hcju/isabelleps huggingface dataset:
module_hom_zero: shows "Vector_Spaces.linear (*s) (*s) (\<lambda>x. 0)"
sep_conjI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and Q :: "'a \<Rightarrow> bool"   and y :: "'a"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "P x"   and "Q y"   and "sep_disj x y"   and "h = plus x y" shows "sep_algebra.sep_conj plus sep_disj P Q h"
bl1_finalVal: fixes h :: "'heap"   and t :: "'thread_id"   and v :: "'addr val"   and xs :: "'addr val list" assumes "hconf h"   and "preallocated h" shows "bisim1_list1 t h (Val v, xs) [] None []"
between_inverse: fixes p :: "hyp2"   and v :: "(real, 2) vec"   and q :: "hyp2" assumes "B\<^sub>\<real> hyp2_rep p v hyp2_rep q" shows "hyp2_rep (hyp2_abs v) = v"
bounded_by_bound_of: fixes P :: "'a \<Rightarrow> real" assumes "Expectations.bounded P" shows "bounded_by (bound_of P) P"
chine_assoc_props_4: shows "fgh.Prj\<^sub>0\<^sub>0 \<cdot> fgh.chine_assoc = fgh.Prj\<^sub>0"
comm_diagrams: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e"   and U :: "'c set"   and V :: "'c set"   and x :: "'d" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U"   and "is_open V"   and "V \<subseteq> U"   and "x \<in> \<FF> U" shows "(\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x"
INF_shift: fixes fa :: "nat \<Rightarrow> ennreal"   and j :: "nat" assumes "decseq fa" shows "Inf (range fa) = (INF i. fa (i + j))"
assms_3: shows "DIM('b) \<le> DIM('a)"
lookup'': shows "\<forall>tp'\<in>Mapping.keys a2_map''. case Mapping.lookup a2_map'' tp' of Some m \<Rightarrow> table n R (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I))"
correctCompositionKeys_def: fixes x :: "specID" shows "correctCompositionKeys x \<equiv> subcomponents x \<noteq> {} \<longrightarrow> specKeys x = \<Union> (specKeys ` subcomponents x)"
in_hhomI: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "D.arr \<mu>"   and "src\<^sub>D \<mu> = a"   and "trg\<^sub>D \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"
decr_grading_p_plus: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "decr_grading_p d n (p + q) = decr_grading_p d n p + decr_grading_p d n q"
transfer_WHILEIT_esc: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'm"   and b :: "'a \<Rightarrow> bool"   and x :: "'a"   and I :: "'a \<Rightarrow> bool" assumes "\<And>x. return (f x) \<le> F x" shows "return (while b f x) \<le> WHILEIT I b F x"
insert_mem_set_rel_iff: fixes A :: "('a \<times> 'b) set"   and x :: "'a"   and xs :: "'a list"   and XXS :: "'b set" assumes "single_valued A" shows "((insert x (set xs), XXS) \<in> \<langle>A\<rangle>set_rel) = (\<exists>X XS. (x, X) \<in> A \<and> (set xs, XS) \<in> \<langle>A\<rangle>set_rel \<and> XXS = insert X XS)"
a0: shows "i < dim_row (Matrix.mat (2 ^ Suc n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ Suc n)))"
assms_1: shows "LeftDerivationLadder \<alpha> D L \<beta>"
classic_Local: fixes G :: "'b triple set"   and P :: "'b assn"   and Y :: "loc"   and a :: "state \<Rightarrow> nat"   and c :: "com"   and Q :: "'b assn" assumes "\<forall>v. G|-{\<lambda>Z s. P Z (s[Loc Y::=v]) \<and> s<Y> = a (s[Loc Y::=v])}. c .{\<lambda>Z s. Q Z (s[Loc Y::=v])}" shows "G|-{P}. LOCAL Y:=a IN c .{Q}"
simps_1: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and P :: "'e \<Rightarrow> bool"   and U :: "'f set" shows "ipurge_tr_aux_foldr I D P U [] = []"
Suc_1: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "I < n" shows "det (swap_row_to_front A I) = (- (1::'a)) ^ I * det A"
shuffle_to_top_def: fixes Y :: "'x set" shows "shuffle_to_top Y = (\<lambda>d. Pd d - dX Y d \<times> UNIV \<union> (Domain (Pd d) \<union> dX Y d) \<times> dX Y d)"
integrable_on_const: fixes S :: "'c set"   and c :: "'d" assumes "S \<in> lmeasurable" shows "(\<lambda>x. c) integrable_on S"
finite_inf_Sup: fixes a :: "'a"   and A :: "'a set" shows "a \<sqinter> \<Squnion> A = \<Squnion> {a \<sqinter> b |b. b \<in> A}"
Gromov_converging_at_boundary_def: fixes u :: "nat \<Rightarrow> 'a" shows "Gromov_converging_at_boundary u = (\<forall>a M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n))"
external_to_internal_invariant_vimage: fixes S :: "('b, 'a) spec"   and Q :: "'a set"   and P :: "'b set" assumes "oreach S \<subseteq> Q"   and "obs S -` Q \<subseteq> P" shows "reach S \<subseteq> P"
dH_equal_def: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and H :: "('e \<times> 'a) set"   and m :: "'b \<Rightarrow> 'c"   and m' :: "'b \<Rightarrow> 'c" shows "WHATWHERE.dH_equal E DA d H m m' \<equiv> WHATWHERE.d_equal DA d m m' \<and> (\<forall>(d', e)\<in>H. d' \<le> d \<longrightarrow> E e m = E e m')"
le_prod_def: fixes p1 :: "'c \<times> 'd"   and p2 :: "'c \<times> 'd" shows "(p1 \<sqsubseteq> p2) = (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
map_injective: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'h list" assumes "map f xs = map f ys"   and "inj f" shows "xs = ys"
C11_var: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> (y \<cdot> x\<^sup>\<star>)\<^sup>\<star>"
order_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "x \<le>p y"   and "y \<le>p z" shows "x \<le>p z"
param_nth: fixes i' :: "nat"   and l' :: "'a list"   and i :: "nat"   and l :: "'b list"   and R :: "('b \<times> 'a) set" assumes "i' < length l'"   and "(i, i') \<in> nat_rel"   and "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "(l ! i, l' ! i') \<in> R"
Class_eq: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). x \<sim> y}" shows "rel.Class a = rel.Class b"
d_def: shows "d = b' - (if a = - \<infinity> then b' - 1 else real_of_ereal a)"
Overriding: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and accclass :: "qtname"   and C :: "qtname"   and new :: "mdecl"   and old :: "qtname \<times> mdecl"   and supr :: "qtname" assumes "G \<turnstile> membr member_of class"   and "G \<turnstile> Class class accessible_in pid accclass"   and "membr = (C, mdecl new)"   and "G \<turnstile> (C, new) overrides\<^sub>S old"   and "G\<turnstile>class\<prec>\<^sub>C supr"   and "G \<turnstile>Method old of supr accessible_from accclass" shows "G \<turnstile> membr of class accessible_from accclass"
MClkReplyS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $ImpInv rmhist p \<and> <MClkReply memCh crCh cst p>_(c p) \<longrightarrow> $S6 rmhist p"
bl1_Normal: fixes t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'addr frame list"   and Ts :: "ty list"   and T :: "ty"   and body :: "'addr expr1"   and D :: "String.literal"   and e :: "'addr expr1"   and xs :: "'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list" assumes "compTP P \<turnstile> t:(xcp, h, (stk, loc, C, M, pc) # frs) \<surd>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D"   and "P,blocks1 0 (Class D # Ts) body,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "max_vars e \<le> length xs"   and "list_all2 (bisim1_fr P h) exs frs" shows "bisim1_list1 t h (e, xs) exs xcp ((stk, loc, C, M, pc) # frs)"
scale_sum_left: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'c" shows "sum f A * x = (\<Sum>a\<in>A. f a * x)"
conversep_MP_Rel_OO_MP_Rel: shows "MP_Rel\<inverse>\<inverse> OO MP_Rel = (=)"
is_subset_MST_empty: fixes g :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "Undirected_Graph.connected g" shows "is_subset_MST w g {}"
add_neg_numeral_special_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus (numeral.numeral one plus m)) = neg_numeral.sub minus plus one num.One m"
f1: shows "|v\<rangle> \<Otimes> |v\<rangle>\<^sup>\<dagger> * (|w\<rangle> \<Otimes> |w\<rangle>) = (\<langle>|v\<rangle>| \<Otimes> \<langle>|s\<rangle>|) * (|w\<rangle> \<Otimes> |s\<rangle>)"
length_rev_upt: fixes i :: "nat"   and j :: "nat" shows "length [i>..j] = i - j"
eq': shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then \<Sum>n. f (p ^ n) else (1::'a)) = (\<lambda>n. \<Prod>p\<le>n. if prime p then inverse ((1::'a) - f p) else (1::'a))"
iteratei_abs_simple_rule_P: assumes "set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"   and "\<And>xc. \<lbrakk>invar xc; \<alpha> xc \<in> S0\<rbrakk> \<Longrightarrow> f xc = f' (\<alpha> xc)"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. True; \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> I (S - {x}) (f' x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
nat_mult_1_right: fixes n :: "nat" shows "n \<cdot> 1 = n"
Suc_3: fixes x :: "'a" assumes "x \<in> S" shows "f x = g x"
annil: fixes x :: "'a" shows "(0::'a) \<parallel> x = (0::'a)"
skl1_inv5I: fixes x :: "skl1_state" assumes "\<And>gny A B Ra Nr. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}; sklvl1.guessed_frame Ra xgny = Some gny; sklvl1.guessed_frame Ra xnr = Some Nr; can_signal x A B\<rbrakk> \<Longrightarrow> 0 < signalsResp x (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)" shows "x \<in> skl1_inv5"
transpose_infmatrix: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "transpose_infmatrix P = (\<lambda>j i. P i j)"
find_sub_in_sound: fixes X :: "'b set"   and xs :: "'b set list"   and b :: "bool"   and ys :: "'b set list"   and z :: "'b set"   and zs :: "'b set list" assumes "find_sub_in X xs b = Some (ys, z, zs)" shows "xs = ys @ z # zs \<and> (z \<subseteq> X \<or> b \<and> X \<subseteq> z)"
division: fixes d :: "'a set set"   and a :: "'a"   and b :: "'a" assumes "d division_of cbox a b" shows "sum content d = content (cbox a b)"
assms: shows "P\<^sub>L \<turnstile>\<^sub>L \<phi>"
WRP_Succ: fixes i :: "tm"   and y :: "tm" shows "{OrdP i, WRP i y} \<turnstile> WRP (SUCC i) (Q_Succ y)"
multiplicative_function'_def: fixes f :: "nat \<Rightarrow> 'b"   and f_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and f_prime :: "nat \<Rightarrow> 'b" shows "multiplicative_function' f f_prime_power f_prime \<equiv> multiplicative_function f \<and> multiplicative_function'_axioms f f_prime_power f_prime"
right_eq: shows "{p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (y, b)} = (\<lambda>y. twoC (y, 1)) ` {0..1}"
representation_eqI: fixes basis :: "'b set"   and v :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "independent basis"   and "v \<in> span basis"   and "\<And>b. f b \<noteq> (0::'a) \<Longrightarrow> b \<in> basis"   and "finite {b. f b \<noteq> (0::'a)}"   and "(\<Sum>b | f b \<noteq> (0::'a). f b *s b) = v" shows "representation basis v = f"
fresh_atm_2: fixes a :: "coname"   and b :: "coname" shows "a \<sharp> b = (a \<noteq> b)"
assms: shows "\<forall>(k, v)\<in>set xs. k \<in> process_names ss proc \<and> variable_inv v"
linear: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<or> y \<le> x"
offs_next_def: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" shows "offs_next ns ub xs index key mi ma i \<equiv> if offs_set_next ns xs index key mi ma i = {} then ub else ns ! Min (offs_set_next ns xs index key mi ma i)"
supdistl_iso: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup \<circ> (`) f \<le> f \<circ> Sup) = order.mono less_eq f"
constant_image_pos: fixes f :: "'a \<Rightarrow> real" assumes "\<forall>x\<in>space M. 0 < f x"   and "\<exists>c. \<forall>x\<in>space M. f x = c" shows "0 < constant_image f"
assms_4: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<noteq> (0::'rows)"
IH: shows "f 0 = x \<and> f (Suc n) = y \<and> (\<forall>i\<le>n. P (f i) (f (Suc i)))"
z1'_def: shows "z1' = of_nat (fst z1) + \<i>\<^sub>\<int> * of_nat (snd z1)"
GreatestI_ex: fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (Greatest P)"
adj_sur_inv: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "f \<stileturn> g" shows "top_pres (\<P> f) = (f \<circ> g = id)"
inject: fixes x21 :: "'k"   and x22 :: "'k list"   and y21 :: "'k"   and y22 :: "'k list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
enumTab_def: shows "enumTab \<equiv> tabulate2 9 9 enumerator"
inv_cancelR_2: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "y = x \<otimes>\<^bsub>R\<^esub> z" shows "y \<otimes>\<^bsub>R\<^esub> inv\<^bsub>R\<^esub> x = z"
\<pi>_def: shows "\<pi> = (\<lambda>(a, bs). (a, tl bs))"
R_consts_3: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n true\<^sub>n R\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>"
memory_read_def: fixes asi :: "word8"   and addr :: "word32"   and state :: "'b sparc_state" shows "memory_read asi addr state \<equiv> let asi_int = uint asi in if asi_int = 1 then let r1 = load_word_mem state addr (word_of_int 8) in if r1 = None then let r2 = load_word_mem state addr (word_of_int 10) in if r2 = None then (None, state) else (r2, state) else (r1, state) else if asi_int = 2 then if uint addr = 0 then (Some (sys_reg_val CCR state), state) else if uint addr = 8 then (Some (sys_reg_val ICCR state), state) else if uint addr = 12 then (Some (sys_reg_val DCCR state), state) else (None, state) else if asi_int \<in> {8, 9} then let ccr_val = sys_reg state CCR in if ccr_val AND 1 \<noteq> 0 then let data = load_word_mem state addr asi in case data of None \<Rightarrow> (None, state) | Some w \<Rightarrow> (Some w, add_instr_cache state addr w 15) else (load_word_mem state addr asi, state) else if asi_int \<in> {10, 11} then let ccr_val = sys_reg state CCR in if ccr_val AND 1 \<noteq> 0 then let data = load_word_mem state addr asi in case data of None \<Rightarrow> (None, state) | Some w \<Rightarrow> (Some w, add_data_cache state addr w 15) else (load_word_mem state addr asi, state) else if asi_int = 13 then let cache_result = read_instr_cache state addr in case cache_result of None \<Rightarrow> (None, state) | Some w \<Rightarrow> (Some w, state) else if asi_int = 15 then let cache_result = read_data_cache state addr in case cache_result of None \<Rightarrow> (None, state) | Some w \<Rightarrow> (Some w, state) else if asi_int \<in> {16, 17} then (None, state) else if asi_int \<in> {20, 21} then (None, state) else if asi_int = 24 then (None, state) else if asi_int = 25 then (mmu_reg_val (mmu state) addr, state) else if asi_int = 28 then (mem_val_w32 asi (ucast addr) state, state) else if asi_int = 29 then (None, state) else (None, state)"
restrictPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "r a b"   and "a \<in> A"   and "b \<in> A" shows "(r |` A) a b"
abs_low_mds_eq_dma\<^sub>C_eq: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and mds :: "Mode \<Rightarrow> 'e set"   and mem\<^sub>1\<^sub>C :: "'e \<Rightarrow> 'd"   and mem\<^sub>2\<^sub>C :: "'e \<Rightarrow> 'd" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "sifum_security_init.low_mds_eq dma\<^sub>A \<C>_vars\<^sub>A (sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds) (sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>1\<^sub>C) (sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>2\<^sub>C)" shows "dma\<^sub>C mem\<^sub>1\<^sub>C = dma\<^sub>C mem\<^sub>2\<^sub>C"
ideD_3: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.cod a = a"
assms_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> 0 \<le> y"
infsetsum_cong_neutral: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'b)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'b)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "infsetsum f A = infsetsum g B"
mset_nempty_set_nempty: fixes A :: "'a multiset" shows "(A \<noteq> {#}) = (set_mset A \<noteq> {})"
height\<^sub>F_0_imp_Leaf_freq\<^sub>F_in_set: fixes ts :: "'a tree list"   and a :: "'a" assumes "consistent\<^sub>F ts"   and "height\<^sub>F ts = 0"   and "a \<in> alphabet\<^sub>F ts" shows "Leaf (freq\<^sub>F ts a) a \<in> set ts"
little_Fermat_int: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "\<not> int p dvd a" shows "[a ^ p = a] (mod int p)"
\<V>: shows "\<V> \<subseteq> Collect (finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (Y i) U) relative_to topspace (product_topology Y I))"
hextI: fixes h :: "'heap"   and h' :: "'heap" assumes "\<And>a C. typeof_addr h a = \<lfloor>Class_type C\<rfloor> \<Longrightarrow> typeof_addr h' a = \<lfloor>Class_type C\<rfloor>"   and "\<And>a T n. typeof_addr h a = \<lfloor>Array_type T n\<rfloor> \<Longrightarrow> typeof_addr h' a = \<lfloor>Array_type T n\<rfloor>" shows "h \<unlhd> h'"
indVfbasis_def: shows "indVfbasis \<equiv> negHorbit_list H_rcoset_reps induced_vector Vfbasis"
cotrace_trace_cos: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event llist"   and i :: "nat" assumes "cotrace c t" shows "trace c (Co_Snapshot.ltake i t) (local.cos c t i)"
no_car: shows "\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot"
that: fixes r :: "'a \<Rightarrow> 'a" assumes "homotopic_with_canon (\<lambda>x. True) S S id r"   and "retraction S {a} r" shows "thesis"
bind_alt1: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'e"   and m' :: "'e"   and f :: "'d \<Rightarrow> 'e" assumes "monad_alt return bind alt" shows "bind (alt m m') f = alt (bind m f) (bind m' f)"
SpecStuck: fixes s :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> (Spec r # cs, css, Normal s) \<rightarrow> (cs, css, Stuck)"
pair_spmf_return_spmf: fixes x :: "'a"   and y :: "'b" shows "pair_spmf (return_spmf x) (return_spmf y) = return_spmf (x, y)"
AddSA_CompFun_PseudoHA_ran2: fixes SA1 :: "('d, 'e, 'f) seqauto"   and RootSA :: "('d, 'e, 'f) seqauto"   and SA2 :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and D :: "'f data" assumes "States SA1 \<inter> States RootSA = {}"   and "States SA2 \<inter> (States RootSA \<union> States SA1) = {}"   and "S \<in> States RootSA" shows "ran (CompFun (PseudoHA RootSA D [++] (S, SA1) [++] (S, SA2))) = {{}, {SA2, SA1}}"
repeat_mset_cancel2: fixes a :: "nat"   and A :: "'b multiset"   and b :: "nat" shows "(repeat_mset a A = repeat_mset b A) = (a = b \<or> A = {#})"
h: shows "h \<in> {v. [v ^ CARD('a) = v] (mod f)}"
obs_singleton: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and n :: "'g"   and S :: "'g set" assumes "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S) = {}"
lemma5: fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "is_path r p (Suc i) j k"   and "\<not> is_path r p i j k" shows "(\<exists>q. is_path r q i j i) \<and> (\<exists>q'. is_path r q' i i k)"
transform_simps_3: fixes a :: "Arity"   and x :: "Vars.var" shows "\<T>\<^bsub>a\<^esub> (Var x) = Var x"
bal_tree: fixes n :: "nat"   and t :: "'b tree" shows "Root_Balanced_Tree.bal_tree n t = Balance.bal_tree n t"
inverse_inverse_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (inverse a) = a"