hcju/isabelleps huggingface dataset:
abs_prod: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "abs (comm_monoid_mult.prod times one f A) = comm_monoid_mult.prod times one (\<lambda>x. abs (f x)) A"
only_vertical_division: shows "only_vertical_division \<gamma> two_chain"
children_eq: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
case_split: fixes I :: "'c"   and a :: "'b"   and b :: "'b"   and J :: "'c"   and c :: "'b"   and d :: "'b"   and x :: "'b" assumes "Q I a b"   and "Q J c d" shows "(x = a \<or> x = c \<longrightarrow> P x I J) \<and> (\<not> (x = a \<or> x = b \<or> x = c \<or> x = d) \<longrightarrow> P x I J)"
subst_var_inv_def: fixes \<delta> :: "'a \<Rightarrow> ('c, 'a) Term.term"   and X :: "'a set" shows "subst_var_inv \<delta> X \<equiv> \<lambda>x. if Var x \<in> \<delta> ` X then Var (inv_into X \<delta> (Var x)) else Var x"
Guarantee: fixes f :: "'d"   and F :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'd) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and g :: "'f set"   and P :: "'f set"   and c :: "('f, 'e, 'd) com"   and Q :: "'f set"   and A :: "'f set" assumes "f \<in> F"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
measurable_compose_rev: fixes f :: "'c \<Rightarrow> 'd"   and L :: "'c measure"   and N :: "'d measure"   and g :: "'e \<Rightarrow> 'c"   and M :: "'e measure" assumes "f \<in> L \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M L" shows "(\<lambda>x. f (g x)) \<in> M \<rightarrow>\<^sub>M N"
positive_resolvent_is_resolvent: shows "less_restrictive resolvent ordered_positive_resolvent"
u_def: shows "u = prio_emb_step prio (prio_emb_step prio t s) s"
dom_comp: fixes g :: "'b"   and f :: "'b" assumes "C'.seq g f" shows "C'.dom (C' g f) = C'.dom f"
prems_3: shows "P \<turnstile> \<langle>blocks (p # ps', Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"
Cn_imp: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(imp_PL p q \<in> Cn A) = (Cn {q} \<subseteq> Cn (A \<union> {p}))"
eval_list_conj: fixes F :: "atom fm list"   and L :: "real list" shows "eval (list_conj F) L = (\<forall>f\<in>set F. eval f L)"
distinct_40: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> x41 ;; x42"
sum_update_rightmost: fixes i :: "nat"   and j :: "nat"   and as :: "'b list"   and x :: "'b" assumes "0 < i"   and "i \<le> j"   and "length as = j" shows "SWA.sum as i j + x = SWA.sum (butlast as @ [last as + x]) i j"
RFC_4291_format: fixes as :: "16 word option list" shows "(parse_ipv6_address_compressed as \<noteq> None) = (length (filter (\<lambda>p. p = None) as) = 1 \<and> length (filter (\<lambda>p. p \<noteq> None) as) \<le> 7)"
case_cong: fixes bool :: "bool"   and bool' :: "bool"   and f1 :: "'a"   and g1 :: "'a"   and f2 :: "'a"   and g2 :: "'a" assumes "bool = bool'"   and "bool' = True \<Longrightarrow> f1 = g1"   and "bool' = False \<Longrightarrow> f2 = g2" shows "(case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (case bool' of True \<Rightarrow> g1 | False \<Rightarrow> g2)"
not_symKeys_privateKey: fixes b :: "bool"   and A :: "agent" shows "invKey (publicKey b A) \<notin> symKeys"
div_nonpos_pos_le0: fixes a :: "int"   and b :: "int" assumes "a \<le> 0"   and "0 < b" shows "a div b \<le> 0"
equivalence_comp_left_complement: fixes x :: "'a" assumes "(top \<le> x \<and> x \<sqinter> x \<le> x) \<and> local.comp_inf.symmetric x" shows "x \<sqinter> - x = - x"
simps: fixes n :: "nat"   and m :: "nat"   and c :: "int"   and bs :: "int list" shows "bin_rsplitl_aux n m c bs = (if m = 0 \<or> n = 0 then bs else let (a, b) = bin_split (min m n) c in bin_rsplitl_aux n (m - n) a (b # bs))"
assms_4: fixes c' :: "'a set" assumes "x \<in> c'"   and "c' \<subseteq> S"   and "path_connected c'" shows "c' \<subseteq> c"
rep_ocircline_def: shows "rep_ocircline \<equiv> quot_type.rep Rep_ocircline"
one_reg_lan_def: shows "1 \<equiv> Abs_reg_lan 1"
LeA_def: 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"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F \<equiv> \<exists>P. Tarski_neutral_dimensionless.InAngle Bet P D E F \<and> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E P"
less_irrefl_TC: fixes x :: "V" shows "\<not> x \<sqsubset> x"
UnOp: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and unop :: "unop" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>eval_unop unop v\<midarrow>n\<rightarrow> s1"
moebius_similarity_uminus: fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> 0" shows "- moebius_similarity a b = moebius_similarity (1 / a) (- b / a)"
simps_10: fixes var :: "nat"   and uv :: "atom fm" shows "grab_atoms var (ExQ uv) = None"
pair_in_Id_conv: fixes a :: "'h"   and b :: "'h" shows "((a, b) \<in> Id) = (a = b)"
lr_def: shows "(l, r) = split_at (n div 2) xs"
ideal_zero: fixes A :: "nat set"   and add :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and mult :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "ideal A {0} add mult 0 0" shows "0 \<in> A"
assms_2: shows "b \<sqsubseteq> a \<cdot> \<theta>"
R0: fixes t :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat" shows "t \<in> gcsort_set index key p t"
that: fixes a :: "'edge" assumes "valid_edge a"   and "inner_node (targetnode a)"   and "sourcenode a = (_Entry_)" shows "thesis"
not_numeral_le_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq (numeral.numeral one plus n) zero"
h1: shows "((\<Sum>i = 0..<n + 1. f i / B ^ i) = 0) = ((\<Sum>i = 0..<n + 1. f i / B ^ i) * B ^ n = 0)"
simps_24: fixes va :: "'b formula" shows "safe_formula (Neg (Exists va)) = False"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "type_wf = DocumentClass.type_wf"   and "set_tag_name = l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name"   and "set_tag_name_locs = l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name_locs" shows "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs"
Mp_0_smult_sdiv_poly: fixes f :: "int poly" assumes "Mp f = 0" shows "smult p (sdiv_poly f p) = f"
lnths_upt_eq_ltake: fixes xs :: "'a llist"   and n :: "nat" shows "lnths xs {..<n} = ltake (enat n) xs"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int" assumes "\<And>x n. (\<lbrakk>\<not> (x < 0 \<or> n < 0); \<not> x ^ p \<le> n\<rbrakk> \<Longrightarrow> P ((n div x ^ pm + x * int pm) div int p) n) \<Longrightarrow> P x n" shows "P a0 a1"
card_subformulae: shows "card (set (subformulae F)) \<le> size F"
list_induct2': fixes P :: "'e list \<Rightarrow> 'f list \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list" assumes "P [] []"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
simps_4: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p X\<^sub>p \<phi> = w [1..] \<Turnstile>\<^sub>p \<phi>"
nat_of_ord_succ: fixes x :: "hf" assumes "Ord x" shows "nat_of_ord (succ x) = Suc (nat_of_ord x)"
rel_cong: fixes x :: "'e llist"   and ya :: "'e llist"   and y :: "'f llist"   and xa :: "'f llist"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Ra :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> lset ya; yb \<in> lset xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "llist_all2 R x y = llist_all2 Ra ya xa"
\<phi>_local_bij: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "bij_betw (S.UP \<circ> inl) (C.hom b a) (Hom.set (b, a))"
CptnEnv: fixes P :: "'b com option"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list"   and s :: "'b" assumes "(P, t) # xs \<in> cptn" shows "(P, s) # (P, t) # xs \<in> cptn"
scene_compat_refl: fixes X :: "'a scene" shows "X ##\<^sub>S X"
arrI: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and a :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and b :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" shows "S.arr f"
index_mat_multrow_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul k a A $$ (i, j) = (if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
listrel1_subset_listrel: fixes r :: "('g \<times> 'g) set"   and r' :: "('g \<times> 'g) set" assumes "r \<subseteq> r'"   and "refl r'" shows "listrel1 r \<subseteq> listrel r'"
exhaust: fixes y :: "fm" obtains x11 :: "tm" and x12 :: "tm" where "y = x11 IN x12"   | x21 :: "tm" and x22 :: "tm" where "y = x21 EQ x22"   | x31 :: "fm" and x32 :: "fm" where "y = x31 OR x32"   | x4 :: "fm" where "y = Neg x4"   | x51 :: "name" and x52 :: "fm" where "y = SyntaxN.Ex x51 x52"
reflpD: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "reflp r" shows "r x x"
multiplicity_eq_zero_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and x :: "'c"   and p :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "(factorial_semiring.multiplicity times one p x = 0) = (\<not> dvd.dvd times p x)"
set_child_nodes_is_l_set_child_nodes: shows "l_set_child_nodes DocumentClass.type_wf set_child_nodes set_child_nodes_locs"
closedin_Un_complement_component: fixes u :: "'a set"   and S :: "'a set"   and c :: "'a set" assumes "locally connected u"   and "closedin (top_of_set u) S"   and "c \<in> components (u - S)" shows "closedin (top_of_set u) (S \<union> c)"
Quotient_rel_rep: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep b) = (a = b)"
Inv_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Inv (Inv t) = t"
sum_nonneg_leq_bound: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and B :: "'b multiset"   and i :: "'a" assumes "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> {#} \<subseteq># f i"   and "comm_monoid_add.sum (+) {#} f s = B"   and "i \<in> s" shows "f i \<subseteq># B"
transcendental_consistent: fixes K :: "'a set" assumes "subring K R" shows "transcendental = ring.transcendental (R\<lparr>carrier := K\<rparr>)"
coms_exists_inv: fixes x :: "char list"   and G :: "diagram"   and c :: "command" assumes "coms_ass (Exists_dia x G) c" obtains "coms_dia G c"
member_of_is_classD: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "is_class G C"
select_convs_1: fixes CatDom :: "('a, 'b, 'c) Category_scheme"   and CatCod :: "('d, 'e, 'f) Category_scheme"   and MapM :: "'b \<Rightarrow> 'e"   and more :: "'g" shows "CatDom \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatDom"
order_iff_strict: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># b) = (a \<subset># b \<or> a = b)"
ss0_lt_ss: shows "ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ss"
simps_1: fixes r :: "proc \<times> val" shows "\<bottom> \<bullet> r = Some (snd r)"
distinct_disc_14: fixes event :: "('a, 'b, 'c) event" assumes "isRecvMarker event" shows "\<not> isSend event"
pred_pmf_code: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a pmf" shows "pred_pmf P p = (\<forall>x\<in>set_pmf p. P x)"
conv_radius_nonneg: fixes f :: "nat \<Rightarrow> 'b" shows "0 \<le> conv_radius f"
sqrt_int_ceiling_code: fixes x :: "int" shows "sqrt_int_ceiling x = (if 0 \<le> x then sqrt_int_ceiling_pos x else - sqrt_int_floor_pos (- x))"
tensor\<^sub>F\<^sub>M\<^sub>C_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g \<equiv> if partial_magma.arr (free_monoidal_category.comp C) f \<and> partial_magma.arr (free_monoidal_category.comp C) g then free_monoidal_category.mkarr C (free_monoidal_category.rep C f \<^bold>\<otimes> free_monoidal_category.rep C g) else partial_magma.null (free_monoidal_category.comp C)"
assms_2: assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "P"
fair_run_is_run: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep"   and "df_system.is_fair_run s\<^sub>0 lstep s" shows "df_system.is_run s\<^sub>0 lstep s"
select_convs_2: fixes Field1 :: "nat message_af"   and Field2 :: "int message_af"   and Field3 :: "int message_af"   and more :: "'b" shows "Field2 \<lparr>Field1 = Field1, Field2 = Field2, Field3 = Field3, \<dots> = more\<rparr> = Field2"
Abs_cl_op_im_inverse: fixes y :: "'b" assumes "y \<in> range cl_op" shows "Rep_cl_op_im (Abs_cl_op_im y) = y"
exp_scaleR_has_vector_derivative_left: fixes A :: "'a"   and t :: "real" shows "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative A * exp (t *\<^sub>R A)) (at t)"
coerce_functor: shows "COERCE('d\<cdot>'e, 'f\<cdot>'e) = fmap\<cdot>COERCE('d, 'f)"
bspec: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
max2_equals2: fixes a :: "'b"   and b :: "'b" assumes "a \<in> Field s"   and "b \<in> Field s" shows "(s.max2 a b = b) = ((a, b) \<in> s)"
simps: fixes s :: "atom_sort"   and n :: "nat" shows "nat_of (Atom s n) = n"
additive: shows "Measure_Space.additive (Pow A) (\<lambda>a. if finite a then of_nat (card a) else \<infinity>)"
resolute_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "resolute_on A f = (\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> (\<exists>a. f B = {a}))"
mpoly_Const_add: fixes a :: "'b"   and b :: "'b" shows "Const (a + b) = Const a + Const b"
regular_closed_pp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_p_algebra.regular (\<ominus> \<ominus> x)"
abs_eq_iff: fixes x :: "'a llist \<times> 'b"   and y :: "'a llist \<times> 'b" shows "((case x of (xs, a) \<Rightarrow> tllist_of_llist a xs) = (case y of (xs, a) \<Rightarrow> tllist_of_llist a xs)) = (case x of (xs, a) \<Rightarrow> \<lambda>(ys, b). xs = ys \<and> (lfinite ys \<longrightarrow> a = b)) y"
runit'_naturality: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "(\<mu> \<star>\<^sub>B src\<^sub>B \<mu>) \<cdot>\<^sub>B B.runit' (B.dom \<mu>) = B.runit' (B.cod \<mu>) \<cdot>\<^sub>B \<mu>"
m1eq: fixes i :: "nat" assumes "i < state_sig.d b.dims0" shows "digit_encode a.dims0 (digit_decode b.dims1 (nths (digit_encode b.dims0 i) b.vars1')) = nths (digit_encode b.dims0 i) b.vars1'"
perm_restrict_def: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" shows "perm_restrict f S x \<equiv> if x \<in> S then f x else x"
simps_2: fixes m :: "nat"   and n :: "nat" shows "Suc m * n = n + m * n"
unx_some_lemmas_3: fixes u :: "'a"   and y :: "name" assumes "unconst u = Some y" shows "unapp u = None"
equivalence_map_def: fixes f :: "'c" shows "C.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. C.internal_equivalence f g \<eta> \<epsilon>"
perfect_hiding: fixes \<A> :: "(('g \<times> 'a) \<times> ('h + 'b) \<Rightarrow> (('e \<times> 'e) \<times> 'm) spmf) \<times> ('j \<times> 'd \<Rightarrow> 'm \<Rightarrow> bool spmf)" shows "OR_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<A>"
Op_castOp: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and s :: "'e"   and X :: "('b, 'c, 'd, 'a, 'f) qTerm set" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.isOp Delta s X" shows "FixVars.Op (fst3 (FixSyn.castOp Delta s X)) (snd3 (FixSyn.castOp Delta s X)) (trd3 (FixSyn.castOp Delta s X)) = X"
assms: shows "s' = snd (memory_read asi addr s) \<and> get_S (cpu_reg_val PSR s) = 0"
assms_2: shows "(s1, s2) \<in> set (half_segments_of_aform X)"
inverse_arrows_sym: fixes f :: "'s"   and g :: "'s" shows "S.inverse_arrows f g = S.inverse_arrows g f"
sgn_1: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'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" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn one = one"
square_free_m_cong: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.square_free_m m f"   and "poly_mod.Mp m f = poly_mod.Mp m g" shows "poly_mod.square_free_m m g"
stern_brocot_Hinze_iterate_def: shows "stern_brocot_Hinze_iterate = map_tree mediant (tree_iterate (\<lambda>s. s \<otimes> UR) (\<lambda>s. s \<otimes> LL) F)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_cons: fixes x :: "'a \<times> ('b, 'c) stateful_strand_step"   and S :: "('a \<times> ('b, 'c) stateful_strand_step) list" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (x # S) = setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p x \<union> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t S"
full_multiset_induct: fixes P :: "'c multiset \<Rightarrow> bool"   and B :: "'c multiset" assumes "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" shows "P B"
subset_code_1: fixes xs :: "'a list"   and B :: "'a set" shows "(list.set xs \<subseteq> B) = (\<forall>x\<in>list.set xs. x \<in> B)"
composable: shows "src \<mu> = trg (\<nu> \<star> \<pi>)"
iffalse: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and \<iota> :: "nat"   and c1 :: "('b, 'c) MWLsCom"   and c2 :: "('b, 'c) MWLsCom" assumes "BMap (E b m) = False" shows "((if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi, m), [], Some c2, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
assms: shows "Abs x s \<rightarrow>\<^sub>\<beta>\<^sup>* t"
n_dist_var1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<oplus> !y) \<cdot> (t x \<oplus> !z) = t x \<cdot> !y \<oplus> !x \<cdot> !z"
jkbpC_prefix_closed: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState" assumes "t \<leadsto> s \<in> SPR.jkbpC" shows "t \<in> SPR.jkbpC"
ZF2Expr_def: shows "ZF2Expr = inv Expr2ZF"
inf_top_right: fixes x :: "'a" shows "x \<sqinter> top = x"
of_nat_diff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
outputPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "(P, Q) \<in> Rel" shows "a{b}.P \<leadsto>\<guillemotleft>Rel\<guillemotright> a{b}.Q"
rel_ad_def: fixes R :: "('a \<times> 'a) set" shows "rel_ad R = {(x, x) |x. \<nexists>y. (x, y) \<in> R}"
assms: shows "\<forall>i\<ge>n. \<exists>j>i. P (S i) (S j)"
intro: fixes M :: "'c measure"   and F :: "'c measure" assumes "finite_measure M"   and "finite_measure_subalgebra_axioms M F" shows "finite_measure_subalgebra M F"
fbrelation_axioms: fixes a :: "V"   and b :: "V" shows "fbrelation (set {[a, b]\<^sub>\<circ>})"
second_safe_dist_expr_def: fixes ve :: "nat"   and ae :: "nat"   and vo :: "nat"   and ao :: "nat" shows "second_safe_dist_expr ve ae vo ao = Add (rel_dist_to_stop_expr ve ae) (Minus (rel_dist_to_stop_expr vo ao))"
vectorspace: fixes U :: "'c set"   and V :: "'c set" assumes "Subspace.subspace U V"   and "vectorspace V" shows "vectorspace U"
stmt_inj_term: fixes x :: "stmt"   and y :: "stmt" shows "\<langle>x\<rangle> = \<langle>y\<rangle> \<equiv> x = y"
Chernoff_on_consistency_on_path_independent_on: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "f_range_on A f" shows "(Chernoff_on A f \<and> consistency_on A f) = path_independent_on A f"
AKcryptSK_analz_insert: fixes K :: "nat"   and K' :: "nat"   and evs :: "event list" assumes "AKcryptSK K K' evs"   and "K \<in> symKeys"   and "evs \<in> kerbV" shows "Key K' \<in> analz (insert (Key K) (knows Spy evs))"
Cnext_solve_implies: fixes K\<^sub>1 :: "clock"   and n :: "nat"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and K\<^sub>2 :: "clock"   and \<Phi> :: "'c TESL_atomic list" shows "{K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>, K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"