hcju/isabelleps huggingface dataset:
measurable_case_nat_Suc: fixes A :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "(\<lambda>\<sigma>. \<sigma> \<circ> Suc) \<in> state_measure (Suc ` A) (case_nat t \<Gamma>) \<rightarrow>\<^sub>M state_measure A \<Gamma>"
SUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd"   and u :: "'c \<times> 'd" assumes "\<And>i. i \<in> A \<Longrightarrow> less_eq_prod (f i) u" shows "less_eq_prod (Sup_prod (f ` A)) u"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b tree23 \<Rightarrow> 'a"   and f2 :: "'b tree23 \<Rightarrow> int \<times> 'b \<Rightarrow> 'b tree23 \<Rightarrow> 'a"   and growth :: "'b growth" shows "P (case growth of Stay x \<Rightarrow> f1 x | Sprout x xa xb \<Rightarrow> f2 x xa xb) = ((\<forall>x1. growth = Stay x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22 x23. growth = Sprout x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
id_dom_cod_2: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Cod\<^bsub>CC\<^esub> (Id\<^bsub>CC\<^esub> A) = A"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc_biconnected G M a"   and "bidel_arc_tail_conn G M a" shows "bidel_arc_biconnected_non_triv G M a"
filter_a2_map_cong: fixes ts' :: "nat"   and tp' :: "nat" assumes "ts' \<in> set (linearize tss)"   and "tp' \<in> {tp - (len - 1)..<tp}" shows "filter_a2_map I ts' tp' a2_map = filter_a2_map I ts' tp' a2_map''"
not_rt_fresh_asI2: fixes dip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<not> rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
length_splice: fixes xs :: "'a list"   and ys :: "'a list" shows "|splice xs ys| = |xs| + |ys|"
length_appr_of_ivl: fixes xs :: "real list"   and ys :: "real list" assumes "list_all2 (\<le>) xs ys" shows "length (appr_of_ivl ops xs ys) = length xs"
covector_complement_closed: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "times top x = x" shows "times top (uminus x) = uminus x"
restrict_apply: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'d" shows "restrict f A x = (if x \<in> A then f x else undefined)"
r_prefixes_total: shows "Partial_Recursive.total r_prefixes"
mulex_on_trans: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and L :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset" assumes "mulex_on P A L M"   and "mulex_on P A M N" shows "mulex_on P A L N"
map_update: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and k :: "nat"   and y :: "'e" shows "map f (xs[k := y]) = (map f xs)[k := f y]"
adversary_def: fixes \<A> :: "'pub_input \<Rightarrow> ('msg \<times> 'challenge \<times> 'response \<times> 'challenge \<times> 'response) spmf"   and x :: "'pub_input" shows "adversary \<A> x = \<A> x \<bind> (\<lambda>(c, e, ez, e', ez'). \<A>ss x (c, e, ez) (c, e', ez'))"
ceq_unit_def: shows "CEQ(unit) \<equiv> Some (=)"
ins_invariant: fixes x :: "'a"   and S :: "'b" assumes "isNode x"   and "invariant S"   and "\<not> memb x S" shows "invariant (ins x S)"
simps: fixes e :: "'c DBMEntry"   and t :: "'c" shows "norm_lower e t = (if e \<prec> Lt t then Lt t else e)"
mat1_vert_wall_right: fixes b :: "wall" assumes "is_tangle_diagram b" shows "rat_poly.matrix_mult (kauff_mat b) (blockmat (make_vert_block (nat (codomain_wall b)))) = kauff_mat b"
changes_alt_itv_smods_rec: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "coprime p q" shows "changes_alt_itv_smods a b p q = cross_alt p q a b + changes_alt_itv_smods a b q (- (p mod q))"
mono_Ndet_sym: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(S |-| P) \<sqsubseteq> (S |-| Q)"
less_le_not_le: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<subset># y) = (x \<subseteq># y \<and> \<not> y \<subseteq># x)"
insertI1: fixes a :: "'e"   and B :: "'e set" shows "a \<in> insert a B"
splitAt_distinct_ram_snd: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs" shows "ram \<notin> set (snd (splitAt ram vs))"
hm_empty_def: shows "hm_empty = (\<lambda>_. hm_empty_const)"
l2_inv3_trans: shows "{l2_inv3} TS.trans l2 {> l2_inv3}"
scalar_product_def: fixes v :: "('b, 'c) vec"   and w :: "('b, 'c) vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $ i * w $ i)"
collapse_4: fixes event :: "('d, 'e, 'f) event" assumes "isSnapshot event" shows "Snapshot (occurs_on event) = event"
all_args_def: fixes f :: "'as \<Rightarrow> ('ap, 'aq, 'ar) prog" shows "all_args f = (\<Union>arg. {f arg})"
brouwer: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "compact S"   and "convex S"   and "S \<noteq> {}"   and "continuous_on S f"   and "f ` S \<subseteq> S" obtains x :: "'a" where "x \<in> S" and "f x = x"
set_delete: fixes xs :: "'b list"   and t :: "'b trie" shows "Trie_Fun.set (Trie_Fun.delete xs t) = Trie_Fun.set t - {xs}"
simps_1: shows "is_\<mu>LTL true\<^sub>n = True"
weakly_preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (F (C'.src \<mu>)) (D'.src (F \<mu>))"
cond_bind_pmf_prob: shows "(\<Sum>\<^sup>+ x. ennreal (F x)) = 1"
du: shows "(f (real_of_float u) - f (real_of_float l) - real_of_float (a * (u - l))) / 2 + be \<in>\<^sub>r divl"
valid_W_inv_no_mark_stores_invD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and p :: "'e process_name" assumes "sys_mem_lock s \<noteq> Some p"   and "valid_W_inv s" shows "filter is_mw_Mark (sys_mem_store_buffers p s) = []"
fsps_chain_bound1: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a"   and "C \<noteq> {}" shows "(\<lambda>x\<in>fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}). THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x)) \<in> mHom R (mdl M (fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}))) N"
subresultant_prs_locale_axioms: shows "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"
sub_add_sum: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> real"   and S :: "'c set" assumes "sub_add t"   and "healthy t"   and "\<And>x. sound (P x)" shows "\<lambda>x. \<Sum>y\<in>S. t (P y) x \<tturnstile> t (\<lambda>x. \<Sum>y\<in>S. P y x)"
transitive_conv_closed: fixes x :: "'a" assumes "transitive x" shows "transitive (x\<^sup>T)"
l2product_rdiff: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S"   and "h square_integrable S" shows "l2product S f (\<lambda>x. g x - h x) = l2product S f g - l2product S f h"
ex_bij_betw_finite_nat: fixes M :: "'a set" assumes "finite M" shows "\<exists>h. bij_betw h M {0..<card M}"
one_smult_vec: fixes v :: "'a vec" shows "(1::'a) \<cdot>\<^sub>v v = v"
S1_and_def: fixes A :: "bool \<times> bool"   and \<sigma> :: "bool" shows "S1_and A \<sigma> = (let (a1, a2) = A; s0 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False); s1 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True); s2 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False); s3 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True) in S1_14_OT (s0, s1, s2, s3) () \<bind> (\<lambda>V. return_spmf ((a1, a2), \<sigma>, V)))"
impl_of_inject: fixes x :: "('c, 'd) mapping_rbt"   and y :: "('c, 'd) mapping_rbt" shows "(impl_of x = impl_of y) = (x = y)"
deltaG_Bonk_Schramm_extension_def: fixes uu :: "'c Bonk_Schramm_extension itself" shows "deltaG uu = deltaG TYPE('c)"
euler'_def: fixes e :: "int"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "euler' e f = grid_function (discrete_evolution (euler_increment' e f))"
wfG: shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
is_aligned_weaken: fixes w :: "'b word"   and x :: "nat"   and y :: "nat" assumes "is_aligned w x"   and "y \<le> x" shows "is_aligned w y"
Chaos_def: shows "Chaos \<equiv> action_of (R (false \<turnstile> true))"
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "arr (T f) = CC.arr f"
t'O: shows "spr_jview agent t = spr_jview agent t'"
focus_coeffs_subset_Polys': fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and Y :: "'c set"   and X :: "'c set" assumes "p \<in> P[Y]" shows "range (lookup (focus X p)) \<subseteq> P[Y - X]"
\<tau>Runs2: shows "trsys2.\<tau>Runs_table s2 stlsss2"
simps: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real" shows "nnfisp f M a = (\<exists>u x y. a = y \<and> u\<up>f \<and> (\<forall>xa. sfisp (u xa) M (x xa)) \<and> x\<up>y)"
sums_hurwitz_zeta: fixes a :: "real"   and s :: "complex" assumes "0 < a"   and "1 < Re s" shows "(\<lambda>n. (of_nat n + complex_of_real a) powr - s) sums hurwitz_zeta a s"
assms_2: shows "finite (collect_clkvt (trans_of A))"
simps_5: fixes n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "lang n (Plus r s) = lang n r \<union> lang n s"
nabla_preloeb: fixes x :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)" shows "PreLoebian x"
bit_ucast_iff: fixes a :: "'c word"   and n :: "nat" shows "bit (ucast a) n = (n < LENGTH('b) \<and> bit a n)"
subset_iff_psubset_eq: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = (A \<subset> B \<or> A = B)"
fail_bindE: fixes f :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool" shows "fail >>=E f = fail"
let_to_bind_conv: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a nres" shows "Let x f = RETURN x \<bind> f"
var: fixes xs :: "'varSort"   and x :: "'var" shows "phi (Var xs x)"
smaller_supp: fixes a :: "atom"   and p :: "perm" assumes "a \<in> supp p" shows "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p"
mdc_leq_ldc: fixes 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"   and R :: "char list \<times> char list list" assumes "is_subobj P R" shows "P \<turnstile> mdc R \<preceq>\<^sup>* ldc R"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Dom \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = Dom t \<^bold>\<star> Dom u \<^bold>\<star> Dom v"
a: shows "!x \<cdot> y \<le> !x \<cdot> y \<cdot> !x"
seqE: fixes g :: "'a"   and f :: "'a" assumes "R.seq g f" obtains "R.arr f" and "R.arr g" and "R.dom g = R.cod f"
homeomorphic_sym: fixes s :: "'c set"   and t :: "'d set" shows "s homeomorphic t = t homeomorphic s"
D_LUB: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "D (lim_proc (range S)) = \<Inter> (D ` range S)"
surj_simplex_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexIsomorphism X Y f" shows "f \<turnstile> X = Y"
unrest_all_var: fixes e :: "('b, 'a) uexpr"   and x :: "'c \<Longrightarrow> 'a" assumes "\<Sigma> \<sharp> e" shows "x \<sharp> e"
bounded_linear_mult_left: fixes y :: "'c" shows "bounded_linear (\<lambda>x. x * y)"
quotientD: fixes X :: "'a set"   and R :: "('a \<times> 'a) set"   and A :: "'a set"   and x :: "'a" assumes "equiv X R"   and "A \<in> X // R"   and "x \<in> A" shows "A = R `` {x}"
ex_set_conv: fixes xs :: "'a list" shows "(\<exists>x. x \<in> set xs) = (xs \<noteq> [])"
Disjunctive_fun_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "Disjunctive_fun R = (\<forall>i. R i \<in> Apply.Disjunctive)"
update_def: fixes f :: "'b \<Rightarrow> 'b" shows "State_Monad.update f = State_Monad.get \<bind> State_Monad.set \<circ> f"
goto_no_decision: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and rest :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>Rule m (Goto chain) # rest, Undecided\<rangle> \<Rightarrow> Undecided"
exp_less: fixes a :: "nat"   and b :: "nat" assumes "a \<le> b" shows "2 ^ a \<le> 2 ^ b"
image_eq: fixes A :: "nat set" shows "nth_least A ` {i. enat i < esize A} = A"
head_eq_headn0: fixes p :: "poly" shows "head p = headn p 0"
measure_measurable_prob_algebra2: fixes M :: "'d measure"   and A :: "'d \<Rightarrow> 'e set"   and N :: "'e measure"   and L :: "'d \<Rightarrow> 'e measure" assumes "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N)"   and "L \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "(\<lambda>x. Sigma_Algebra.measure (L x) (A x)) \<in> borel_measurable M"
neutralD: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and P :: "'c \<Rightarrow> ('d \<times> 'd) set"   and P' :: "'c \<Rightarrow> ('d \<times> 'd) set"   and x :: "'d"   and y :: "'d"   and z :: "'d"   and w :: "'d" assumes "neutral scf A Is"   and "profile A Is P"   and "profile A Is P'"   and "{x, y, z, w} \<subseteq> A"   and "\<And>i. i \<in> Is \<Longrightarrow> (x \<^bsub>(P i)\<^esub>\<preceq> y) = (z \<^bsub>(P' i)\<^esub>\<preceq> w)"   and "\<And>i. i \<in> Is \<Longrightarrow> (y \<^bsub>(P i)\<^esub>\<preceq> x) = (w \<^bsub>(P' i)\<^esub>\<preceq> z)" shows "(x \<^bsub>(scf P)\<^esub>\<preceq> y) = (z \<^bsub>(scf P')\<^esub>\<preceq> w) \<and> (y \<^bsub>(scf P)\<^esub>\<preceq> x) = (w \<^bsub>(scf P')\<^esub>\<preceq> z)"
induced_id_hom_surj_1: fixes S :: "'b set"   and R :: "'b list set" assumes "GroupWithGeneratorsRelators S R" shows "GroupHom (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id)"
set_ofD: fixes x :: "'a"   and y :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "(x, y) \<in> set_of P" shows "P x y"
numeral_neq_neg_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "numeral.numeral one plus n \<noteq> uminus one"
cert_stepsize_lemma: fixes h :: "real"   and X0 :: "'d set"   and l :: "'d"   and u :: "'d"   and Res :: "'d set"   and Res_ivl :: "'d set"   and PHI' :: "'d set"   and PHI'' :: "'d set" assumes "0 < h"   and "X0 \<subseteq> {l..u}"   and "Res \<subseteq> Csafe"   and "Res_ivl \<subseteq> Csafe"   and "{l..u} \<subseteq> Csafe"   and "nonempty PHI'"   and "nonempty Res"   and "\<forall>x0\<in>X0. x0 \<in> Csafe \<longrightarrow> h \<in> existence_ivl0 x0 \<longrightarrow> (\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> PHI') \<longrightarrow> x0 + h *\<^sub>R ode x0 \<in> PHI' \<longrightarrow> flow0 x0 h \<in> Res"   and "nonempty Res_ivl"   and "\<forall>x0\<in>X0. x0 \<in> Csafe \<longrightarrow> (\<forall>h\<in>{0..h}. h \<in> existence_ivl0 x0 \<longrightarrow> (\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> PHI') \<longrightarrow> x0 + h *\<^sub>R ode x0 \<in> PHI' \<longrightarrow> flow0 x0 h \<in> Res_ivl)"   and "compact PHI'"   and "PHI'' \<subseteq> PHI'"   and "RETURN (Some PHI'') \<le> Picard_step X0 0 h PHI'" shows "flowpipe0 X0 h h Res_ivl Res"
pickFT_theFT: fixes cl :: "('a, 'b, 'c) cmd list" assumes "WtFT cl = 1" shows "pickFT cl \<in> theFT cl"
deutsch_transform_coeff_4: shows "U\<^sub>f $$ (3, 2) = of_nat (f 1)"
poincare_distance_zero_opposite: fixes z :: "complex" assumes "of_complex z \<in> unit_disc" shows "poincare_distance 0\<^sub>h (of_complex (- z)) = poincare_distance 0\<^sub>h (of_complex z)"
embed_measure_count_space': fixes f :: "'k \<Rightarrow> 'l"   and A :: "'k set" assumes "inj_on f A" shows "embed_measure (count_space A) f = count_space (f ` A)"
clinear_iff: fixes f :: "'b \<Rightarrow> 'c" shows "clinear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>C x) = c *\<^sub>C f x))"
sigma_sets_empty_eq: fixes A :: "'b set" shows "sigma_sets A {} = {{}, A}"
stalk_axioms: shows "stalk Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec {U. is_zariski_open U \<and> \<pp> \<in> U} \<pp>"
power_gt_expt: fixes n :: "nat"   and k :: "nat" assumes "Suc 0 < n" shows "k < n ^ k"
wens_ensures: fixes act :: "('b \<times> 'b) set"   and F :: "'b program"   and B :: "'b set" assumes "act \<in> Acts F" shows "F \<in> wens F act B ensures B"
more_temp_simps2_8: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>\<box>\<diamond>F \<equiv> \<box>\<diamond>F"
Card_order_singl_ordLeq: fixes r :: "('a \<times> 'a) set"   and b :: "'b" assumes "Card_order r"   and "Field r \<noteq> {}" shows "|{b}| \<le>o r"
WS_weak_sim_trans: fixes ms\<^sub>1 :: "'node list"   and s\<^sub>1 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms\<^sub>2 :: "'node list"   and s\<^sub>2 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and as :: "'edge list"   and ms\<^sub>1' :: "'node list"   and s\<^sub>1' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> WS S"   and "S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) =as\<Rightarrow>* (ms\<^sub>1',s\<^sub>1')"   and "as \<noteq> []" shows "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>1', transfers (slice_kinds S as) s\<^sub>2) \<in> WS S \<and> S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as\<Rightarrow>* (ms\<^sub>1',transfers (slice_kinds S as) s\<^sub>2)"
norm_cblinfun_bound: fixes b :: "real"   and f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "0 \<le> b"   and "\<And>x. norm (cblinfun_apply f x) \<le> b * norm x" shows "norm f \<le> b"
bit_eq_rec: 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 a :: "'b"   and b :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b \<and> divide a (numeral.numeral one plus (num.Bit0 num.One)) = divide b (numeral.numeral one plus (num.Bit0 num.One)))"
IsoInvId2: fixes f :: "'b" assumes "ciso f" shows "f ;; Cinv f = id (dom f)"
fst2: fixes A :: "'a trm"   and B :: "'a trm" shows "Fst (SimplyTyped.Pair A B) \<rightarrow>\<beta> A"
rel_conversep: fixes R1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_prod R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_prod R1 R2)\<inverse>\<inverse>"
abstract_def: fixes disc :: "'i common_primitive negation_type \<Rightarrow> bool" shows "abstract disc \<equiv> optimize_matches (abstract_primitive disc)"
arbitrary_union_of_complement: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "(arbitrary union_of P) S = (arbitrary intersection_of (\<lambda>S. P (- S))) (- S)"
divide_pos_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less y zero" shows "less (divide x y) zero"
in_set_zip: fixes p :: "'d \<times> 'e"   and xs :: "'d list"   and ys :: "'e list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
f_shrink_empty_conv: fixes k :: "nat"   and xs :: "'b message_af list" assumes "0 < k" shows "(xs \<div> k = []) = (length xs < k)"
sin_2pi_minus: fixes x :: "real" shows "sin (2 * pi - x) = - sin x"
diff_add_cancel: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
map2_val_raw_singleton_eq_update_by_fun_raw: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'e"   and g :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and h :: "('a \<times> 'f) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and k :: "'a"   and v :: "'f"   and ko :: "'o" assumes "oalist_inv xs"   and "\<And>k x. f k x (0::'f) = x"   and "\<And>zs. oalist_inv zs \<Longrightarrow> g zs = zs"   and "\<And>ko. h ([(k, v)], ko) = map_val_raw (\<lambda>k. f k (0::'e)) ([(k, v)], ko)" shows "map2_val_raw f g h xs ([(k, v)], ko) = update_by_fun_raw k (\<lambda>x. f k x v) xs"
dnf_eval: fixes \<phi> :: "atom fm"   and xs :: "real list" shows "(\<exists>(al, fl)\<in>set (dnf \<phi>). (\<forall>a\<in>set al. aEval a xs) \<and> (\<forall>f\<in>set fl. eval f xs)) = eval \<phi> xs"
domA_def: fixes h :: "('d \<times> 'e) list" shows "domA h = fst ` set h"
implies_sub: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c\<^sub>1 implies c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c\<^sub>1 implies c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<nexists>x1 x2. prod = (x1, x2) \<and> \<not> P (f x1 x2))"
IdFtor'Obj: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>CatDom FId' C\<^esub>" shows "FId' C @@ X = X"
assms: fixes x :: "'a" assumes "x \<in> poly_vars p" shows "\<alpha> x = \<beta> x"
orthogonal_matrix_fst_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) ** fst (QR_decomposition A) = mat 1"
relabel: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "((f, gap_relabel_effect f l u), f, l) \<in> gap_algo_rel"