hcju/isabelleps huggingface dataset:
cyclic_group: shows "cyclic_group G = (\<exists>x\<in>carrier G. carrier G = range (([^]) x))"
pair4_eqD: fixes a1 :: "'a"   and a2 :: "'b"   and a3 :: "'c"   and a4 :: "'d"   and b1 :: "'a"   and b2 :: "'b"   and b3 :: "'c"   and b4 :: "'d" assumes "(a1, a2, a3, a4) = (b1, b2, b3, b4)" shows "a3 = b3"
poincare_map_returns: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x"   and "closed P" shows "poincare_map P x \<in> P"
P_def: shows "P = (\<Sum>x\<in>{p. prime p \<and> p dvd n}. complex_of_real (ln (real x) / real x))"
c: shows "((\<lambda>x. \<MM> x - ln x) \<longlongrightarrow> c) at_top"
in_set_simps_2: fixes x :: "'b"   and y :: "'b" shows "(x \<in> set [y]) = (x = y)"
add_nonpos_eq_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq x zero"   and "less_eq y zero" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
conv_radius_fps_expansion: fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex"   and r :: "ereal" assumes "f holomorphic_on eball z0 r" shows "r \<le> fps_conv_radius (fps_expansion f z0)"
delay_bisimulation_final_base_axioms: shows "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2"
Syntax_with_Connectives_False_Disj_axioms: shows "Syntax_with_Connectives_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj"
ab_2: shows "r = rat_of_int a / rat_of_nat b"
that: fixes u :: "'a" assumes "s \<lesssim> u"   and "t \<lesssim> u" shows "thesis"
bij_betw_imp_surj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "f ` A = B"
Inf_Sup_var: fixes X :: "'a set" shows "\<Sqinter> \<Inter> (ord.downset (\<lambda>x y. y \<le> x) ` X) = \<Squnion> X"
ltake_is_lprefix: fixes n :: "enat"   and xs :: "'a llist" shows "lprefix (ltake n xs) xs"
language_def: shows "N.language \<equiv> {xs. N.nextl (nfa.init \<lparr>nfa.states = {0}, init = {0}, final = {0}, nxt = \<lambda>q x. {}, eps = {}\<rparr>) xs \<inter> nfa.final \<lparr>nfa.states = {0}, init = {0}, final = {0}, nxt = \<lambda>q x. {}, eps = {}\<rparr> \<noteq> {}}"
sums_pre_newman: fixes s :: "complex" assumes "1 / 2 < Re s" shows "(\<lambda>p. if prime p then B p s else 0) sums pre_newman s"
I_minWeightBasisI: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and T :: "'c set"   and E :: "'c set"   and weight :: "'c \<Rightarrow> 'd" assumes "weighted_matroid carrier indep"   and "indep T"   and "\<And>e. e \<in> carrier - E - T \<Longrightarrow> \<not> indep (insert e T)"   and "E \<subseteq> carrier"   and "\<And>x y. \<lbrakk>x \<in> T; y \<in> E\<rbrakk> \<Longrightarrow> weight x \<le> weight y"   and "T \<subseteq> carrier"   and "weighted_matroid.no_smallest_element_skipped carrier indep weight E T" shows "weighted_matroid.I_minWeightBasis carrier indep weight (T, E)"
cons_1: fixes as1 :: "'a list"   and as2 :: "'a list"   and f1 :: "'a \<Rightarrow> bool"   and f2 :: "'a \<Rightarrow> 'a" assumes "as1 @ as2 = filter f1 (map f2 p)" shows "\<exists>p_1 p_2. p_1 @ p_2 = p \<and> as1 = filter f1 (map f2 p_1) \<and> as2 = filter f1 (map f2 p_2)"
tendsto_add: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
hom_1_iff: fixes x :: "'a" shows "([:x:] = 1) = (x = (1::'a))"
of_int_le_1_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) one = (z \<le> 1)"
four_block_one_mat: fixes n1 :: "nat"   and n2 :: "nat" shows "four_block_mat (1\<^sub>m n1) (0\<^sub>m n1 n2) (0\<^sub>m n2 n1) (1\<^sub>m n2) = 1\<^sub>m (n1 + n2)"
intro: fixes M :: "'b measure" assumes "prob_space M" shows "shift_operator_ergodic M"
l2_inv7I: fixes x :: "l2_state" assumes "secret x \<subseteq> {Exp (Exp Gen (NonceF (R $ N))) (NonceF (R' $ N')) |N N' R R'. R = test \<and> R' \<in> partners \<and> (N = nx \<or> N = ny) \<and> (N' = nx \<or> N' = ny)}" shows "x \<in> l2_inv7"
is_lasso_pre_prpl_of_lasso: fixes m :: "'a"   and L :: "'Q lasso" shows "graph.is_lasso_prpl_pre (to_gbg_ext m) (prpl_of_lasso L) = graph.is_lasso_pre (to_gbg_ext m) L"
IH: assumes "m < n"   and "infinite S" shows "pick S m < pick S n"
extensionalityI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Right.arr f"   and "Right.dom f = a"   and "Right.cod f = b" shows "Right.in_hom f a b"
bounded_linear_inner_left: fixes b :: "'c" shows "bounded_linear (\<lambda>a. a \<bullet> b)"
index_update_mat2: fixes i' :: "nat"   and A :: "'b mat"   and j' :: "nat"   and ij :: "nat \<times> nat"   and a :: "'b" assumes "i' < dim_row A"   and "j' < dim_col A"   and "(i', j') \<noteq> ij" shows "(A |\<^sub>m ij \<mapsto> a) $$ (i', j') = A $$ (i', j')"
nn_integral_cong_simp: fixes M :: "'k measure"   and u :: "'k \<Rightarrow> ennreal"   and v :: "'k \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M =simp=> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
invariant: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Graph_Invariants E P Q"   and "P a"   and "E a b" shows "Q b"
comp_assoc: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VV.comp (B.VV.comp h g) f = B.VV.comp h (B.VV.comp g f)"
sel_2: fixes x1 :: "'b \<Rightarrow> 'b set"   and x2 :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "V (Model x1 x2) = x2"
BoxS: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> in v] = (\<forall>v. [\<phi> in v])"
indexl_def: fixes s :: "('a, 'b) state" shows "\<I>\<^sub>l s = fst \<circ> the \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>l s)"
asym: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subset># b"   and "b \<subset># a" shows "False"
half_open_segment_closed_segmentI: fixes t :: "'b"   and t0 :: "'b"   and t1 :: "'b" assumes "t \<in> {t0--t1}"   and "t \<noteq> t1" shows "t \<in> {t0--<t1}"
add2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> k + b" shows "a + B \<equiv> k + (a + b)"
total_def: fixes valuation :: "'b"   and variables :: "nat set" shows "CNF.total valuation variables \<equiv> variables \<subseteq> vars valuation"
mQmp_mHom: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "Mp\<^bsub>M  H,N\<^esub> \<in> mHom R (M /\<^sub>m H) (M /\<^sub>m N)"
fcont: shows "continuous_on (cbox (u, w) (v, z)) f"
ex_per_cong: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "X \<noteq> Y"   and "Col A B C"   and "\<not> Col A B D" shows "\<exists>P. Per P C A \<and> Cong P C X Y \<and> A B OS P D"
igSubstAllIPresIGWlsAll_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstAllIPresIGWlsAll MOD \<equiv> igSubstIPresIGWls MOD \<and> igSubstAbsIPresIGWlsAbs MOD"
trans_inv': fixes f :: "nat \<Rightarrow> real"   and x :: "real"   and p :: "nat"   and np' :: "nat" shows "sum (fiX (\<lambda>y. f y + x) p) {..<np'} = sum (fiX f p) {..<np'} + real np' * x"
enabled_disj_rew: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "Enabled (F \<or> G) \<equiv> Enabled F \<or> Enabled G"
inf_left_idem: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (inf x (y::'b::type)) = inf x y"
satC_def: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'h" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Struct.satC intF intP \<xi> \<equiv> list_ex (Struct.satL intF intP \<xi>)"
funpow_simps_right_1: fixes f :: "'a \<Rightarrow> 'a" shows "f ^^ 0 = id"
cases: fixes a1 :: "'pub0"   and a2 :: "'pub1"   and a3 :: "'witness0 + 'witness1" assumes "((a1, a2), a3) \<in> Rel_OR" obtains x0 :: "'pub0" and w0 :: "'witness0" and x1 :: "'pub1" where "a1 = x0a__" and "a2 = x1a__" and "a3 = Inl w0" and "(x0a__, w0) \<in> Rel0 \<and> x1a__ \<in> valid_pub1"   | x1 :: "'pub1" and w1 :: "'witness1" and x0 :: "'pub0" where "a1 = x0" and "a2 = x1" and "a3 = Inr w1" and "(x1, w1) \<in> Rel1 \<and> x0 \<in> valid_pub0"
pmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "pmpt M T \<equiv> fmpt M T \<and> prob_space M"
core_argument: fixes p :: "int \<times> int"   and ps :: "(int \<times> int) list"   and \<delta> :: "real"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int" assumes "distinct (p # ps)"   and "sorted_snd (p # ps)"   and "0 \<le> \<delta>"   and "set (p # ps) = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>q\<in>set (p # ps). real_of_int l - \<delta> < real_of_int (fst q) \<and> real_of_int (fst q) < real_of_int l + \<delta>"   and "\<forall>q\<in>ps\<^sub>L. fst q \<le> l"   and "\<forall>q\<in>ps\<^sub>R. l \<le> fst q"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R" shows "length (filter (\<lambda>q. real_of_int (snd q - snd p) \<le> \<delta>) ps) \<le> 7"
IH: fixes n :: "nat"   and m :: "nat" assumes "\<And>n. n < length xs - 1 \<Longrightarrow> (xs ! Suc n, xs ! n) \<in> R"   and "n < length xs"   and "m < n" shows "(xs ! n, xs ! m) \<in> R\<^sup>+"
valid_path_imp_path: fixes g :: "real \<Rightarrow> 'a" assumes "valid_path g" shows "path g"
simps_15: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Protected of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f3"
fps_of_ratfps_dvd: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "(fps_of_ratfps x dvd fps_of_ratfps y) = (x dvd y)"
simps_93: fixes x41 :: "'a formula"   and x42 :: "'a formula" shows "atoms (x41 \<^bold>\<and> x42) = atoms x41 \<union> atoms x42"
lift_vars_monom: fixes i :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "real" shows "vars (liftPoly i j (MPoly_Type.monom m a)) = (\<lambda>x. if i \<le> x then x + j else x) ` vars (MPoly_Type.monom m a)"
v1_def: shows "V1 = \<lparr>ext = l1, lan = lan v, own = own v\<rparr>"
insertI2: fixes a :: "'d"   and B :: "'d set"   and b :: "'d" assumes "a \<in> B" shows "a \<in> insert b B"
assms: shows "(\<tau>trsys.silent_move (init_fin t) init_fin_\<tau>move)\<^sup>*\<^sup>* ((s, x), m) ((s', x'), m')"
reflp_ge_eq: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R" shows "(=) \<le> R"
ordering2_def: fixes f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ordering2 f ord X \<equiv> (\<forall>n. (finite X \<longrightarrow> n < card X) \<longrightarrow> f n \<in> X) \<and> (\<forall>x\<in>X. \<exists>n. (finite X \<longrightarrow> n < card X) \<and> f n = x) \<and> (\<forall>n n' n''. (finite X \<longrightarrow> n'' < card X) \<and> Suc n = n' \<and> Suc n' = n'' \<longrightarrow> ord (f n) (f n') (f n''))"
3: shows "v \<in> F \<and> w \<in> G \<and> x = v \<sqinter> w"
Merge_insert: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and x :: "'a"   and A :: "'a set" assumes "merkle_interface h bo m" shows "merkle_interface.Merge m (insert x A) = (if A = {} then Some x else merkle_interface.Merge m A \<bind> m x)"
simps_3: fixes a :: "'b"   and b :: "'b"   and as :: "'b list" shows "allConsec (a # b # as) = insert (a, b) (allConsec (b # as))"
pure_prod: fixes P1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "is_pure P1"   and "is_pure P2" shows "is_pure (P1 \<times>\<^sub>a P2)"
beta_l_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "grover_state.beta_l n f l = complex_of_real (sin ((real l + 1 / 2) * grover_state.\<theta> n f))"
map_ds_def: fixes f :: "'c \<Rightarrow> 'd" shows "map_ds f = ds \<circ> f \<circ> SSup"
g_independent: shows "\<forall>a m\<^sub>1 m\<^sub>2. a \<in> G \<longrightarrow> g m\<^sub>1 a = g m\<^sub>2 a"
weight_le_def: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "weight_le s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s \<le> weight t"
ins_xlate: fixes t :: "('c, 'd) BsSkewBinomialTree"   and q :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (ins t q) = SkewBinomialHeapStruc.ins (bsmapt t) (map bsmapt q)"
sup_continuous_mult_right: fixes c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "0 \<le> c"   and "c < \<infinity>"   and "sup_continuous f" shows "sup_continuous (\<lambda>x. f x * c)"
\<J>_subset: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "indexle k' u' k u" shows "\<J> k' u' \<subseteq> \<J> k u"
Id_l_unit: fixes G :: "('e, 'f) Group_scheme"   and f :: "'e \<Rightarrow> 'e" assumes "Group G"   and "gbij\<^bsub>G,G\<^esub> f" shows "I\<^bsub>G\<^esub> \<circ>\<^bsub>G\<^esub> f = f"
arr_char: fixes f :: "'a" shows "S.arr f = (arr f \<and> src f = a \<and> trg f = a)"
arr_Rel_ArrCod_in_Vset: shows "S\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
dflt_size_2: shows "Suc 0 \<le> dflt_size"
extend_act_def: fixes h :: "'c \<times> 'e \<Rightarrow> 'd" shows "extend_act h = (\<lambda>act. \<Union>(s, s')\<in>act. \<Union>y. {(h (s, y), h (s', y))})"
inv_comp_right_2: fixes g :: "'a"   and f :: "'a" assumes "\<phi>.A.iso (g \<cdot>\<^sub>A f)"   and "\<phi>.A.iso f" shows "\<phi>.A.iso g"
less_numeral_extra_3: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
elims: fixes x :: "32 Word.word"   and xa :: "32 Word.word"   and y :: "32 Word.word" assumes "wmin x xa = y" obtains w1 :: "32 Word.word" and w2 :: "32 Word.word" where "x = w1" and "xa = w2" and "y = (if w1 <s w2 then w1 else w2)"
leadsto_init: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "\<turnstile> Init F \<and> (F \<leadsto> G) \<longrightarrow> \<diamond>G"
wt_init_comp_ty': fixes Env :: "env"   and T :: "ty" assumes "is_acc_type (prg Env) (pkg Env) T" shows "Env\<turnstile>init_comp_ty T\<Colon>\<surd>"
triple_valid_def2: fixes G :: "prog"   and n :: "nat"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" shows "G\<Turnstile>n:{P} t\<succ> {Q} = (\<forall>Y s Z. P Y s Z \<longrightarrow> (\<exists>L. (normal s \<longrightarrow> (\<exists>C T A. \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T \<and> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>t\<guillemotright> A)) \<and> s\<Colon>\<preceq>(G, L)) \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s') \<longrightarrow> Q Y' s' Z))"
hyps_2: shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* s\<^sup>\<bullet> \<and> s\<^sup>\<bullet> \<rightarrow>\<^sub>\<beta>\<^sup>* t\<^sup>\<bullet>"
assms_2: shows "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"
partition_on_partition_on_unique: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c" assumes "partition_on A P"   and "x \<in> A" shows "\<exists>!X. x \<in> X \<and> X \<in> P"
acquire: fixes t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and l :: "'l" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "0 < ln $ l" shows "\<exists>l'. 0 < ln $ l' \<and> \<not> may_lock (locks s $ l') t"
init_fin_bisim_simps_5: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and s1 :: "status"   and x1 :: "'a"   and m1 :: "'d"   and x2 :: "'b"   and m2 :: "'e" shows "FWbisimulation_base.init_fin_bisim final1 final2 bisim t ((s1, x1), m1) ((Running, x2), m2) = (s1 = Running \<and> bisim t (x1, m1) (x2, m2))"
scale_zero_left: fixes x :: "'t \<Rightarrow>\<^sub>0 'c" shows "0 \<odot> x = 0"
0: shows "exu x \<phi> = cnj (exi x \<phi>) (exi z (all x (imp \<phi> (eql (Var x) (Var z)))))"
II: shows "Q_invar Qi \<and> M_invar \<pi>i \<and> prim_invar2 (Q_\<alpha> Qi) (M_lookup \<pi>i)"
sum_integral_ubound_decreasing': fixes m :: "nat"   and n :: "nat"   and g :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "m \<le> n"   and "\<And>x. x \<in> {real m - 1..real n} \<Longrightarrow> (g has_real_derivative f x) (at x)"   and "\<And>x y. \<lbrakk>real m - 1 \<le> x; x \<le> y; y \<le> real n\<rbrakk> \<Longrightarrow> f y \<le> f x" shows "(\<Sum>k = m..n. f (real k)) \<le> g (real n) - g (real m - 1)"
rel_gpv_Done: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" shows "rel_gpv A C (Generative_Probabilistic_Value.Done x) (Generative_Probabilistic_Value.Done y) = A x y"
hisPrev_prop_1: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> Blockchain.his active cmp bc mining t n nid" shows "Blockchain.hisPred active cmp bc mining t n nid n' < n'"
length_zip: fixes xs :: "'g list"   and ys :: "'h list" shows "length (zip xs ys) = min (length xs) (length ys)"
ij_le2_2: shows "j < dim_col (matricize I (tensor0 ds))"
relcompI: fixes a :: "'g"   and b :: "'h"   and r :: "('g \<times> 'h) set"   and c :: "'i"   and s :: "('h \<times> 'i) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
convert_term_id: fixes t :: "'c" assumes "no_abs t" shows "convert_term t = t"
branch0: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "branch subs Gamma f" shows "f 0 = Gamma"
less_fun_strongI: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "\<And>c. 0 < c \<Longrightarrow> \<exists>n. \<forall>m>n. c * f m < g m" shows "f \<prec> g"
elims_2: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a stream" assumes "holds x xa" obtains P :: "'a \<Rightarrow> bool" and xs :: "'a stream" where "x = P" and "xa = xs" and "P (shd xs)"
length_tl_queue_rep: fixes q :: "'a queue" assumes "\<not> Optimized_MTL.is_empty q" shows "length (linearize (tl_queue q)) < length (linearize q)"
sum_unfold_sum_mset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "sum f A = \<Sum>\<^sub># (image_mset f (mset_set A))"
squarefree_def: fixes n :: "'b" shows "squarefree n = (\<forall>x. x\<^sup>2 dvd n \<longrightarrow> x dvd (1::'b))"
odiff_add_cancel: fixes x :: "V"   and y :: "V" shows "odiff (x + y) x = y"
max_le: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z"   and "y \<le> z" shows "max x y \<le> z"
lg_cong_lg: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.QCong Cong l"   and "l A B"   and "Cong A B C D" shows "l C D"
singleton_insert_inj_eq': fixes a :: "'g"   and A :: "'g set"   and b :: "'g" shows "(insert a A = {b}) = (a = b \<and> A \<subseteq> {b})"
it_to_it_map_fold: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'a" assumes "proper_it it it'" shows "it_to_it (\<lambda>c f. it c (f \<circ> f')) = (\<lambda>c f. it' c (f \<circ> f'))"