hcju/isabelleps huggingface dataset:
prems_3: fixes x :: "'a node \<times> 'a node set" assumes "((\<mu> R\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst x) \<union> new (fst x) \<or> \<eta> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nd'\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>r. \<forall>nd\<in>snd r. \<forall>nd'\<in>snd r. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd')"
predicate2I_obj: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x y. P x y \<longrightarrow> Q x y" shows "P \<le> Q"
flatten_pi_2_intersection: fixes N :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "N \<inter> subformulas\<^sub>\<nu> \<phi> \<subseteq> S" shows "\<phi>[N \<inter> S]\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2"
a: shows "integral\<^sup>N M (induced_function A f) = (\<Sum>n. set_nn_integral M (D n) (induced_function A f))"
red_labeled_clauses: fixes C :: "'f"   and N :: "('f \<times> 'l) set"   and L :: "'l" assumes "C \<in> no_labels.Red_F_\<G>_empty (fst ` N) \<or> (\<exists>C'\<in>fst ` N. C' \<prec>\<cdot> C) \<or> (\<exists>(C', L')\<in>N. L' \<sqsubset>L L \<and> C' \<preceq>\<cdot> C)" shows "(C, L) \<in> Red_F_\<G> N"
dickson_lessD2: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_less d m s t" shows "d t \<le> m"
jumpF_add: fixes p :: "real poly"   and q :: "real poly" assumes "poly p x = 0"   and "coprime p q" shows "jumpF_polyL q p x = jumpF_polyL 1 (q * p) x"
homI: fixes G :: "('e, 'f) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and H :: "('g, 'h) monoid_scheme" assumes "\<And>x. x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> hom G H"
flip: shows "bounded_bilinear (\<lambda>x y. blinop_apply y x)"
att_def: fixes cf :: "CAttId" shows "att cf = (case cf of UndoCounter'UndoCounter'save \<Rightarrow> UndoCounter'save | CounterImpl'Counter'dummy \<Rightarrow> Counter'dummy | UndoCounter'Counter'dummy \<Rightarrow> Counter'dummy | _ \<Rightarrow> CounterImpl'value)"
two_powr_minus_nat_float: fixes i :: "nat" shows "2 powr - real i \<in> float"
set_interval_defs_2: fixes u :: "'a" shows "{..u} \<equiv> {x. x \<le> u}"
empty_eq_union: fixes M :: "'c multiset"   and N :: "'c multiset" shows "({#} = M + N) = (M = {#} \<and> N = {#})"
boundedI: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b"   and "a \<subseteq># c" shows "a \<subseteq># b \<inter># c"
ring_simprules_3: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
essentially_equalE: fixes fs1 :: "'a list"   and fs2 :: "'a list" assumes "essentially_equal G fs1 fs2" obtains fs1' :: "'a list" where "fs1 <~~> fs1'" and "fs1' [\<sim>] fs2"
a_minus_def: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
aext_imp: fixes P :: "'b upred"   and Q :: "'b upred"   and x :: "'b \<Longrightarrow> 'a" shows "(P \<Rightarrow> Q) \<oplus>\<^sub>p x = (P \<oplus>\<^sub>p x \<Rightarrow> Q \<oplus>\<^sub>p x)"
DESTRimpl_rule1: fixes s :: "'s"   and ni :: "'ni" assumes "I s"   and "(ni, Trueif) \<in> R s" shows "ospec (DESTRimpl ni s) (\<lambda>r. r = TD)"
hcomp_arr_obj_2: fixes \<mu> :: "'a"   and a :: "'a" assumes "arr \<mu>"   and "obj a"   and "src \<mu> = a" shows "\<r>[cod \<mu>] \<cdot> (\<mu> \<star> a) \<cdot> \<r>\<^sup>-\<^sup>1[local.dom \<mu>] = \<mu>"
eq_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and M :: "'b multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z M = fold_mset f z M"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a" shows "(case LNil of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f1"
filter_M_holds_for_result: fixes h :: "'l"   and P :: "'n \<Rightarrow> ('l, 'm, bool) prog"   and xs :: "'n list"   and ys :: "'n list"   and x :: "'n" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "x \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> P x \<rightarrow>\<^sub>r True"
hom_one: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
cont_vec_field: shows "\<forall>s\<in>S. continuous_on T (\<lambda>t. f t s)"
comp_null_2: fixes f :: "'a \<times> 'a" shows "H.VV.comp f H.VV.null = H.VV.null"
has_domain_char: fixes f :: "'c \<times> 'c" shows "(CopxC.domains f \<noteq> {}) = (Cop.arr (fst f) \<and> C.arr (snd f))"
\<tau>moves2_iff: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and es :: "'addr expr1 list"   and pc :: "nat"   and xcp :: "'addr option" shows "\<tau>moves2 P h stk es pc xcp = (pc < length (compEs2 es) \<and> (xcp = None \<longrightarrow> \<tau>instr P h stk (compEs2 es ! pc)))"
index_free_def: fixes P :: "'c multiset \<Rightarrow> 'a"   and i :: "'c" shows "index_free P i = (\<forall>m. i \<in># m \<longrightarrow> P m = \<zero>)"
span_zero: fixes S :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "0 \<in> pmdl S"
gcd_is_Max_divisors_int: fixes n :: "int"   and m :: "int" assumes "n \<noteq> 0" shows "gcd m n = Max {d. d dvd m \<and> d dvd n}"
step: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and a :: "'f"   and \<sigma>' :: "'a"   and p' :: "('b, 'c, 'd, 'e) seqp"   and l' :: "'e" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> p'"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', l')"
simps_12: fixes Vs :: "char list list"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (if (e) e\<^sub>1 else e\<^sub>2) = if (compE\<^sub>1 Vs e) compE\<^sub>1 Vs e\<^sub>1 else compE\<^sub>1 Vs e\<^sub>2"
BINDING_imply_SNa_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)" shows "SNa M"
is_lowest_element_def: fixes x :: "'b"   and S :: "'b set" shows "is_lowest_element x S = (x \<in> S \<and> (\<forall>y\<in>S. y \<le> x \<longrightarrow> y = x))"
equal_pair_count_correct: fixes i :: "nat"   and ps :: "('b \<times> 'b) list" assumes "i < equal_pair_count ps" shows "equal_pair (ps ! i)"
in_rtrancl_on_in_F_1: fixes a :: "'a"   and b :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> rtrancl_on F r" shows "a \<in> F"
less_eq_prod_simp: fixes x1 :: "'a"   and y1 :: "'b"   and x2 :: "'a"   and y2 :: "'b" shows "((x1, y1) \<le> (x2, y2)) = (x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2)"
positively_invariant_iff: fixes M :: "'a set" shows "rev.positively_invariant M = ((\<Union>x\<in>M. rev.flow0 x ` (rev.existence_ivl0 x \<inter> {0..})) \<subseteq> M)"
emeasure_bind_const: fixes M :: "'c measure"   and X :: "'d set"   and N :: "'d measure" assumes "space M \<noteq> {}"   and "X \<in> sets N"   and "subprob_space N" shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X * emeasure M (space M)"
SeqQuoteP_graph_aux_def: shows "SeqQuoteP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqQuoteP_aux l s k sl sl' m n sm sm' sn sn' t u. x1 = (t, u, s, k) \<and> x2 = LstSeqP s k (HPair t u) AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (Var sl EQ Zero AND Var sl' EQ Zero OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn'))))))))))) \<and> atom l \<sharp> (s, k, sl, sl', m, n, sm, sm', sn, sn') \<and> atom sl \<sharp> (s, sl', m, n, sm, sm', sn, sn') \<and> atom sl' \<sharp> (s, m, n, sm, sm', sn, sn') \<and> atom m \<sharp> (s, n, sm, sm', sn, sn') \<and> atom n \<sharp> (s, sm, sm', sn, sn') \<and> atom sm \<sharp> (s, sm', sn, sn') \<and> atom sm' \<sharp> (s, sn, sn') \<and> atom sn \<sharp> (s, sn') \<and> atom sn' \<sharp> s)"
corr_2: shows "(\<N>\<lbrace>\<Gamma>\<rbrace>) f|` domA \<Gamma> \<sqsubseteq> (\<N>\<lbrace>\<Delta>\<rbrace>) f|` domA \<Gamma>"
mset_list_tree: fixes l :: "'c list" shows "Heap.multiset (of_list_tree l) = mset l"
rbt_del_rbt_less: fixes v :: "'a"   and lt :: "('a, 'c) rbt"   and x :: "'a" assumes "lt |\<guillemotleft> v" shows "rbt_del x lt |\<guillemotleft> v"
map_comp0: fixes g1 :: "'r \<Rightarrow> 'o"   and f1 :: "'l \<Rightarrow> 'r"   and g2 :: "'s \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 's"   and g3 :: "'t \<Rightarrow> 'q"   and f3 :: "'n \<Rightarrow> 't" shows "F'map (g2 \<circ> f2) (g3 \<circ> f3) (g1 \<circ> f1) = F'map g2 g3 g1 \<circ> F'map f2 f3 f1"
SOC_wrt_preorder_iff_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel)) \<and> (\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel))) \<and> preorder TRel \<and> strong_reduction_simulation TRel Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel) \<and> preorder Rel \<and> strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel)))"
sub_subst: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and \<rho> :: "'b \<Rightarrow> ('a, 'b) tm" assumes "sub s t" shows "sub (subst \<rho> s) (subst \<rho> t)"
split_mult_neg_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b" shows "less_eq (times a b) zero"
fbox_g_orbital: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and Q :: "'b \<Rightarrow> bool" shows "|x\<acute>=f & G on U S @ t\<^sub>0] Q = (\<lambda>s. \<forall>X\<in>Sols f U S t\<^sub>0 s. \<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (X \<tau>)) \<longrightarrow> Q (X t))"
y1: shows "(\<Union>xa\<in>R `` {y}. {per_union R x y `` {xa}}) = {per_union R x y `` {y}}"
nat_of_ipv6addr_def: fixes a :: "128 word" shows "nat_of_ipv6addr a = unat a"
dim_span_eq_card_independent: fixes B :: "('a, 'b) vec set" assumes "vec.independent B" shows "vec.dim (vec.span B) = card B"
assms_5: shows "\<forall>x\<in>#arr_mset arr l\<^sub>0 m. \<forall>y\<in>#arr_mset arr m r\<^sub>0. \<not> x < y"
simps_1: fixes p :: "fm" shows "not (NOT p) = p"
integrable_mult_right_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and c :: "real" shows "integrable M (\<lambda>x. f x * c) = (c = 0 \<or> integrable M f)"
Comm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   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 Q :: "('a, 'b, 'c) psi"   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 "\<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 "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "xvec \<sharp>* Q" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"
P7: shows "X \<noteq> P \<and> S \<noteq> X \<and> Q \<noteq> Y \<and> S \<noteq> Y"
inessential_spheremap_2_aux: fixes a :: "'a"   and r :: "real"   and f :: "'a \<Rightarrow> complex" assumes "2 < DIM('a)"   and "continuous_on (sphere a r) f"   and "f ` sphere a r \<subseteq> sphere 0 1" obtains c :: "complex" where "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere 0 1) f (\<lambda>x. c)"
sp_lift_instr: fixes instr :: "('c, 'd, 'fun, 'op, 'opinl) Inca.instr"   and \<Sigma>1 :: "Unboxed.type option list"   and instr' :: "('c, 'd, 'fun, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'fun \<Rightarrow> 'g fundef option" assumes "lift_instr instr \<Sigma>1 = Ok (instr', \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "Subx.sp_instr F instr' \<Sigma>1 = Ok \<Sigma>2"
r_step_Cn: fixes e :: "nat" shows "eval r_step_Cn [e] \<down>= estep_Cn e"
maximalI: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>c. 0 \<le> c \<Longrightarrow> t (\<lambda>_. c) = (\<lambda>_. c)" shows "maximal t"
rev_sing: fixes x :: "'b" shows "rev [x] = [x]"
distinct_1: shows "UMinusOp \<noteq> AbsOp"
transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_fset A {} {||}"
existence_ivl_eq_domain: fixes t0 :: "real"   and x0 :: "('b, 'sz) vec" assumes "t0 \<in> UNIV"   and "x0 \<in> UNIV"   and "\<And>tm tM t x. \<lbrakk>tm \<in> UNIV; tM \<in> UNIV\<rbrakk> \<Longrightarrow> \<exists>M L. \<forall>t\<in>{tm..tM}. \<forall>x\<in>UNIV. norm (\<chi>i. (0::'b)) \<le> M + L * norm x"   and "is_interval UNIV"   and "UNIV = UNIV" shows "ll.existence_ivl t0 x0 = UNIV"
map_seq: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.seq (\<nu> (\<tau>.A.cod f)) (\<tau> f)"
intensional_simps_29: fixes P :: "'b \<Rightarrow> bool" shows "\<not> P \<or> P \<equiv> #True"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod m \<noteq> 0"   and "1 < m" shows "degree_m f = degree f"
t_fourtimes_compile_def: shows "t_fourtimes_compile = tm_of abc_fourtimes @ shift (mopup 1) (length (tm_of abc_fourtimes) div 2)"
index_transpose_mat_3: fixes A :: "'b mat" shows "dim_col A\<^sup>T = dim_row A"
fold_congs_1: fixes r :: "('c, 'd) Graph_scheme"   and r' :: "('c, 'd) Graph_scheme"   and v' :: "'c set"   and f :: "'c set \<Rightarrow> 'c set"   and f' :: "'c set \<Rightarrow> 'c set" assumes "r = r'"   and "V\<^bsub>r'\<^esub> = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "verts_update f r = verts_update f' r'"
list_emb_Cons: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "list_emb P xs ys" shows "list_emb P xs (y # ys)"
dg_prod_2_dg_op_dg_Obj: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "(\<AA> \<times>\<^sub>D\<^sub>G op_dg \<BB>)\<lparr>Obj\<rparr> = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
less_iff_diff_less_0: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a b = less (minus a b) zero"
Seq_append: fixes s1 :: "hf"   and k1 :: "hf"   and s2 :: "hf"   and k2 :: "hf" assumes "Seq s1 k1"   and "Seq s2 k2" shows "Seq (seq_append k1 s1 s2) (k1 @+ k2)"
assms_3: fixes x :: "'a" assumes "P x" shows "G (proj \<phi> x) = f x"
assms: shows "(sup (\<rightarrow>\<^sub>\<beta>) (\<rightarrow>\<^sub>\<eta>))\<^sup>*\<^sup>* s t"
ms_closure_embed': fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool" assumes "is_expansion_aux xs f basis" shows "ms_closure P xs f basis"
neq_Nil_conv: fixes xs :: "'b list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
outs_\<I>_uniform: fixes A :: "'a set"   and B :: "'b set" shows "outs_\<I> (\<I>_uniform A B) = (if B = {} then {} else A)"
cases: fixes R :: "('a, 'b) ring_scheme"   and a1 :: "nat"   and a2 :: "'a set"   and a3 :: "'a set" assumes "ring R"   and "ring.dimension R a1 a2 a3" obtains K :: "'a set" where "a1 = 0" and "a2 = K__" and "a3 = {\<zero>\<^bsub>R\<^esub>}"   | v :: "'a" and E :: "'a set" and n :: "nat" and K :: "'a set" where "a1 = Suc n" and "a2 = K" and "a3 = ring.line_extension R K v E" and "v \<in> carrier R" and "v \<notin> E" and "ring.dimension R n K E"
del_arc_in: fixes a :: "'b" assumes "a \<notin> arcs G" shows "G.del_arc a = G"
fix_const: fixes c :: "'a" shows "(\<mu> x. c) = c"
y_2: shows "\<forall>x\<ge>y. \<psi> j x \<up>"
IfT: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and pc :: "int"   and NEXT :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and os :: "Val list"   and s :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "get_ins M l = Some (iftrue pc)"   and "NEXT = (os, s, h)" shows "(M, l, (TRUE # os, s, h), 1, pc, NEXT) \<in> Step"
row_bra_vec: fixes v :: "complex Matrix.vec" shows "Matrix.row (bra_vec v) 0 = Matrix.vec (dim_vec v) (\<lambda>i. cnj (v $ i))"
gen_image_def: fixes it1 :: "'h \<Rightarrow> ('i \<Rightarrow> bool) \<Rightarrow> ('j \<Rightarrow> 'k \<Rightarrow> 'l) \<Rightarrow> 'm \<Rightarrow> 'g"   and emp2 :: "'m"   and ins2 :: "'n \<Rightarrow> 'k \<Rightarrow> 'l"   and f :: "'j \<Rightarrow> 'n"   and s1 :: "'h" shows "gen_image it1 emp2 ins2 f s1 \<equiv> it1 s1 (\<lambda>_. True) (\<lambda>x. ins2 (f x)) emp2"
assms_1: shows "s2 = (x * y + q - s1) mod q"
id_hom: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Id\<^bsub>CC\<^esub> A \<in> Hom\<^bsub>CC\<^esub> A A"
transp_on_def: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" shows "transp_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. P x y \<and> P y z \<longrightarrow> P x z)"
vlrestriction_atD: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" shows "r\<lparr>a\<rparr> = b"
psen_fin_word: fixes s :: "'state"   and w :: "'action list"   and p :: "'process" assumes "s \<in> nodes"   and "path w s"   and "pac p \<inter> set w = {}" shows "psen p (target w s) = psen p s"
all_neq_Inl_ex_eq_Inr: fixes x :: "'c + 'd" assumes "\<forall>l. x \<noteq> Inl l" shows "\<exists>r. x = Inr r"
max_ge_x: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.non_strict_order less_eq" shows "less_eq x (ord.max less_eq x y)"
nat_pow_eone: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)" shows "x [^]\<^bsub>UP Zp\<^esub> 1 = x"
emeasure_finite: fixes A :: "'a set" shows "emeasure M A \<noteq> top"
intros_10: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>"   and "\<Gamma> \<Rightarrow> \<BB> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<AA> \<^bold>\<and> \<BB> # \<Theta>"
t1: shows "space_as_set A \<subseteq> space_as_set B"
inj_on: fixes a :: "nat"   and b :: "nat"   and p :: "'a mpoly" shows "inj_on (swap0 a b) (monomials p)"
reflexive_axioms: shows "reflexive P (\<sqsubseteq>)"
charts_submanifold_def: fixes S :: "'a set" shows "charts_submanifold S = restrict_chart S ` charts"
oracle_encrypt1''_def: shows "oracle_encrypt1'' = (\<lambda>(k_prf, k_upf) b _ (msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> spmf_of_set prf_domain \<bind> (\<lambda>x. let p = prf_fun k_prf x; c = p [\<oplus>] (if b then msg1 else msg0); t = upf_fun k_upf (x @ c) in return_spmf (Some (x, c, t), ())) | False \<Rightarrow> return_spmf (None, ()))"
euclidean_representation_sum_fun: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<lambda>x. comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner (f x) b) b) Basis) = f"
c2: shows "q \<in> set qs \<longrightarrow> q dvd prod_list_var qs"
rotate_inv_eq: fixes k :: "nat"   and xs :: "'b list" shows "rotater k (rotate k xs) = xs \<and> rotate k (rotater k xs) = xs \<and> rotater k (rotate k xs) = xs \<and> rotate k (rotater k xs) = xs"
scaleR2_rep_coll_def: fixes X :: "('d \<times> 'e) set" shows "scaleR2_rep_coll X = sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN ((0, 0), op_empty_coll)) (\<lambda>X. scaleR2_rep X \<bind> (\<lambda>((l, u), Y). RETURN ((l, u), mk_coll Y))) (\<lambda>((l, u), Y) ((l', u'), Y'). RETURN ((inf l' l, sup u' u), Y' \<union> Y)))"
assms_1: shows "foldl1 f1 (map (h1 k i) js) = One"
l_neg: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "\<ominus>\<^bsub>poly_ring R\<^esub> x \<oplus>\<^bsub>poly_ring R\<^esub> x = \<zero>\<^bsub>poly_ring R\<^esub>"