hcju/isabelleps huggingface dataset:
lx: shows "real_of_float (float_plus_down prec lx (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_down prec (lb_pi prec) else float_round_up prec (ub_pi prec)))) \<le> x - real_of_int k * (2 * pi)"
rtyping_imp_typing: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type" assumes "e \<turnstile>\<^sub>R t : T" shows "e \<turnstile> t : T"
OUT_f_SOURCE: fixes n :: "nat" shows "d_OUT (f n) SOURCE = d_IN (h n) b - d_IN (h 0) b"
substitute_T_delete: fixes f :: "'a \<Rightarrow> 'a ttree"   and x :: "'a"   and T :: "'a set"   and t :: "'a ttree" assumes "f x = TTree.empty" shows "substitute f (T - {x}) t = substitute f T t"
iMOD_card_if: fixes r :: "nat"   and m :: "nat" shows "card [ r, mod m ] = (if m = 0 then Suc 0 else 0)"
G_sel: fixes x :: "'v"   and y :: "'v" shows "edge G x y = (0 < cap' (x, y))"
union_assoc: fixes M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" shows "M + N + K = M + (N + K)"
sink_rev_step_1: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and q' :: "'d"   and \<nu> :: "'c" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q"   and "q = \<delta> q' \<nu>"   and "\<nu> \<in> \<Sigma>" shows "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q'"
all_non_empty: shows "\<forall>x\<in>S. cl_ecl x \<noteq> {}"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>Adj'.\<epsilon> f : Adj'.FG.map a \<rightarrow>\<^sub>C Adj.C.map b\<guillemotright>"
card_partition_on_size1_eq_iverson: fixes A :: "'c set"   and k :: "nat" assumes "finite A" shows "card {P. partition_on A P \<and> card P \<le> k \<and> (\<forall>X\<in>P. card X = 1)} = iverson (card A \<le> k)"
"3_1": shows "v\<^sub>2 \<le>\<^sub>F\<^sub>I w"
signed_take_bit_eq_iff_take_bit_eq: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and "and" :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and or :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and xor :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and mask :: "nat \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and not :: "'k \<Rightarrow> 'k"   and n :: "nat"   and a :: "'k"   and b :: "'k" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "(ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n b) = (take_bit (Suc n) a = take_bit (Suc n) b)"
sorted_wrt_quicksort_by_rel: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (quicksort_by_rel R [] xs)"
PiE_subset: shows "Pi\<^sub>E A B' \<subseteq> Pi\<^sub>E A B"
two_comparisons_into_case_order_2: fixes x :: "'b"   and y :: "'b"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if c1.le x y then if y = x then P else Q else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
if_is_pr3: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and q1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and q2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p \<in> PrimRec3"   and "q1 \<in> PrimRec3"   and "q2 \<in> PrimRec3" shows "(\<lambda>x y z. if p x y z = 0 then q1 x y z else q2 x y z) \<in> PrimRec3"
3: shows "s \<star> Cons r' rs = \<bottom> \<star> Cons r' rs'"
seq_incr: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "seq i \<subseteq> seq j"
disc_1: fixes xs :: "'f llist"   and b :: "'g" assumes "lnull xs" shows "is_TNil (tllist_of_llist b xs)"
assms_4: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F"   and "x \<noteq> y" shows "D \<le> dist x y"
fold_add_vars'_FV_pre: fixes Hs :: "Core.term list"   and acc :: "(variable \<times> typ) list" shows "set (fold add_vars' Hs acc) = set acc \<union> FV (set Hs)"
simps_7: fixes t :: "'a local.term" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<rr> \<lbrace>t\<rbrace>"
llist_all2_LNil1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d llist" shows "llist_all2 P LNil xs = (xs = LNil)"
unats_def: fixes n :: "nat" shows "unats n = {i. i < 2 ^ n}"
cod_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.cod (f \<otimes> g) = \<F>C.cod f \<otimes> \<F>C.cod g"
test_kat_31: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q"   and "p \<cdot> x \<cdot> !q = (0::'a)" shows "p \<cdot> x = p \<cdot> x \<cdot> q"
star_decompose_1: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.left_kleene_algebra star one times sup less_eq less bot" shows "star (sup x y) = star (times (star x) (star y))"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_rexp_of''_norm: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (norm \<phi>)" shows "\<Phi>.lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = lang enum_class.enum n (rexp_of'' n (norm \<phi>))"
single_non_qAFreshAll_ordLess_var: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> |{x. \<not> qAFresh xs x X}| <o |UNIV|) \<and> (qGoodAbs A \<longrightarrow> |{x. \<not> qAFreshAbs xs x A}| <o |UNIV|)"
locally_compact_closedin_Un: fixes S :: "'a set"   and T :: "'a set" assumes "locally compact S"   and "locally compact T"   and "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T" shows "locally compact (S \<union> T)"
less_eq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x + y = y)"
image_empty: fixes f :: "'c \<Rightarrow> 'b" shows "bot_pres ((`) f)"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'b list"   and a3 :: "'a" shows "rtrancl3p r a1 a2 a3 = ((\<exists>a. a1 = a \<and> a2 = [] \<and> a3 = a) \<or> (\<exists>a bs a' b a''. a1 = a \<and> a2 = bs @ [b] \<and> a3 = a'' \<and> rtrancl3p r a bs a' \<and> r a' b a''))"
exhaust: fixes y :: "'d option" obtains "y = None"   | x2 :: "'d" where "y = Some x2"
semiring_of_sets_axioms: shows "semiring_of_sets \<Omega> (sigma_sets \<Omega> A)"
orbit_cong: fixes t :: "'c"   and f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "t \<in> orbit f t"   and "\<And>s. s \<in> orbit f t \<Longrightarrow> g s = f s" shows "orbit g t = orbit f t"
in_card_fset_not_0: fixes A :: "'b fset"   and a :: "'b" assumes "a |\<in>| A" shows "card_fset A \<noteq> 0"
Skolem_list_nth: fixes k :: "nat"   and P :: "nat \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. |xs| = k \<and> (\<forall>i<k. P i (xs ! i)))"
npOne: fixes R :: "('c, 'd) Ring_scheme"   and n :: "nat" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub>^\<^bsup>R n\<^esup> = 1\<^sub>r\<^bsub>R\<^esub>"
swap_id_eq: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
hide_sync_D2: fixes A :: "'a set"   and C :: "'a set"   and P :: "'a process"   and Q :: "'a process" assumes "A \<inter> C = {}" shows "D (P \ A \<lbrakk>C\<rbrakk> Q \ A) \<subseteq> D ((P \<lbrakk>C\<rbrakk> Q) \ A)"
SKIP_in_stable: fixes A :: "'c set" shows "\<bottom> \<in> stable A"
simps: fixes f :: "nat \<Rightarrow> complex"   and n :: "nat"   and k :: "nat" shows "power_sum_puzzle_existence.e' f n k = (if k = 0 then 1 else if n < k then 0 else - (\<Sum>i = 1..k. (- 1) ^ i * power_sum_puzzle_existence.e' f n (k - i) * f i) / of_nat k)"
callee_invariant_run_resource: fixes \<I> :: "('a, 'b) \<I>" shows "callee_invariant_on run_resource (WT_resource \<I>) \<I>"
repv_access: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and r :: "real"   and y :: "variable" shows "repv \<omega> x r y = (if x = y then r else \<omega> y)"
mult2_s_one_step: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "ns O s \<subseteq> s"   and "refl ns"   and "trans s" shows "((X, Y) \<in> mult2_s ns s) = (\<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> Y2 \<noteq> {#} \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)))"
smult_smult_assoc: fixes a :: "'a"   and b :: "'a"   and v :: "'a vec" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v v) = a * b \<cdot>\<^sub>v v"
sig_red_tail_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg (\<prec>) F p q" shows "punit.lt (rep_list q) = punit.lt (rep_list p)"
rel_prod_conv: fixes R1 :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
Res: shows "Res = filter (\<lambda>(A, i). A \<noteq> 1) R"
msg_nonempty1: fixes a :: "nat \<Rightarrow> 'd list"   and t :: "nat"   and aa :: "'d"   and l :: "'d list" assumes "msg (Suc 0) a"   and "a t = aa # l" shows "l = []"
kernel_scaleC: fixes a :: "complex"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "a \<noteq> 0" shows "kernel (a *\<^sub>C A) = kernel A"
content_decompose: fixes p :: "'a poly" obtains p' :: "'a poly" where "p = smult (content p) p'" and "content p' = (1::'a)"
ring_mat_def: fixes ty :: "'c itself"   and n :: "nat"   and b :: "'d" shows "ring_mat ty n b \<equiv> \<lparr>carrier = carrier_mat n n, mult = (*), one = 1\<^sub>m n, zero = 0\<^sub>m n n, add = (+), \<dots> = b\<rparr>"
awalk_verts_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "u \<in> verts G"   and "set p \<subseteq> arcs G"   and "v \<in> set (awalk_verts u p)" shows "v \<in> verts G"
IH: fixes j :: "nat" assumes "\<And>i. \<lbrakk>i \<noteq> j; i < length As\<rbrakk> \<Longrightarrow> As ! i = tensor0 ds"   and "dims (As ! j) = ds"   and "j < length As" shows "listsum ds As = As ! j"
g4: shows "\<turnstile> \<box>[F \<longrightarrow> \<circle>F]_v"
Red1FAccNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and F :: "String.literal"   and D :: "String.literal"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<bullet>F{D},s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
inv_is_inverse: fixes f :: "'a" assumes "A.iso f" shows "A.inverse_arrows f (A.inv f)"
at_within_altdef: fixes x :: "'a"   and A :: "'a set" shows "at x within A = (INF S\<in>{S. open S \<and> x \<in> S}. principal (S \<inter> (A - {x})))"
rs_1: shows "set rs = (Rel succs)\<^sup>* `` set ws"
sup_unique: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. x \<le> f x y"   and "\<And>x y. y \<le> f x y"   and "\<And>x y z. \<lbrakk>y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> f y z \<le> x" shows "x \<squnion> y = f x y"
1: shows "x \<sqinter> z = bot \<and> y \<squnion> w = Filters.filter.Abs_filter UNIV \<longrightarrow> phi z \<le> y"
facts_5: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "1 / real (card (uverts H) ^ card (uverts H)) * (real n ^ card (uverts H) * p n ^ card (uedges H)) \<le> prob_space_with_indicators.\<mu> (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))})"
moebius_inv_rotation_rotation: fixes \<phi> :: "real"   and z :: "complex_homo" shows "moebius_pt (moebius_rotation (- \<phi>)) (moebius_pt (moebius_rotation \<phi>) z) = z"
2: shows "gtarget (stake n r) v \<in> gunodes A w"
PM_def: shows "PM \<equiv> (\<forall>x. P x x) \<and> (\<forall>x y z. P x y \<longrightarrow> P y z \<longrightarrow> P x z) \<and> (\<forall>x y. O x y = (\<exists>z. P z x \<and> P z y))"
tObsC_ii_abs_def: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "tObsC_ii_abs t \<equiv> {tLast t' |t'. t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t}"
resl_distr: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<leftarrow> x \<squnion> y = (z \<leftarrow> x) \<sqinter> (z \<leftarrow> y)"
steps_def: fixes s :: "char list \<Rightarrow> int"   and c :: "com"   and n :: "nat" shows "steps s c n = (step {} ^^ n) (step {s} (annotate (\<lambda>p. {}) c))"
rref_condition2_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" assumes "reduced_row_echelon_form A"   and "\<not> is_zero_row i A" shows "A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)"
m_parity_def: fixes x :: "parity" shows "m_parity x = (if x = Either then 0 else 1)"
mono: fixes a :: "'a Word.word"   and c :: "'a Word.word"   and b :: "'a Word.word"   and d :: "'a Word.word" assumes "a \<le>s c"   and "b \<le>s d" shows "signed.min a b \<le>s signed.min c d"
hyps: assumes "i \<le> j"   and "j < min n m" shows "A $$ (i, i) dvd A $$ (j, j)"
abs_eq: fixes xa :: "'c poly"   and x :: "'c poly" shows "xa + x = Abs_poly (\<lambda>n. poly.coeff xa n + poly.coeff x n)"
assms_1: shows "stake (length \<pi>) \<sigma> = \<pi>"
subseq_Cons2': fixes x :: "'d"   and xs :: "'d list"   and ys :: "'d list" assumes "subseq (x # xs) (x # ys)" shows "subseq xs ys"
simps_10: fixes v :: "fm"   and va :: "fm" shows "not (And v va) = NOT (And v va)"
istopology_base_alt: fixes P :: "'c set \<Rightarrow> bool" shows "istopology (arbitrary union_of P) = (\<forall>S T. (arbitrary union_of P) S \<and> (arbitrary union_of P) T \<longrightarrow> (arbitrary union_of P) (S \<inter> T))"
sconjunctive_def: fixes S :: "'a \<Rightarrow> 'b" shows "sconjunctive S = (\<forall>Q. (\<exists>x. x \<in> Q) \<longrightarrow> S (Inf Q) = Inf (S ` Q))"
sum_not_inf_dest: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a + b \<noteq> \<infinity>" shows "a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
connected_ball: fixes x :: "'a"   and e :: "real" shows "connected (ball x e)"
meet_def: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "x \<sqinter>\<^bsub>L\<^esub> y = \<Sqinter>\<^bsub>L\<^esub>{x, y}"
simps: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c llist" assumes "Digraph G" shows "Digraph.valid_path G a = (a = LNil \<or> (\<exists>v. a = LCons v LNil \<and> v \<in> V\<^bsub>G\<^esub>) \<or> (\<exists>v w Ps. a = LCons v Ps \<and> v \<in> V\<^bsub>G\<^esub> \<and> w \<in> V\<^bsub>G\<^esub> \<and> v \<rightarrow>\<^bsub>G\<^esub> w \<and> Digraph.valid_path G Ps \<and> \<not> lnull Ps \<and> lhd Ps = w))"
Rep_cfun_inject: fixes x :: "'a \<rightarrow> 'b"   and y :: "'a \<rightarrow> 'b" shows "(Rep_cfun x = Rep_cfun y) = (x = y)"
size_2: fixes x :: "'b \<Rightarrow> nat"   and x21 :: "'b"   and x22 :: "'b list" shows "size_list x (x21 # x22) = x x21 + size_list x x22 + Suc 0"
that: fixes g :: "'a \<Rightarrow> real" assumes "g \<in> lspace (lebesgue_on S) (ennreal p)"   and "bounded (g ` S)"   and "lnorm (lebesgue_on S) p (f - g) < e" shows "thesis"
finite_image_of_finite2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "finite {f x y |x y. P x y}"
dfeq: shows "(\<lambda>t. dterm_sem I (f1 fid2 vid1) (mk_v I (OVar vid1) (sol 0, b) (sol t))) = (\<lambda>t. sterm_sem I (f1 fid2 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol t))))"
lr_def: shows "(l, r) = split_at (length (x # y # xs) div 2) (x # y # xs)"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_right_move m rs (l, r) = (\<exists>ml mr nl nr rn. l = Bk \<up> nl @ Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Bk \<up> nr @ Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr \<and> 0 < nl + nr)"
assms: fixes t :: "'a" assumes "t \<in> space M" shows "f t = g t"
that: fixes t0 :: "real"   and t1 :: "real" assumes "0 < t0"   and "t0 \<le> t1"   and "t1 \<in> existence_ivl0 x"   and "flow0 x t1 \<in> P"   and "\<And>t. \<lbrakk>0 < t; t < t0\<rbrakk> \<Longrightarrow> flow0 x t \<notin> P" shows "thesis"
notIn_Un_bij_betw: fixes b :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "b \<notin> A"   and "f b \<notin> A'"   and "bij_betw f A A'" shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
normalize_rules_fst: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr list"   and r :: "'b rule"   and rs :: "'b rule list" shows "normalize_rules f (r # rs) = normalize_rules f [r] @ normalize_rules f rs"
T_mtf_competitive: fixes n :: "nat" assumes "s0 \<noteq> []"   and "\<forall>i<n. rs ! i \<in> set s0" shows "real_of_int (T_mtf n) \<le> (2 - 1 / real (length s0)) * real_of_int (T_A n)"
DERIV_isconst3: fixes a :: "real"   and b :: "real"   and x :: "real"   and y :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "x \<in> {a<..<b}"   and "y \<in> {a<..<b}"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> (f has_real_derivative 0) (at x)" shows "f x = f y"
lspasl_starl: fixes Gamma :: "bool"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and Delta :: "bool" assumes "(\<exists>h1 h2. Gamma \<and> (h1,h2\<triangleright>h0) \<and> A h1 \<and> B h2) \<longrightarrow> Delta" shows "Gamma \<and> (A \<and>* B) h0 \<longrightarrow> Delta"
lex_pp_zero_min: fixes s :: "('a, 'b) pp" shows "lex_pp 0 s"
exec_gpv_lift_resumption: fixes r :: "('b, 'call, 'ret) Resumption.resumption"   and s :: "'s" shows "exec_gpv callee (lift_resumption r) s = local.execp_resumption r s"
subset: shows "\<forall>e\<in>lotteries_on out. set_pmf e \<subseteq> out"
KbaseTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<exists>f. KbaseP K P n f"
relcompI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" assumes "r a b"   and "s b c" shows "(r OO s) a c"
ord_p_punit_def: fixes cmp_term :: "'a nat_term_order" shows "ord_p_punit cmp_term \<equiv> ordered_term.ord_p (lt_of_nat_term_order cmp_term)"
R1_SOME_3: fixes f :: "nat \<Rightarrow> nat option"   and r :: "recf"   and x :: "nat" assumes "f \<in> \<R>"   and "r = (SOME r'. recfn 1 r' \<and> Partial_Recursive.total r' \<and> f = (\<lambda>x. eval r' [x]))" shows "f x = eval r [x]"
wf_trm_subst: fixes \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and t :: "('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m t = wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> \<delta>)"
smult_closed: fixes a :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "x \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
points_in_chain: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and Q :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.fin_long_chain betw f x y z Q" shows "x \<in> Q \<and> y \<in> Q \<and> z \<in> Q"
ordered_keys_update_1: fixes k :: "'b"   and m :: "('b, 'c) mapping"   and v :: "'c" assumes "k \<in> keys m" shows "ordered_keys (update k v m) = ordered_keys m"
assms_1: shows "subdivision (G, rev_G) (H, rev_H)"
set_attribute_def: shows "set_attribute \<equiv> l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute"
int_simps_27: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<or> P) = P"
intra_path_to_matching_method_exit: fixes n' :: "'node"   and n :: "'node" assumes "method_exit n'"   and "get_proc n = get_proc n'"   and "valid_node n" obtains as :: "'edge list" where "n -as\<rightarrow>\<^sub>\<iota>* n'"
ctr: fixes a :: "'d" shows "Generative_Probabilistic_Value.Done a = GPV (return_spmf (Pure a))"
maddux_141: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z = 0) = (x\<^sup>\<smile> ; z \<cdot> y = 0)"
mono1: fixes y :: "'c" shows "monotone orda (\<le>) (\<lambda>x. f x y)"
sdom_cons_xx: fixes x :: "nat"   and \<rho> :: "(nat \<times> exp) list" shows "sdom ((x, EVar x) # \<rho>) = sdom \<rho> - {x}"
f_Exec_eq_f_Exec_Stream_last: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun xs c = last (c # f_Exec_Comp_Stream trans_fun xs c)"
monom_mult_monomial: fixes c :: "'c"   and s :: "'b"   and v :: "'b"   and d :: "'c" shows "MPoly_Type_Class.punit.monom_mult c s (monomial d v) = monomial (c * d) (MPoly_Type_Class.punit.splus s v)"
vp_maximal: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "maximal_ideal (Vr K v) (vp K v)"
nths_nths: fixes xs :: "'a list"   and A :: "nat set"   and B :: "nat set" shows "nths (nths xs A) B = nths xs {i \<in> A. \<exists>j\<in>B. card {i' \<in> A. i' < i} = j}"
valid_W_invD_2: fixes r :: "'c"   and fl :: "bool"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and p :: "'d process_name" assumes "mw_Mark r fl \<in> set (sys_mem_store_buffers p s)"   and "valid_W_inv s" shows "fl = sys_fM s \<and> r \<in> ghost_honorary_grey (s p) \<and> sys_mem_lock s = Some p \<and> white r s \<and> filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]"
extf_snoc_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and xs :: "('s, 'v) tm list"   and x :: "('s, 'v) tm" assumes "\<delta>\<^sub>h = \<epsilon>\<^sub>h" shows "extf f gt (xs @ [x]) xs"
less_integer_code_7: fixes k :: "num" shows "(Code_Numeral.Neg k < 0) = True"
Constrains_imp_subset: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> A Co A'" shows "reachable F \<inter> A \<subseteq> A'"
case_cong: fixes sum :: "'a + 'b"   and sum' :: "'a + 'b"   and f1 :: "'a \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "sum = sum'"   and "\<And>x1. sum' = Inl x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x2. sum' = Inr x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case sum of Inl x1 \<Rightarrow> f1 x1 | Inr x2 \<Rightarrow> f2 x2) = (case sum' of Inl x1 \<Rightarrow> g1 x1 | Inr x2 \<Rightarrow> g2 x2)"
HPhase1or2Write_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2c s"   and "HPhase1or2Write s s' p d"   and "Inv2a s" shows "Inv2c s'"
leg1_composite_2: shows "HHH\<mu>\<nu>\<pi>\<rho>.dom.leg1 = \<mu>\<nu>.leg1 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>1"
ap_list_def: fixes fs :: "('d \<Rightarrow> 'c) list"   and xs :: "'d list" shows "Applicative.ap fs xs = List.bind fs (\<lambda>f. List.bind xs (\<lambda>x. [f x]))"
F_def: fixes n :: "nat" shows "F n = \<Inter> (A ` {..n})"