hcju/isabelleps huggingface dataset:
guar_def: fixes X :: "'d program set"   and Y :: "'d program set" shows "X guarantees Y \<equiv> {F. \<forall>G. F ok G \<longrightarrow> F \<squnion> G \<in> X \<longrightarrow> F \<squnion> G \<in> Y}"
has_integral_component_lbound: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and a :: "'c"   and b :: "'c"   and B :: "real"   and k :: "'d" assumes "(f has_integral i) (cbox a b)"   and "\<forall>x\<in>cbox a b. B \<le> f x \<bullet> k"   and "k \<in> Basis" shows "B * content (cbox a b) \<le> i \<bullet> k"
isEmpty_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "isEmpty t = (t = Empty)"
timpls_transformable_to_mono: fixes TI :: "('e set \<times> 'e set) list"   and TI' :: "('e set \<times> 'e set) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "set TI \<subseteq> set TI'"   and "timpls_transformable_to TI s t" shows "timpls_transformable_to TI' s t"
nat_power_0_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.nat_power one times plus zero nat_power"   and "0 < n" shows "nat_power n zero = one"
scf_anonymous_neutral: fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and R1 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and R2 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R1"   and "is_pref_profile R2"   and "anonymous_profile R1 = image_mset (map ((`) \<sigma>)) (anonymous_profile R2)" shows "scf R1 = \<sigma> ` scf R2"
lm01: fixes P :: "('c \<times> 'd) set" shows "card P = card (P\<inverse>)"
pred_alt_def_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vwellformed (Vrecabs css name \<Gamma>) = (wellformed_venv \<Gamma> \<and> pred_fmap wellformed_clauses css \<and> name |\<in>| fmdom css \<and> True)"
SourcesA2_L0: shows "Sources level0 sA2 = {sA1, sA2, sA3, sA4}"
a_ge_0: fixes a :: "real" assumes "a \<in> set as" shows "0 \<le> a"
encode_problem_parallel_complete_vi_b: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and l :: "nat"   and op :: "'b strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l"   and "l < length \<pi>" shows "\<not> valuation_for_plan \<Pi> \<pi> (Operator l (index (\<Pi>\<^sub>\<O>) op))"
polysizes_def: fixes p :: "nat"   and g :: "graph" shows "polysizes p g \<equiv> let lb = squanderLowerBound g in filter (\<lambda>n. lb + \<d> n < squanderTarget) [3..<Suc (maxGon p)]"
llist_all2_lsetD1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and x :: "'c" assumes "llist_all2 P xs ys"   and "x \<in> lset xs" shows "\<exists>y\<in>lset ys. P x y"
simulates_trace_subset: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" assumes "trace_simulates e1 e2" shows "T e1 \<subseteq> T e2"
B_def: shows "B \<equiv> \<langle>lb, b, rb\<rangle>"
hcomplex_eq_cancel_iff1: fixes xa :: "real star"   and x :: "real star"   and y :: "real star" shows "(hcomplex_of_hypreal xa = HComplex x y) = (xa = x \<and> y = 0)"
ocrit_spec_def: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" shows "ocrit_spec crit = (\<forall>d m data hs ps B p q. dickson_grading d \<longrightarrow> set hs \<subseteq> B \<longrightarrow> fst ` B \<subseteq> dgrad_p_set d m \<longrightarrow> unique_idx (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps) data \<longrightarrow> (\<forall>p' q'. (p', q') \<in>\<^sub>p snd ` set ps \<longrightarrow> fst p' \<noteq> 0 \<longrightarrow> fst q' \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')) \<longrightarrow> p \<in> B \<longrightarrow> q \<in> B \<longrightarrow> fst p \<noteq> 0 \<longrightarrow> fst q \<noteq> 0 \<longrightarrow> crit data hs ps p q \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q))"
proj2_line_through_unique: fixes p :: "proj2"   and q :: "proj2"   and l :: "proj2_line" assumes "p \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l" shows "l = proj2_line_through p q"
infiniteTr_sub: fixes tr :: "'a treeFI" assumes "infiniteTr tr" shows "\<exists>tr'\<in>set (sub tr). infiniteTr tr'"
conga_pseudo_refl: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> B" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C C B A"
comp_inf_covector: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y \<sqinter> top * z) = x * y \<sqinter> top * z"
C: shows "Pri_AgrK b' \<in> items (A \<union> spies evsFR4) \<or> Pri_AgrK g \<in> items (A \<union> spies evsFR4) \<longrightarrow> Pri_AgrK b' \<in> items (A \<union> spies evsFR4) \<and> Pri_AgrK g \<in> items (A \<union> spies evsFR4)"
ij_le3_1: shows "i < prod_list (nths (dims B) I)"
Suc_n_less_card: shows "Suc n < nrows A"
ipSuc: fixes I :: "'c set"   and R :: "('c, 'd) Ring_scheme"   and n :: "nat" shows "I \<^bsup>\<diamondsuit>R Suc n\<^esup> = I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> I \<^bsup>\<diamondsuit>R n\<^esup>"
inj_on_image_iff: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'a" assumes "\<forall>x\<in>A. \<forall>y\<in>A. (g (f x) = g (f y)) = (g x = g y)"   and "inj_on f A" shows "inj_on g (f ` A) = inj_on g A"
split: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
igAbsCongS_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igAbsCongS MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
abs: fixes cs :: "(Term_Class.term \<times> sterm) list" shows "is_value (Sabs cs)"
x0_4: fixes x' :: "real" assumes "min tg (t0 + e / 2) \<le> x'"   and "x' < x0" shows "v x' < w x'"
face_of_Int_supporting_hyperplane_ge_strong: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "convex (S \<inter> {x. a \<bullet> x = b})"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> a \<bullet> x" shows "S \<inter> {x. a \<bullet> x = b} face_of S"
SucSuc: fixes m :: "nat" assumes "P m" shows "P (Suc (Suc m))"
image_add_atLeastLessThan': fixes k :: "'a"   and i :: "'a"   and j :: "'a" shows "(\<lambda>n. n + k) ` {i..<j} = {i + k..<j + k}"
is_natural_1: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.map f \<bullet> V.map (V.dom f) = V.map f"
diff_closed: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G" shows "g - h \<in> G"
sorted_filter: fixes f :: "'b \<Rightarrow> 'a Word.word"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "signed.sorted (map f xs)" shows "signed.sorted (map f (filter P xs))"
if_loc_reach: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos, mds, mem\<rangle>" shows "c' = Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos \<and> mds' = mds \<or> (\<exists>mem''. \<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>1, mds \<oplus> annos, mem''\<rangle>) \<or> (\<exists>mem''. \<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>2, mds \<oplus> annos, mem''\<rangle>)"
heap_upds_ok_upd: fixes \<Gamma> :: "(Vars.var \<times> 'a) list"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "heap_upds_ok (\<Gamma>, Upd x # S)" shows "x \<notin> domA \<Gamma> \<and> x \<notin> upds S"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<tau> f) = Cop.arr f"
plus_mult_def: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a" shows "plus_mult id f zer g inver \<equiv> Matrix_Tensor.mult id f \<and> plus_mult_axioms f zer g inver"
add_diff_cancel_left': fixes a :: "'a"   and b :: "'a" shows "a + b - a = b"
converse_galois: fixes f :: "'c \<Rightarrow> ('e \<times> 'd) set"   and g :: "'c \<Rightarrow> ('d \<times> 'e) set" shows "(\<smile> \<circ> f = g) = (\<smile> \<circ> g = f)"
nth_drop_def: fixes i :: "nat"   and xs :: "'a list" shows "nth_drop i xs \<equiv> take i xs @ drop (Suc i) xs"
tuple_iso_tuple_def: shows "tuple_iso_tuple \<equiv> Tuple_Isomorphism id id"
sup_absorb2: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "sup x y = y"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>2\<^sub>' {x1} x2 {x3}"   and "\<And>Pa. P (\<lambda>s. eSuc (Pa s)) SKIP Pa"   and "\<And>Pa a x. P (\<lambda>s. eSuc (Pa (s[a/x]))) (x ::= a) Pa"   and "\<And>Pa b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. Pa s + \<up> (bval b s)} c\<^sub>1 {Q}; P (\<lambda>s. Pa s + \<up> (bval b s)) c\<^sub>1 Q; \<turnstile>\<^sub>2\<^sub>' {\<lambda>s. Pa s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}; P (\<lambda>s. Pa s + \<up> (\<not> bval b s)) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P (\<lambda>s. eSuc (Pa s)) (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>2\<^sub>' {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>2\<^sub>' {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>I b c. \<lbrakk>\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}; P (\<lambda>s. I s + \<up> (bval b s)) c (\<lambda>t. I t + 1)\<rbrakk> \<Longrightarrow> P (\<lambda>s. I s + 1) (WHILE b DO c) (\<lambda>s. I s + \<up> (\<not> bval b s))"   and "\<And>Pa c Q k P' Q'. \<lbrakk>\<turnstile>\<^sub>2\<^sub>' {Pa} c {Q}; P Pa c Q; \<And>s. Pa s \<le> enat k * P' s; \<And>s. enat k * Q' s \<le> Q s; 0 < k\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
IDirProds_is_subgroup: fixes S :: "'d \<Rightarrow> 'a set"   and I :: "'d set" assumes "\<Union> (S ` I) \<subseteq> carrier G" shows "subgroup (IDirProds G S I) G"
span_insertD: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "'c"   and r :: "'c"   and B :: "'c set" assumes "module scale"   and "p \<in> module.span scale (insert r B)"   and "r \<in> module.span scale B" shows "p \<in> module.span scale B"
FDOM_eff_subset_prob_dom_pair: fixes a :: "('b, 'c) fmap \<times> ('b, 'd) fmap"   and PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" assumes "a \<in> PROB" shows "fmdom' (snd a) \<subseteq> prob_dom PROB"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.arr f = (arr (fst f) \<and> VV.arr (snd f) \<and> I.map (fst f) = I.map (fst (snd f)))"
OrdSum_first: fixes a :: "'b" shows "(a, 1::'c) \<in> OrdSum"
Sup_bool_def: fixes A :: "bool set" shows "\<Sqinter>A = (True \<in> A)"
intersect_mult_set_block_subset_iff: fixes blv :: "'a set"   and p :: "'a set" assumes "blv \<in># \<B>"   and "p \<in># (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2})" shows "p \<subseteq> blv"
minus_plus_minus: fixes s :: "'b"   and t :: "'b"   and u :: "'b"   and v :: "'b" assumes "s adds t"   and "u adds v" shows "t - s + (v - u) = t + v - (s + u)"
int_iffI: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G"   and "\<turnstile> G \<longrightarrow> F" shows "\<turnstile> F = G"
intro: fixes T :: "'b \<Rightarrow> 'b"   and M :: "'b measure" assumes "T \<in> measure_preserving M M" shows "mpt_axioms M T"
permutation_compose: fixes p :: "'a \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a" assumes "permutation p"   and "permutation q" shows "permutation (p \<circ> q)"
bres_canc2: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<rightarrow> x \<cdot> y"
setToListFresh_1: fixes xvec :: "name list"   and yvec :: "name list" shows "xvec \<sharp>* set yvec = xvec \<sharp>* yvec"
HaddP_Zero2: fixes H :: "fm set"   and x :: "tm" shows "H \<turnstile> HaddP x Zero x"
unit_simps_5: fixes a :: "'b" assumes "B.obj a" shows "C.cod (H.unit a) = H a"
unit_simps_5: fixes a :: "'c" assumes "C.obj a" shows "C.cod \<i>\<^sub>C[a] = a"
val_ab: shows "valof (a + b) = valof (zerosign (if is_zero a \<and> is_zero b \<and> sign a = sign b then sign a else 0) (IEEE.round To_nearest (valof a + valof b)))"
g: shows "v = g * q \<and> \<bar>g\<bar> = 1"
valuate_poly_of_vec: fixes w :: "rat vec"   and n :: "nat"   and v :: "rat vec" assumes "w \<in> carrier_vec n"   and "v \<in> carrier_vec n" shows "poly_of_vec v \<lbrace> val_of_vec w \<rbrace> = v \<bullet> w"
Q: shows "partition_on B' Q \<and> partition_on B'' Q'"
inj: fixes j :: "'a" assumes "j \<in> J" shows "f' (f j) = j"
preserves_cod: fixes f :: "J.Arr" assumes "J.arr f" shows "C.cod (\<chi>.A.map f) = \<chi>.A.map (J.cod f)"
vsubset_vinsert_vinsertI: fixes A :: "V"   and b :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> vinsert b B" shows "vinsert b A \<subseteq>\<^sub>\<circ> vinsert b B"
line_integral_degenerate_path: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and c :: "'a" assumes "finite basis" shows "line_integral F basis (\<lambda>x. c) = 0"
big_small_asymmetric: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> o[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'b)"
assms_3: shows "\<Union> (deps ` M) \<subseteq> I ` insert_messages M"
A: shows "four_block_mat A1 A2 A3 A4 \<in> carrier_mat (n + m) (n + m)"
vlist_of_vfsequence_vempty: shows "vlist_of_vfsequence []\<^sub>\<circ> = []"
Rxa_one: shows "R \<diamondsuit>\<^sub>p 1\<^sub>r = carrier R"
split_in_read_only_reads: fixes a :: "nat"   and \<O> :: "nat set"   and xs :: "'b memref list" assumes "a \<in> read_only_reads \<O> xs" shows "\<exists>t v ys zs. xs = ys @ Read\<^sub>s\<^sub>b False a t v # zs \<and> a \<notin> acquired True ys \<O>"
Rec_zero: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" shows "Rec e r zero e"
app2: fixes l :: "nat"   and d :: "val"   and ds :: "val list"   and x :: "lambda" assumes "App l d ds \<in> calls x" shows "set ds \<subseteq> vals x"
arctan_upper_43_def: shows "arctan_lower_42 \<equiv> \<lambda>x. (256 * x ^ 6 + 5943 * x ^ 4 + 19250 * x\<^sup>2 + 15015) * x / (35 * (35 * x ^ 6 + 315 * x ^ 4 + 693 * x\<^sup>2 + 429))"
one_xor_eq: fixes a :: "'a" shows "(1::'a) XOR a = a + of_bool (even a) - of_bool (odd a)"
psubset_imp_ex_mem: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subset> B" shows "\<exists>b. b \<in> B - A"
dist_triangle5: fixes x :: "'a"   and u :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" shows "dist x u \<le> dist x y + dist y z + dist z t + dist t u"
vec_Suc: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "vec (Suc n) f = vCons (f 0) (vec n (f \<circ> Suc))"
aadd_assoc_m: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x \<in> Z\<^sub>-\<^sub>\<infinity>"   and "y \<in> Z\<^sub>-\<^sub>\<infinity>"   and "z \<in> Z\<^sub>-\<^sub>\<infinity>" shows "x + y + z = x + (y + z)"
wf1: shows "mat nr nc (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
preorder_on_axioms: shows "Order_Predicates.preorder_on (f -` carrier) (map_relation f le)"
eq: shows "fst ` {x \<in> set bs. fst x \<noteq> 0} = fst ` set bs - {0}"
colours_distinct_2: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "black r s" shows "\<not> white r s"
RTranCl_conv: fixes g :: "'c"   and h :: "'c"   and succs :: "'c \<Rightarrow> 'c list" shows "(g [succs]\<rightarrow>* h) = ((g, h) \<in> (Rel succs)\<^sup>*)"
indRelTEQ_impl_TRel_is_weak_reduction_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_contrasimulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_contrasimulation (TRel\<^sup>*) Target"
cos_treble_cos: fixes x :: "'a" shows "cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x"
i': shows "length (map f (takeWhile (\<lambda>x. t < f x) xs)) + i < length (map f xs)"
of_weak_ranking_Collect_ge_def: fixes xs :: "'c set list"   and x :: "'c" shows "of_weak_ranking_Collect_ge xs x = {y. of_weak_ranking xs y x}"
prems: shows "mat (row_length ((a # M1) @ M2)) (length ((a # M1) @ M2)) ((a # M1) @ M2)"
le_imp_neg_le: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "- b \<le> - a"
simps: fixes a1 :: "'node"   and a2 :: "'node"   and a3 :: "nat" shows "distance a1 a2 a3 = (\<exists>n as n' x. a1 = n \<and> a2 = n' \<and> a3 = x \<and> n -as\<rightarrow>\<^sub>\<iota>* n' \<and> length as = x \<and> (\<forall>as'. n -as'\<rightarrow>\<^sub>\<iota>* n' \<longrightarrow> x \<le> length as'))"
diagseq_holds: fixes k :: "nat" assumes "\<And>r s n. \<lbrakk>strict_mono r; \<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^sub>F n) \<longlonglongrightarrow> l\<rbrakk> \<Longrightarrow> \<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^sub>F n) \<longlonglongrightarrow> l" shows "\<exists>l. (\<lambda>i. ((f \<circ> (diagseq \<circ> (+) (Suc k))) i)\<^sub>F k) \<longlonglongrightarrow> l"
of_weak_ranking_Cons': fixes x :: "'b set"   and xs :: "'b set list"   and a :: "'b"   and b :: "'b" assumes "is_weak_ranking (x # xs)"   and "a \<in> \<Union> (set (x # xs))"   and "b \<in> \<Union> (set (x # xs))" shows "of_weak_ranking (x # xs) a b = (b \<in> x \<or> a \<notin> x \<and> of_weak_ranking xs a b)"
sync_order_change_prefix: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat"   and n :: "enat"   and E' :: "('b \<times> ('c, 'b) obs_event action) llist" assumes "P,E \<turnstile> a \<le>so a'"   and "ltake n E [\<approx>] ltake n E'"   and "enat a < n"   and "enat a' < n" shows "P,E' \<turnstile> a \<le>so a'"
ovalid_NF_case_prod: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and v :: "'a \<times> 'b" assumes "\<And>x y. ovalidNF (P x y) (B x y) Q" shows "ovalidNF (case v of (x, y) \<Rightarrow> P x y) (case v of (x, y) \<Rightarrow> B x y) Q"
eval_fmla_graph_aux_def: shows "eval_fmla_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>eval_fmla_aux e t u. x1 = (e, t EQ u) \<and> x2 = (eval_trm e t = eval_trm e u)) \<or> (\<exists>eval_fmla_aux e A B. x1 = (e, A OR B) \<and> x2 = (((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, A) \<or> ((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, B)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, A)) \<and> p (e, B) (((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, B))) \<or> (\<exists>eval_fmla_aux e A. x1 = (e, neg A) \<and> x2 = (\<not> ((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, A)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (e, A))) \<or> (\<exists>eval_fmla_aux k e A. x1 = (e, exi k A) \<and> x2 = (\<exists>x. ((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (finfun_update e k x, A)) \<and> atom k \<sharp> e \<and> HOL.induct_forall (\<lambda>x. p (finfun_update e k x, A) (((\<lambda>uu_. uu_) \<circ> eval_fmla_aux) (finfun_update e k x, A)))))"
convergent_powser_stl: fixes c :: "real"   and cs :: "real msllist" assumes "convergent_powser (MSLCons c cs)" shows "convergent_powser cs"
f: shows "f \<in> ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"
wf_rcv_append''': fixes V :: "'d set"   and S :: "('e, 'd) strand_step list"   and t :: "('e, 'd) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
strategy_attracts_viaE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts_via p \<sigma> v0 A W" shows "visits_via P' A W"
unifiers_occur_left_not_Var: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" assumes "x \<in> vars_term t"   and "t \<noteq> Var x" shows "unifiers (insert (Var x, t) E) = {}"
a_mult_idem: fixes x :: "('b \<times> 'b) set" shows "rel_ar x O rel_ar x = rel_ar x"
spr_simAbs_inj_on: shows "inj_on spr_simAbs (rel_ext (\<lambda>x. spr_simAbs x \<in> (\<Union>a. SPRdet.sim_equiv_class a ` SPR.jkbpC)))"
IH: fixes B :: "'a set" assumes "B \<subset> A"   and "B \<subseteq> B" shows "map_pmf (qs_cost R) (pmf_of_set (permutations_of_set B)) = rqs_cost (card B)"
inres_simps_2: fixes X :: "'c set" shows "inres (RES X) = (\<lambda>x. x \<in> X)"
compress_crule_set_def: shows "compress_crule_set = fcompress \<circ> (|`|) (map_prod id fcompress)"
simps_5: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a" shows "(case \<infinity> of eint x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
surjec_finitely_gen: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "R module N"   and "f \<in> mHom R M N"   and "surjec\<^bsub>M,N\<^esub> f"   and "M fgover R" shows "N fgover R"
rep_eq: fixes x :: "nat"   and xa :: "uint64" shows "Rep_uint64 (push_bit x xa) = push_bit x (Rep_uint64 xa)"
lists_length_eq_Suc_eq_empty_iff: fixes A :: "'a set"   and n :: "nat" shows "({xs. set xs \<subseteq> A \<and> length xs = Suc n} = {}) = (A = {})"
find_consistent_signs_at_roots_R: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0" shows "set (find_consistent_signs_at_roots_R p qs) = set (characterize_consistent_signs_at_roots p qs)"
fbnorm_def: fixes q :: "real" shows "fbnorm q \<equiv> 1 < q"
dgrad_p_set_le_snd_crit_pair: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d" shows "dgrad_p_set_le d {snd (crit_pair p q)} {p, q}"
two_chain_horizontal_boundary_def: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "two_chain_horizontal_boundary twoChain \<equiv> \<Union> (horizontal_boundary ` twoChain)"