High  \<longrightarrow> (\<forall>mds. mds_consistent mds \<Gamma> \<longrightarrow> c\<^sub>1 \<sim>\<^bsub>mds\<^esub> c\<^sub>2) \<and> (\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High)"   and "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'"   and "\<turnstile> \<Gamma> {c\<^sub>2} \<Gamma>'" shows "\<turnstile> \<Gamma> {Stmt.If e c\<^sub>1 c\<^sub>2} \<Gamma>'"
word_n1_ge: fixes y :: "'b word" shows "y \<le> - 1"
induced_automorphism_ntrivial: shows "\
 \<noteq> id"
stable_on_range': fixes ds :: "'d set"   and X :: "'x set"   and x :: "'x" assumes "stable_on ds X"   and "x \<in> X" shows "x \<in> Field (Pd (Xd x))"
extend_run: fixes w :: "'b list"   and x :: "'a"   and y :: "'a" assumes "set w \<subseteq> set \<Sigma>"   and "x = foldl \<delta> q\<^sub>0 w"   and "y \<in> set (map (\<delta> x) \<Sigma>)" shows "\<exists>w'. set w' \<subseteq> set \<Sigma> \<and> y = foldl \<delta> q\<^sub>0 w'"
src_dom: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "src (local.dom \<mu>) = src \<mu>"
mor_incl: fixes B1 :: "'d set"   and B1' :: "'d set"   and B2 :: "'e set"   and B2' :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "B1 \<subseteq> B1'"   and "B2 \<subseteq> B2'" shows "mor B1 B2 s1 s2 B1' B2' s1 s2 id id"
is_unit_gcd: fixes a :: "'a"   and b :: "'a" shows "is_unit (gcd a b) = algebraic_semidom_class.coprime a b"
that_1: shows "f \<midarrow>x\<rightarrow> l"
abs_eq: fixes x :: "'b" shows "fpxs_const x = Abs_fpxs (\<lambda>n. if n = 0 then x else (0::'b))"
map_acom_Seq: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d acom"   and c1' :: "'c acom"   and c2' :: "'c acom" shows "(ACom.map_acom f c = c1';; c2') = (\<exists>c1 c2. c = c1;; c2 \<and> ACom.map_acom f c1 = c1' \<and> ACom.map_acom f c2 = c2')"
DJcooper: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = Ifm bs (DJ cooper p) \<and> qfree (DJ cooper p)"
vector_all_zero_from_index_def: fixes A' :: "nat \<times> 'a iarray" shows "vector_all_zero_from_index A' = (let i = fst A'; A = snd A' in IArray.all (\<lambda>x. A !! x = (0::'a)) (IArray [i..<IArray.length A]))"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case R of R \<Rightarrow> f1 | B \<Rightarrow> f2) = f1"
fres_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<leftarrow> z \<le> y \<leftarrow> z"
mult_poly_raw_simp_1: fixes q :: "(char list list \<times> int) list" shows "PAC_Polynomials_Operations.mult_poly_raw [] q = []"
N: fixes n :: "nat" assumes "N \<le> n" shows "\<bar>orthonormal_coeff {- pi..pi} trigonometric_set g n\<bar> < e / 2"
update_convs_2: fixes p_oiface' :: "char list \<Rightarrow> char list"   and p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'b word"   and p_dst :: "'b word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'c" shows "p_oiface_update p_oiface' \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = \<lparr>p_iiface = p_iiface, p_oiface = p_oiface' p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
totient_prod_coprime: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set" assumes "pairwise coprime (f ` A)"   and "inj_on f A" shows "totient (prod f A) = (\<Prod>a\<in>A. totient (f a))"
splitAt_1: fixes ram :: "'c" shows "splitAt ram [] = ([], [])"
IH: assumes "insert_ops (xs @ [x])"   and "insert_ops ys"   and "set ys \<subseteq> set (xs @ [x])"   and "x \<in> set ys" shows "x = last ys"
rell_FGcontra_def: fixes L1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool" shows "rell_FGcontra L1 = rel_FGcontra L1 (=) (=) (=) (=) (=) (=) (=) (=) (=) (=)"
bij_G_perm: fixes g :: "'a" assumes "g \<in> G" shows "bij (G_perm g)"