hcju/isabelleps huggingface dataset:
inverse_functors_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" shows "inverse_functors_axioms A B F G \<equiv> G \<circ> F = identity_functor.map B \<and> F \<circ> G = identity_functor.map A"
same_prefix_nil: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs @ ys \<le> xs) = (ys = [])"
upt_0: fixes i :: "nat" shows "[i..<0] = []"
Range_snd: fixes r :: "('c \<times> 'b) set" shows "Range r = snd ` r"
assms_2: shows "\<not> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
before_dist_not2: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs"   and "ram1 \<in> set vs"   and "ram2 \<in> set vs"   and "ram1 \<noteq> ram2"   and "\<not> before vs ram1 ram2" shows "before vs ram2 ram1"
rupture_is_field_iff_pirreducible: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "field (Rupt\<^bsub>L\<^esub> K p) = pirreducible\<^bsub>L\<^esub> K p"
\<delta>_4: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
cases: fixes x :: "('s, 'v) tm" obtains \<zeta> :: "('s, 'v) hd" and ss :: "('s, 'v) tm list" where "x = apps (Hd \<zeta>) ss"
fun_star_unfoldr_eq: fixes f :: "'b nd_fun" shows "1 \<squnion> qstar f \<cdot> f = qstar f"
neutr1: fixes x :: "'a" shows "f\<cdot>z\<cdot>x = x"
freeword_inverse: fixes x :: "'a freeword" shows "Abs_freeword (freeword x) = x"
subspace_substandard: fixes P :: "'b \<Rightarrow> bool" shows "subspace {x. \<forall>i\<in>Basis. P i \<longrightarrow> x \<bullet> i = 0}"
dg_prod_2_ObjI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and a :: "V"   and b :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
new_keys_not_analzd: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_pur" shows "K \<notin> keysFor (analz (knows Spy evs))"
left_commute: fixes b :: "'a upred"   and a :: "'a upred"   and c :: "'a upred" shows "(b \<and> a \<and> c) = (a \<and> b \<and> c)"
Response1: fixes p :: "'b"   and ou :: "'d"   and i :: "nat" assumes "Resp p ou s s' \<and> t = t'"   and "i < id1"   and "a = Response i p ou" shows "thesis"
Iso_def: shows "{\<cong>} \<equiv> {(F\<^sub>1, F\<^sub>2). F\<^sub>1 \<cong> F\<^sub>2}"
eq_neg_iff_add_eq_0: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(x = \<ominus>\<^bsub>R\<^esub> y) = (x \<oplus>\<^bsub>R\<^esub> y = \<zero>\<^bsub>R\<^esub>)"
countable_nat_UN: fixes A :: "nat \<Rightarrow> 'd set"   and M :: "'d measure" assumes "range A \<subseteq> sets M" shows "\<Union> (range A) \<in> sets M"
assms_6: shows "set cands \<subseteq> S (lu, []) vs d"
surj_directional_derivative: fixes k :: "enat"   and a :: "'c" assumes "k = \<infinity>" shows "range (directional_derivative k a) = manifold_eucl.tangent_space k a"
nchotomy: shows "\<forall>nat. nat = 0 \<or> (\<exists>x2. nat = Suc x2)"
span_data_ext_def: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c" shows "\<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons span_data_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple Leg0 Leg1) more"
simps_6: fixes vb :: "'b"   and vc :: "'b" shows "separate [DenyAllFromTo vb vc] = [DenyAllFromTo vb vc]"
intro: fixes \<AA> :: "('a \<times> 'a) set" assumes "Card_order \<AA> \<and> infinite (Field \<AA>)" shows "setcat \<AA>"
prems_3: shows "num_args (subst \<rho> w) = num_args w"
subcls_notin_has_fields: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and F :: "char list"   and D :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "((F, D), b, T) \<in> set FDTs" shows "(D, C) \<notin> (subcls1 P)\<^sup>+"
le_inf_iff: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "(x \<subseteq># y \<inter># z) = (x \<subseteq># y \<and> x \<subseteq># z)"
le_zero_eq: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
kat_3': fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<cdot> y \<cdot> ad z = 0" shows "d x \<cdot> y = d x \<cdot> y \<cdot> d z"
assoc'_is_natural_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "bicategory.\<a>' V H \<a> src trg \<mu> \<nu> \<tau> = V (bicategory.\<a>' V H \<a> src trg (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>) (partial_magma.cod V \<tau>)) (H \<mu> (H \<nu> \<tau>))"
dom_one: fixes x :: "'a" shows "x ; 1 = d x ; 1"
compile_consE: fixes name :: "name"   and rhs' :: "sterm"   and rest :: "(name \<times> sterm) list"   and rs :: "(name \<times> pterm) fset" assumes "(name, rhs') # rest = Rewriting_Sterm.compile rs"   and "is_fmap rs" obtains rhs :: "pterm" where "rhs' = pterm_to_sterm rhs" and "(name, rhs) |\<in>| rs" and "rest = Rewriting_Sterm.compile (rs |-| {|(name, rhs)|})"
intro: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>x. F x x = x" shows "idem_overrider_axioms F"
prod_lb: fixes X :: "'a list set"   and m :: "nat"   and Y :: "'a list set"   and n :: "nat" assumes "\<forall>w\<in>X. m \<le> |w|"   and "\<forall>w\<in>Y. n \<le> |w|" shows "\<forall>w\<in>X \<cdot> Y. m + n \<le> |w|"
subtract_Some_set_of_res: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree"   and t :: "'b tree" assumes "subtract t\<^sub>1 t\<^sub>2 = Some t" shows "set_of t \<subseteq> set_of t\<^sub>2"
resolvent_def: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "resolvent P1 P2 C \<equiv> \<exists>A. Pos A \<in> P1 \<and> Neg A \<in> P2 \<and> C = P1 - {Pos A} \<union> (P2 - {Neg A})"
f_is_measurable: shows "f \<in> borel_measurable lborel"
restrict_merge_1: fixes I :: "'b set"   and J :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "I \<inter> J = {}" shows "restrict (merge I J (x, y)) I = restrict x I"
submanifold_atlasI: fixes c :: "('b, 'b) chart" assumes "c \<in> manifold_eucl.dest.atlas" shows "restrict_chart (codomain c) c \<in> sub.atlas"
lookup_env_restr_eq: fixes S :: "'d set"   and m :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(m f|` S) x = (if x \<in> S then m x else \<bottom>)"
transfer: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.semiring_1 one times plus zero" shows "rel_fun pcr_word (=) (semiring_1.of_nat one plus zero \<circ> nat \<circ> take_bit LENGTH('l)) (semiring_1.unsigned one plus zero)"
nocommonPP: shows "set (PPc c1) \<inter> set (PPc c2) = {}"
ac_operator_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "ac_operator f \<equiv> (\<forall>a b c. f (f a b) c = f a (f b c)) \<and> (\<forall>a b. f a b = f b a)"
weight_exec_gpv: fixes \<I> :: "('b, 'c) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and gpv :: "('d, 'b, 'c) gpv"   and s :: "'a" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (callee s x)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s. \<I> \<turnstile>c callee s \<surd>" shows "weight_gpv \<I> gpv \<le> weight_spmf (exec_gpv callee gpv s)"
Sup_trans_least2: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set"   and b :: "real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "\<forall>t\<in>S. \<forall>P. nneg P \<longrightarrow> bounded_by b P \<longrightarrow> t P \<tturnstile> u P"   and "\<forall>u\<in>S. \<forall>P. nneg P \<and> bounded_by b P \<longrightarrow> nneg (u P) \<and> bounded_by b (u P)"   and "nneg P"   and "bounded_by b P"   and "\<And>P. \<lbrakk>nneg P; bounded_by b P\<rbrakk> \<Longrightarrow> nneg (u P)" shows "Sup_trans S P \<tturnstile> u P"
size_tree_Leaf: shows "Root_Balanced_Tree.size_tree \<langle>\<rangle> = 0"
nn_integral_substitution: fixes g :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "set_borel_measurable borel {g a..g b} f"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "continuous_on {a..b} g'"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> g' x"   and "a \<le> b" shows "\<integral>\<^sup>+ x. ennreal (f x * indicat_real {g a..g b} x) \<partial>lborel = \<integral>\<^sup>+ x. ennreal (f (g x) * g' x * indicat_real {a..b} x) \<partial>lborel"
path_is_empty_impl_def: fixes s :: "'c list \<times> nat list \<times> ('c \<Rightarrow> node_state option) \<times> (nat \<times> 'c set) list" shows "path_is_empty_impl s \<equiv> GS.S s = []"
COMP_def: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "g \<cdot> f \<equiv> if (f \<noteq> Null \<and> Dom f \<in> {A. |A| <o \<AA>} \<and> Cod f \<in> {A. |A| <o \<AA>} \<and> Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)) \<and> (g \<noteq> Null \<and> Dom g \<in> {A. |A| <o \<AA>} \<and> Cod g \<in> {A. |A| <o \<AA>} \<and> Map g \<in> extensional (Dom g) \<inter> (Dom g \<rightarrow> Cod g)) \<and> Dom g = Cod f then MkArr (Dom f) (Cod g) (compose (Dom f) (Map g) (Map f)) else Null"
semiring_simprules_3: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y \<in> carrier R"
frechet_derivative_at': fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "(f has_derivative s) (at x)" shows "frechet_derivative f (at x) = s"
closure_bounded_linear_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "linear f"   and "bounded S" shows "f ` closure S = closure (f ` S)"
filters_def: fixes T :: "'b set set" shows "Filters\<^bsub>T\<^esub> = {F. {} \<notin> F \<and> \<Union> F \<subseteq> carrier\<^bsub>T\<^esub> \<and> (\<forall>A B. A \<in> F \<and> B \<in> F \<longrightarrow> A \<inter> B \<in> F) \<and> (\<forall>A B. A \<in> F \<and> A \<subseteq> B \<and> B \<subseteq> carrier\<^bsub>T\<^esub> \<longrightarrow> B \<in> F)}"
monom_eq_iff: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "(Polynomial.monom a n = Polynomial.monom b n) = (a = b)"
rqs_cost_exp_leI: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "rqs_cost_exp m \<le> rqs_cost_exp n"
tendsto_mult_left_iff: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F = (f \<longlongrightarrow> l) F"
atLeast_eq_UNIV_iff: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order_bot bot less_eq less" shows "(ord.atLeast less_eq x = UNIV) = (x = bot)"
simps: fixes z :: "name"   and t :: "tm"   and r :: "tm" assumes "atom z \<sharp> (t, r)" shows "NotInDom t r = SyntaxN.All z (Neg (HPair t (Var z) IN r))"
fbrelation_vinsertI: fixes a :: "V"   and b :: "V"   and aa :: "V"   and ba :: "V" shows "fbrelation (vinsert [a, b]\<^sub>\<circ> (set {[aa, ba]\<^sub>\<circ>}))"
lift_CFGExit: 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" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "CFGExit src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
jozsa_algo_eval_def: shows "jozsa_algo_eval \<equiv> prob0_fst_qubits n jozsa_algo"
assms_4: shows "fst ` set txs \<subseteq> atrm"
select_convs_4: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>G = goal_of"
power_poly_f_mod_i: fixes m :: "'i list \<Rightarrow> 'i list"   and m' :: "'a mod_ring poly"   and f :: "'i list"   and f' :: "'a mod_ring poly"   and n :: "nat" assumes "(poly_rel ===> poly_rel) m (\<lambda>x'. x' mod m')"   and "poly_rel f f'" shows "poly_rel (power_poly_f_mod_i ff_ops m f n) (power_poly_f_mod m' f' n)"
ideD_1: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "arr a"
derived_is_design: assumes "\<And>b. b \<in># remove1_mset bl \<B> \<Longrightarrow> 0 < bl |\<inter>| b" shows "design bl \<B>\<^sup>D"
vec_first_carrier: fixes v :: "'a vec"   and n :: "nat" shows "vec_first v n \<in> carrier_vec n"
intensional_simps_1: fixes x :: "'a \<Rightarrow> 'b" shows "x = x \<equiv> #True"
stream_ap: fixes f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "stream (f \<diamondop> x) = stream f \<diamondop> stream x"
perm: shows "filter ((\<noteq>) (0::'a)) (dense_list_of_pdevs X) <~~> map snd (list_of_pdevs X)"
condition2_part2: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f" assumes "echelon_form_upt_k A k"   and "a < b"   and "\<forall>m>mod_type_class.from_nat (nrows A). A $ m $ mod_type_class.from_nat k = (0::'d)"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) = nrows A" shows "(LEAST n. A $ a $ n \<noteq> (0::'d)) < (LEAST n. A $ b $ n \<noteq> (0::'d))"
irreflexive_complement_reflexive: fixes x :: "'a" shows "irreflexive x = reflexive (- x)"
Cons_1: fixes ap :: "'a list" assumes "length ys \<le> length ap" shows "zip ys ap = zip ys (take (length ys) ap)"
triangular_column_def: fixes j :: "nat"   and A :: "'a mat" shows "triangular_column j A \<equiv> \<forall>i>j. i < dim_row A \<longrightarrow> A $$ (i, j) = (0::'a)"
is_chain_extend: fixes S :: "'b fm set"   and C :: "'b fm set set"   and f :: "nat \<Rightarrow> 'b fm" shows "is_chain (extend S C f)"
countable_INT'': fixes \<Omega> :: "'b set"   and M :: "'b set set"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'b set" assumes "sigma_algebra \<Omega> M"   and "UNIV \<in> M"   and "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M" shows "\<Inter> (F ` I) \<in> M"
attracted_strategy_step: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set"   and v0 :: "'a" assumes "G'.strategy p \<sigma>"   and "G'.strategy_attracts p \<sigma> A W"   and "\<not> G'.deadend v0"   and "v0 \<in> A - W"   and "v0 \<in> G'.VV p" shows "\<sigma> v0 \<in> A \<union> W"
LC: shows "\<forall>ts'. ts \<^bold>\<Rightarrow> ts' \<longrightarrow> (\<forall>x. (\<exists>xa. xa \<noteq> x \<and> (\<exists>va u. move ts ts' v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts' x = ext v \<and> restrict v (clm ts') x = lan v \<and> |lan v| = 1) \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (res ts') xa = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (clm ts') xa = lan v \<and> |lan v| = 1)) \<and> True)) \<and> True))) \<longrightarrow> (\<forall>ts'a. ts' \<^bold>\<midarrow>r( x ) \<^bold>\<rightarrow> ts'a \<longrightarrow> False))"
add_diff_inverse: fixes a :: "'g"   and b :: "'g" assumes "a \<le> b" shows "a + (b - a) = b"
dom_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "local.dom f = (if arr f then if f = One then One else Zero else null)"
trees_cyl_sets_in_space: fixes M :: "'d measure" shows "trees_cyl ` trees (sets M) \<subseteq> Pow (trees (space M))"
eq_iff: fixes a :: "('x, nat) pp"   and b :: "('x, nat) pp" shows "(a = b) = (ord b a \<and> ord a b)"
blinding_of_on_list: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "blinding_of_on A h bo" shows "blinding_of_on {x. set x \<subseteq> A} (map h) (list_all2 bo)"
labeling_consistent: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> R1" shows "s1.a.L a = s1.b.L b"
lconjby_add: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus (plus (plus x y) z) (plus x y) = minus (plus x (minus (plus y z) y)) x"
of_int_0_le_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (ring_1.of_int one plus zero minus z) = (0 \<le> z)"
ideal_ring_multiple: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e"   and r :: "'e" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I"
prv_dsjR_tru: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (dsj tru \<phi>) tru)"
iso_def: fixes G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
emeasure_has_parametrized_subprob_density: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and x :: "'a"   and X :: "'b set" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M"   and "X \<in> sets R" shows "emeasure (N x) X = set_nn_integral R X (f x)"
lossless_resourceD: fixes \<I> :: "('a, 'b) \<I>"   and res :: "('a, 'b) resource"   and a :: "'a" assumes "lossless_resource \<I> res"   and "a \<in> outs_\<I> \<I>" shows "lossless_spmf (run_resource res a) \<and> (\<forall>(x, res')\<in>set_spmf (run_resource res a). lossless_resource \<I> res')"
Min_in: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
balance_left_rbt_greater: fixes k :: "'a"   and a :: "('a, 'c) rbt"   and b :: "('a, 'c) rbt"   and x :: "'a"   and t :: "'c" assumes "k \<guillemotleft>| a"   and "k \<guillemotleft>| b"   and "k < x" shows "k \<guillemotleft>| balance_left a x t b"
leof_add_nofailI: fixes m :: "'e nres"   and m' :: "'e nres" assumes "nofail m \<Longrightarrow> m \<le>\<^sub>n m'" shows "m \<le>\<^sub>n m'"
RTranCl_subset2: fixes s :: "'c"   and g :: "'c"   and f :: "'c \<Rightarrow> 'c list"   and h :: "'c \<Rightarrow> 'c list" assumes "s [f]\<rightarrow>* g"   and "\<And>g. s [f]\<rightarrow>* g \<Longrightarrow> set (f g) \<subseteq> set (h g)" shows "s [h]\<rightarrow>* g"
ex_Suc_conv: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>i<Suc n. P i) = (P 0 \<or> (\<exists>i<n. P (Suc i)))"
l2_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "l2_step4 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> Insec A B (hmac \<langle>Number 1, Exp Gen (NonceF (Rb $ ny)), gnx, Messages.Agent A, Messages.Agent B\<rangle> (hmac Ni (NonceF (Rb $ nr)))) \<in> chan s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp2 s\<rparr>}"
wt0_cong: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and \<zeta> :: "('b, 'a) hd"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and B :: "'a pvar \<Rightarrow> zhmultiset" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "\<forall>x\<in>vars_hd \<zeta>. A (PWt x) = B (PWt x)" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt0 wt_sym \<zeta>)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt0 wt_sym \<zeta>))"
pointbased_mk_regular2: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "pointbased (mk_regular2 P F)"
globally_release: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n G\<^sub>n (\<phi> R\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"
assms_3: shows "x \<noteq> 0\<^sub>h"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (poly_ring R); y \<in> carrier (poly_ring R)\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>poly_ring R\<^esub> y = y \<otimes>\<^bsub>poly_ring R\<^esub> x" shows "Group.comm_monoid (poly_ring R)"
prod_mset_subset_imp_dvd: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subseteq># B" shows "prod_mset A dvd prod_mset B"
Rep_uint8_numeral: fixes n :: "num" shows "Rep_uint8 (numeral n) = numeral n"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "vwalk_to_vpath (x # xs) = (if x \<in> set xs then vwalk_to_vpath (dropWhile (\<lambda>y. y \<noteq> x) xs) else x # vwalk_to_vpath xs)"