hcju/isabelleps huggingface dataset:
r_distr: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "z \<in> carrier Zp" shows "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
exhaust_disc: fixes "term" :: "Core.term" obtains "is_Ct term"   | "is_Fv term"   | "is_Bv term"   | "is_Abs term"   | "is_App term"
add_mset_add_mset_same_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
wf_until_aux_Cons1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and pos :: "bool"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and t :: "nat"   and a1 :: "event_data option list set"   and a2 :: "event_data option list set"   and aux :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and ne :: "nat" assumes "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> ((t, a1, a2) # aux) ne" shows "t = \<tau> \<sigma> ne"
append_simpl4: fixes y :: "'b list"   and i :: "nat"   and z :: "'c"   and zs :: "'c list" shows "length y < i \<longrightarrow> i < length (z # zs) * length y \<longrightarrow> i - length y < length zs * length y"
multiplicity_prime: fixes p :: "'a" assumes "prime_elem p" shows "multiplicity p p = 1"
subgroup_transitive: fixes K :: "'c set"   and H :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G :: "'c set" assumes "subgroup K H composition unit"   and "subgroup H G composition unit" shows "subgroup K G composition unit"
elims: fixes x :: "'a set"   and xa :: "'a list"   and y :: "'a set" assumes "finite_extension x xa = y" obtains K :: "'a set" and xs :: "'a list" where "x = K" and "xa = xs" and "y = foldr (\<lambda>x K'. simple_extension K' x) xs K"
next_propagate'_def: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" shows "next_propagate' c0 c1 loc t \<equiv> t \<in>#\<^sub>z c_work c0 loc \<and> (\<forall>t' loc'. t' \<in>#\<^sub>z c_work c0 loc' \<longrightarrow> \<not> t' < t) \<and> c1 = c0\<lparr>c_imp := (c_imp c0)(loc := c_imp c0 loc + {#t' \<in>#\<^sub>z c_work c0 loc. t' = t#}), c_work := \<lambda>loc'. if loc' = loc then {#t' \<in>#\<^sub>z c_work c0 loc'. t' \<noteq> t#} else c_work c0 loc' + after_summary (frontier_changes (c_imp c0 loc + {#t' \<in>#\<^sub>z c_work c0 loc. t' = t#}) (c_imp c0 loc)) (summary loc loc')\<rparr>"
fls_X_inv_shift_to_one: shows "fls_shift (- 1) fls_X_inv = 1"
star_of_inject: fixes x :: "'c"   and y :: "'c" shows "(star_of x = star_of y) = (x = y)"
g_def: shows "g = (\<lambda>x. f x / (1 + \<bar>psi (real_cond_exp M F X x)\<bar> + \<bar>q (real_cond_exp M F X x)\<bar>))"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "sigma_finite_measure M"   and "qmpt_axioms M T" shows "qmpt M T"
simps_2: fixes i :: "nat"   and j :: "nat"   and uu :: "'b option"   and t :: "'b tree"   and u :: "'b tree" shows "well_shaped (Node i j uu t u) = (i \<le> j \<and> (i = j \<longrightarrow> t = Leaf \<and> u = Leaf) \<and> (i < j \<longrightarrow> t \<noteq> Leaf \<and> u \<noteq> Leaf \<and> well_shaped t \<and> well_shaped u \<and> i = l t \<and> j = r u \<and> Suc (r t) = l u))"
coherence: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and V :: "'a \<Rightarrow> 'b"   and t :: "'a local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C t" shows "D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Cod C t))) (evaluation_map.eval D T \<alpha> \<iota> V t) = D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Diagonalize C t)) (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Dom C t)))"
succeed\<^sub>R_def: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and F :: "'c set"   and i :: "nat" shows "mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F i = {uu_. \<exists>r \<nu> s q. uu_ = (r, \<nu>, s) \<and> r q = Some i \<and> q \<notin> F - {q\<^sub>0} \<and> \<delta> q \<nu> \<in> F}"
t3: shows "\<forall>z\<in>{l (- dr)--l dr}. 0 < f z \<bullet> rot (l (- dr) - l dr)"
simps_2: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (fm.Atom (Less p)) = (var \<notin> vars p)"
unit_char': fixes a :: "'b" assumes "B.obj a" shows "HoG.unit a = H (\<tau>.G.unit a) \<cdot>\<^sub>D H.unit (\<tau>.G.map\<^sub>0 a)"
that_1: fixes c' :: "('a, 'b) chart" assumes "c' \<in> charts" shows "k-smooth_compat c c'"
dominatesE: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "old.dominates g n m" obtains "m \<in> set (\<alpha>n g)" and "\<And>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<Longrightarrow> n \<in> set ns"
conga_diff1: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "A \<noteq> B"
a_\<beta>: fixes p :: "fm"   and l :: "int"   and bbs :: "bool list"   and x :: "int"   and bs :: "int list" assumes "iszlfm p"   and "d_\<beta> p l"   and "0 < l" shows "iszlfm (a_\<beta> p l) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> Ifm bbs (l * x # bs) (a_\<beta> p l) = Ifm bbs (x # bs) p"
left_ac_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b * c) = a * b * c"
lnull_lSup: fixes A :: "'c llist set" shows "lnull (lSup A) = (\<forall>x\<in>A. lnull x)"
asymp_equiv_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<sim>[F] g1"   and "f2 \<sim>[F] g2" shows "(\<lambda>x. f1 x * f2 x) \<sim>[F] (\<lambda>x. g1 x * g2 x)"
integrable_primes_M: fixes x :: "real"   and y :: "real" assumes "2 \<le> x" shows "\<MM> integrable_on {x..y}"
sum_bij: fixes F :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and t :: "'b set" assumes "bij F"   and "\<forall>x\<in>s. f x = g (F x)"   and "F ` s = t" shows "sum f s = sum g t"
epsilon_def: shows "\<epsilon> = star_n (\<lambda>n. inverse (real (Suc n)))"
unitaryD2: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "unitary U" shows "U o\<^sub>C\<^sub>L U* = id_cblinfun"
sumdd: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n (M 0) (M 1) (denote S) n \<rho>) n \<in> carrier_mat d d"
mat_geI: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> less_eq (B $$ (i, j)) (A $$ (i, j))" shows "ord.mat_ge less_eq A B"
diff_numeral_special_12: 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" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus one) (uminus one) = zero"
affine_dim_equal: fixes S :: "'a set"   and T :: "'a set" assumes "affine S"   and "affine T"   and "S \<noteq> {}"   and "S \<subseteq> T"   and "aff_dim S = aff_dim T" shows "S = T"
1: shows "point y \<and> point z \<and> y ; z\<^sup>T \<le> v"
intros_2: fixes k :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body k k xs ys zs"   and "0 < k" shows "Form (2 * k - 1) {xs, ys}"
reachable_V: shows "prod.reachable \<subseteq> prod.V"
base: fixes C :: "'b literal multiset"   and CC :: "'b literal multiset set"   and \<Gamma> :: "'b inference set" assumes "C \<in> CC" shows "C \<in> inference_system.saturate \<Gamma> CC"
parallel_reduction_fstE: fixes P :: "'a trm"   and X :: "'a trm" assumes "Fst P >> X" shows "(\<exists>P'. P >> P' \<and> X = Fst P') \<or> (\<exists>A A' B. P = SimplyTyped.Pair A B \<and> A >> A' \<and> X = A')"
Radius_def: shows "Radius = (MAX Y\<in>Roots'. cmod (root' Y))"
def_while_unfold: fixes f :: "'b \<Rightarrow> 'b"   and test :: "'b \<Rightarrow> bool"   and do :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "f \<equiv> while test do" shows "f x = (if test x then f (do x) else x)"
rComm2: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
succ_imp_graph_alt_def: fixes cnf :: "lit uprod set"   and l :: "lit" shows "succ_imp_graph cnf l = Finite_Set.fold (succs_of_clause l) {} cnf"
WT_elim_cases_17: fixes is_lub' :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'b expr"   and T :: "ty" assumes "is_lub',P,E \<turnstile> a\<bullet>length :: T" obtains Ta :: "ty" where "T = Integer" and "is_lub',P,E \<turnstile> a :: Ta\<lfloor>\<rceil>"
semiring_norm_63: fixes a :: "'a" shows "(0::'a) * a = (0::'a)"
asubg_inc_zero: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set" assumes "aGroup A"   and "A +> H" shows "\<zero>\<^bsub>A\<^esub> \<in> H"
psubst_new_subset: fixes z :: "('a, 'b) form list"   and z' :: "('a, 'b) form list"   and c :: "'a"   and n :: "'a" assumes "set z \<subseteq> set z'"   and "c \<notin> \<Union> (params ` set z)" shows "set z \<subseteq> set (map (psubst (id(c := n))) z')"
a_Ints: fixes i :: "nat"   and l :: "nat"   and k :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)"   and "l \<le> i"   and "k < n" shows "gs.a i l $ k \<in> \<int>"
order_finite_chain: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and l :: "nat" assumes "long_ch_by_ord f X"   and "finite X"   and "0 \<le> i \<and> i < j \<and> j < l \<and> l < card X" shows "[[f i f j f l]]"
lappend_llist_of_llist_of: fixes xs :: "'b list"   and ys :: "'b list" shows "lappend (llist_of xs) (llist_of ys) = llist_of (xs @ ys)"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> yahalom" shows "X \<in> knows Spy evs"
assms_2: shows "all_fulfill (\<lambda>x. \<not> tautology x) S"
assms_3: shows "path_image f \<subseteq> cbox a b"
list_app_typeI: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and Ts :: "LambdaType.type list"   and T :: "LambdaType.type"   and ts :: "dB list" assumes "e \<turnstile> t : Ts \<Rrightarrow> T"   and "e \<tturnstile> ts : Ts" shows "e \<turnstile> t \<degree>\<degree> ts : T"
row_to_poly_zero: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ts :: "'c list" assumes "term_powerprod pair_of_term term_of_pair" shows "row_to_poly ts (0\<^sub>v (length ts)) = 0"
rcosets_mult_eq: fixes M :: "'a set" assumes "M \<in> rcosets derived G (carrier G)" shows "derived G (carrier G) <#> M = M"
set_relD: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x :: "'a" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x \<in> s" shows "\<exists>x'\<in>s'. (x, x') \<in> R"
that: shows "\<forall>i\<in>I. length xs \<le> i"
assms_2: fixes p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "p \<in> set ps" shows "u \<preceq>\<^sub>t sig_of_pair p"
lless_eq: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "partial_order L" shows "(x \<sqsubset>\<^bsub>L\<^esub> y) = (x \<sqsubseteq>\<^bsub>L\<^esub> y \<and> x \<noteq> y)"
assms_3: assumes "\<Phi>" shows "m' \<le> m"
bigMaxU: shows "wmax (wmax (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2)) (wmax (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2)) \<equiv>\<^sub>U max (max (rl1 * rl2) (ru1 * rl2)) (max (rl1 * ru2) (ru1 * ru2))"
assms_2: shows "u adds\<^sub>t v \<ominus> t"
valuation_for_state_variables_is: fixes k :: "nat"   and ks :: "nat list"   and v :: "'b"   and vs :: "'b list"   and s :: "nat \<Rightarrow> 'b \<Rightarrow> bool option" assumes "k \<in> set ks"   and "v \<in> set vs" shows "foldr (\<lambda>(k, v). valuation_for_state vs (s k) k v) (List.product ks vs) \<A>\<^sub>0 (State k (index vs v)) = (s k v = Some True)"
interiorE: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> interior S" obtains T :: "'b set" where "open T" and "x \<in> T" and "T \<subseteq> S"
igSwapClsSTR_def: fixes MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'l, 'm) model_scheme" shows "igSwapClsSTR MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbsSTR MOD \<and> igSwapIGOpSTR MOD"
map_tailrec_def: fixes f :: "'c \<Rightarrow> 'b"   and as :: "'c list" shows "map_tailrec f as = rev (map_tailrec_rev f as [])"
sig_red_zero_sing_regI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red_zero sing_reg G p" shows "sig_red_zero (\<preceq>\<^sub>t) G p"
Aexp_IfThenElse: fixes scrut :: "exp"   and e1 :: "exp"   and a :: "Arity"   and e2 :: "exp" shows "\<A>\<^bsub>0\<^esub> scrut \<squnion> \<A>\<^bsub>a\<^esub> e1 \<squnion> \<A>\<^bsub>a\<^esub> e2 \<sqsubseteq> \<A>\<^bsub>a\<^esub> (scrut ? e1 : e2)"
healthy_feasibleD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "feasible t"
assms_3: shows "B \<subseteq> dgrad_p_set d m"
sum: fixes x :: "'a"   and g :: "'e \<Rightarrow> 'b"   and S :: "'e set" shows "f' y x (sum g S) = (\<Sum>a\<in>S. f' y x (g a))"
weight_less_def: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" shows "weight_less s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s < weight t"
\<sigma>_arbitrary_def: shows "\<sigma>_arbitrary \<equiv> \<lambda>v. SOME w. v \<rightarrow> w"
proj_list_insert_3: fixes i :: "'a"   and j :: "'a"   and d :: "'b"   and D :: "('a strand_label \<times> 'b) list" assumes "i \<noteq> j" shows "proj i (List.insert (ln j, d) D) = proj i D"
split_paired_Ex: fixes P :: "'c \<times> 'd \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>a b. P (a, b))"
Gchain_pre: fixes n :: "nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme" assumes "Gchain (Suc n) g" shows "Gchain n g"
assms_5: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])"
simps_6: fixes r :: "'a rexp" shows "nullable (Star r) = True"
mrSet: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and r :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x p = \<lfloor>r\<rfloor>" shows "r \<in> set p"
One_fract_def: shows "1 = Fraction_Field.Fract (1::'b) (1::'b)"
simps_2: fixes final1 :: "'c \<Rightarrow> bool"   and r1 :: "('a, 'b, 'c, 'd, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and bisim :: "'b \<Rightarrow> 'c \<times> 'd \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('a, 'b, 'c, 'd, 'e, 'f) \<tau>moves"   and s1 :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'a \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and ct :: "'b conditional_action"   and cts :: "'b conditional_action list" shows "FWdelay_bisimulation_base.activate_cond_actions1 final1 r1 final2 bisim \<tau>move1 s1 s2 (ct # cts) = FWdelay_bisimulation_base.activate_cond_actions1 final1 r1 final2 bisim \<tau>move1 (FWdelay_bisimulation_base.activate_cond_action1 final1 r1 final2 bisim \<tau>move1 s1 s2 ct) s2 cts"
scc_of_def: fixes u :: "'a" shows "scc_of u \<equiv> {v. u \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> v \<and> v \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> u}"
path_image_shiftpath: fixes a :: "real"   and g :: "real \<Rightarrow> 'a" assumes "a \<in> {0..1}"   and "pathfinish g = pathstart g" shows "path_image (shiftpath a g) = path_image g"
integral_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" shows "integral {- b..- a} (\<lambda>x. f (- x)) = integral {a..b} f"
identity_functor_axioms: shows "identity_functor (\<cdot>\<^sub>B)"
S_transition_Act_iff: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" shows "P \<rightarrow>\<^sub>S \<langle>S_action.Act \<alpha>,P'\<rangle> = P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"
LNil_fin: fixes A :: "'b set" shows "LNil \<in> A\<^sup>\<star>"
new_tv_subst_scheme_list: fixes n :: "nat"   and S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list" assumes "new_tv n S"   and "new_tv n A" shows "new_tv n ($ S A)"
Q_relax_outgoing_def: fixes u :: "'v"   and du :: "nat"   and V :: "'a"   and Q :: "'b" shows "Q_relax_outgoing u du V Q = fold (\<lambda>(d, v) Q. case Q_lookup Q v of None \<Rightarrow> if M_lookup V v \<noteq> None then Q else Q_update v (du + d) Q | Some d' \<Rightarrow> Q_update v (min (du + d) d') Q) (G_succ g u) Q"
simps_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less [] = True"
evaluate_clock_monotone': fixes eval :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r :: "(v, v) result" assumes "Evaluate_Single.evaluate eval s e = (s', r)" shows "clock s' \<le> clock s"
g_def: fixes a :: "nat"   and b :: "nat" shows "g a b = a + 1"
zip_Cons_Cons: fixes x :: "'d"   and xs :: "'d list"   and y :: "'e"   and ys :: "'e list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
adjacent_finite: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "valid_unMultigraph G"   and "finite (edges G)" shows "finite {n. valid_unMultigraph.adjacent G v n}"
pmf_moreelemlist: fixes n :: "nat"   and x :: "bool"   and xs :: "bool list" shows "pmf (bv (Suc n)) (x # xs) = pmf (bv n) xs * pmf (bernoulli_pmf (5 / 10)) x"
completely_multiplicative_imp_moebius_mu_inverse: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "completely_multiplicative_function f" shows "dirichlet_prod f (\<lambda>n. moebius_mu n * f n) n = (if n = 1 then 1::'a else (0::'a))"
fresh_imp_VarNonOccForm: fixes i :: "name"   and A :: "fm"   and e :: "(name, hf) finfun" assumes "atom i \<sharp> A" shows "VarNonOccForm (q_Var i) \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"
intro: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" assumes "prob_space M"   and "disc_filtr_prob_space_axioms M F" shows "disc_filtr_prob_space M F"
List_non_null: fixes a :: "nat"   and h :: "nat \<Rightarrow> nat option"   and as :: "nat list" assumes "a \<noteq> 0" shows "List h a as = (\<exists>b bs. as = a # bs \<and> h a = Some b \<and> List h b bs)"
gen_iapath_triv_path: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "pair_sym_digraph G"   and "pre_digraph.gen_iapath (with_proj G) (pverts G) u p v" shows "p = [(u, v)]"
dummy_abc_def: fixes k :: "nat" shows "dummy_abc k = [Inc k, Dec k 0, Goto 3]"
transcendental_standard_liouville: fixes q :: "int"   and p :: "nat \<Rightarrow> int" assumes "1 < q"   and "range p \<subseteq> {0..<q}"   and "\<exists>\<^sub>F k in sequentially. p k \<noteq> 0" shows "\<not> algebraic (standard_liouville p q)"
wfvarsoccs\<^sub>s\<^sub>t_def: fixes S :: "('e, 'd) strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfvarsoccs\<^sub>s\<^sub>t\<^sub>p S))"
multiset_remove_induct: fixes P :: "'a multiset \<Rightarrow> bool"   and A :: "'a multiset" assumes "P {#}"   and "\<And>A. \<lbrakk>A \<noteq> {#}; \<And>x. x \<in># A \<Longrightarrow> P (A - {#x#})\<rbrakk> \<Longrightarrow> P A" shows "P A"
image_add_integer_atLeastLessThan: fixes l :: "integer"   and u :: "integer" shows "(\<lambda>x. x + l) ` {0..<u - l} = {l..<u}"
simulation_silents2: fixes s1 :: "'s1"   and s2 :: "'s2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s2 -\<tau>2\<rightarrow>* s2'" shows "\<exists>s1'. s1 -\<tau>1\<rightarrow>* s1' \<and> s1' \<approx> s2'"
sign_r_pos_def: fixes p :: "real poly"   and x :: "real" shows "sign_r_pos p x \<equiv> \<forall>\<^sub>F x in at_right x. 0 < poly p x"
Saturated_def: fixes steps :: "('state \<times> 'rule) stream" shows "Saturated steps \<equiv> \<forall>r\<in>R. saturated r steps"
red: shows "S' -T\<triangleright>TA\<rightarrow> S''"
not_iFROM_iIN_subset: fixes n' :: "nat"   and n :: "nat"   and d :: "nat" shows "\<not> [n'\<dots>] \<subseteq> [n\<dots>,d]"
limsup_root_powser: fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" shows "limsup (\<lambda>n. ereal (root n (norm (f n * z ^ n)))) = limsup (\<lambda>n. ereal (root n (norm (f n)))) * ereal (norm z)"
sort_leq_def: fixes sub :: "('a \<times> 'b) set"   and s1 :: "'a set"   and s2 :: "'b set" shows "sort_leq sub s1 s2 = (\<forall>c\<^sub>2\<in>s2. \<exists>c\<^sub>1\<in>s1. class_leq sub c\<^sub>1 c\<^sub>2)"
cases: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "('c, 'd) label list"   and a3 :: "'c \<Rightarrow> 'd" assumes "ce_star a1 a2 a3" obtains c :: "'c \<Rightarrow> 'd" where "a1 = c" and "a2 = []" and "a3 = c"   | c1 :: "'c \<Rightarrow> 'd" and l :: "('c, 'd) label" and c2 :: "'c \<Rightarrow> 'd" and ls :: "('c, 'd) label list" and c3 :: "'c \<Rightarrow> 'd" where "a1 = c1" and "a2 = l # ls" and "a3 = c3" and "ce c1 l c2" and "ce_star c2 ls c3"
collinear_sym2: fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" shows "collinear z1 z2 z3 = collinear z2 z1 z3"
trace_rel_def: shows "trace_rel \<equiv> Collect (case_prod (\<lambda>(x, y). trace x y))"
OrdNotEqP_OrdP_E: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (OrdP x) (insert (OrdP y) H) \<turnstile> A" shows "insert (x NEQ y) H \<turnstile> A"