hcju/isabelleps huggingface dataset:
tr_par_sem_equiv: fixes D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(l, t, s)\<in>set D. (fv t \<union> fv s) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "ground M"   and "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<lbrakk>M; set (unlabel D) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>; unlabel A\<rbrakk>\<^sub>s \<I> = (\<exists>B\<in>set (tr\<^sub>p\<^sub>c A D). \<lbrakk>M; unlabel B\<rbrakk>\<^sub>d \<I>)"
inject_3: fixes x2 :: "tyvrs"   and x3 :: "ty"   and x1 :: "ty"   and y2 :: "tyvrs"   and y3 :: "ty"   and y1 :: "ty" shows "((\<forall>x2<:x1. x3) = (\<forall>y2<:y1. y3)) = ([x2].x3 = [y2].y3 \<and> x1 = y1)"
hm_is_it_def: fixes m :: "'c \<Rightarrow> 'd option"   and ht :: "('c, 'd) hashtable"   and m' :: "'c \<Rightarrow> 'd option"   and it :: "nat \<times> ('c \<times> 'd) list \<times> ('c, 'd) hashtable" shows "hm_is_it m ht m' it \<equiv> \<exists>\<^sub>Al l'. hm_is_it' l ht l' it * \<up> (map_of (concat l) = m \<and> map_of l' = m')"
dg_prod_2_dg_op_dg_Arr: 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>Arr\<rparr> = (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
ba1_complement: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> - y = z \<squnion> - z) = (x \<squnion> y = x)"
cd_swap: fixes x :: "nat"   and m :: "nat"   and n :: "nat" shows "cd x m n = cd x n m"
length_Basis_list: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Basis_list :: "'e list"   and eucl_down :: "int \<Rightarrow> 'e \<Rightarrow> 'e"   and eucl_truncate_down :: "nat \<Rightarrow> 'e \<Rightarrow> 'e"   and eucl_truncate_up :: "nat \<Rightarrow> 'e \<Rightarrow> 'e" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "length Basis_list = card Basis"
antisym_init_seg_of: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r initial_segment_of s"   and "s initial_segment_of r" shows "r = s"
outstanding_refs_takeWhileD: fixes x :: "nat"   and P :: "'b memref \<Rightarrow> bool"   and P' :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" assumes "x \<in> outstanding_refs P (takeWhile P' sb)" shows "x \<in> outstanding_refs P sb"
not_empty_eq_Ici_eq_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b" assumes "class.preorder less_eq less" shows "{} \<noteq> ord.atLeast less_eq l"
P6_a: fixes x :: "'a"   and y :: "'a" shows "x l\<rightarrow> y r\<rightarrow> x = (1::'a)"
m_def: shows "m = Cn 1 r_amalgamate [at0, at1]"
h_q2_copr: shows "(\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> (\<forall>q. List.member (drop (length qs div 2) qs) q \<longrightarrow> coprime p q)"
split_computation: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c\<^sub>f :: "('a, 'b, 'c) com"   and s\<^sub>f :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c\<^sub>f, s\<^sub>f)"   and "\<not> final (c, s)"   and "final (c\<^sub>f, s\<^sub>f)" shows "\<exists>c' s'. \<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s') \<and> \<Gamma>\<turnstile> (c', s') \<rightarrow>\<^sup>* (c\<^sub>f, s\<^sub>f)"
antisymmetric: fixes x :: "'a"   and y :: "'a" assumes "- x \<le> - y"   and "- y \<le> - x" shows "- x = - y"
luniq: shows "\<omega>_limit_set x \<inter> {a<--<b} = {l}"
j: shows "j_to_jb_index n_as i = (bi, li)"
star_test_export_left: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "x \<cdot> p \<le> p \<cdot> x \<longrightarrow> x\<^sup>\<star> \<cdot> p = p \<cdot> (x \<cdot> p)\<^sup>\<star>"
di: fixes c' :: "'a set set" assumes "\<And>S T. S \<in> c \<and> T \<in> c' \<Longrightarrow> disjnt S T" shows "disjnt (\<Union> c) (\<Union> c')"
cong_prime_prod_zero_int: fixes a :: "int"   and b :: "int"   and p :: "int" assumes "[a * b = 0] (mod p)"   and "prime p" shows "[a = 0] (mod p) \<or> [b = 0] (mod p)"
of_int_eq_1_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = one) = (z = 1)"
finite_dimensional_vector_space_pair_1_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" shows "finite_dimensional_vector_space_pair_1_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
inverse_le_imp_le: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less zero a" shows "less_eq b a"
comp_in_homI: fixes f :: "'s"   and a :: "'s"   and b :: "'s"   and g :: "'s"   and c :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
to_fun_one: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_fun \<one>\<^bsub>P\<^esub> a = \<one>"
cobounded1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<subseteq># a \<union># b"
circ_sup_upper_bound: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<star>"   and "y \<le> z\<^sup>\<star>" shows "x \<squnion> y \<le> z\<^sup>\<star>"
times_prod'_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "times_prod' x y = (fst x \<cdot> fst y, snd x \<cdot> snd y)"
aff_dim_empty: fixes S :: "'c set" shows "(S = {}) = (aff_dim S = - 1)"
induct: fixes x1 :: "('e, 'f, 'g) Tree"   and x2 :: "('e, 'f, 'g) Tree"   and P :: "('e, 'f, 'g) Tree \<Rightarrow> ('e, 'f, 'g) Tree \<Rightarrow> bool" assumes "(x1, x2) \<in> Tree_wf"   and "\<And>t tset. t \<in> set_bset tset \<Longrightarrow> P t (tConj tset)"   and "\<And>t. P t (tNot t)"   and "\<And>t \<alpha>. P t (tAct \<alpha> t)" shows "P x1 x2"
pr1_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "cod (hfsetcat.pr1 a b) = a"
le: shows "\<forall>\<^sub>F n in F. C \<le> X n"
elims_2: fixes x :: "nat"   and xa :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and xb :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" assumes "xa \<rightarrow>\<^bsub>x\<^esub> xb" obtains c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" where "x = 0" and "xa = c__" and "xb = c'__" and "c__ = c'__"   | n :: "nat" and c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" where "x = Suc n" and "xa = c" and "xb = c'" and "\<exists>c''. c \<rightarrow>\<^bsub>n\<^esub> c'' \<and> c'' \<rightarrow> c'"
simps: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and ISeqs :: "'d list set" shows "LS\<^sub>i\<^sub>n M q ISeqs = {xs || ys |xs ys. xs \<in> ISeqs \<and> length xs = length ys \<and> xs || ys \<in> LS M q}"
open_iff: fixes m :: "'a set" shows "(m open) = (m = {})"
ubs_def: fixes R :: "'a set"   and S :: "'a set" shows "ubs R S = Collect (isUb R S)"
empty: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state" shows "evaluate_list ck env s [] (s, Rval [])"
simps: fixes i :: "nat"   and j :: "nat" shows "\<mu>' i j = (if j = 0 then fs ! i \<bullet> fs ! j else \<mu>' (j - 1) (j - 1) * (fs ! i \<bullet> fs ! j) - \<sigma>s (j - 1) i j)"
int_pow_mult: fixes x :: "'c"   and i :: "int"   and j :: "int" assumes "x \<in> carrier H" shows "x [^]\<^bsub>H\<^esub> (i + j) = x [^]\<^bsub>H\<^esub> i \<otimes>\<^bsub>H\<^esub> x [^]\<^bsub>H\<^esub> j"
Assign': fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x :: "char list"   and a :: "aexp"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<forall>s. eSuc (Q (s[a/x])) \<le> P s" shows "\<turnstile>\<^sub>2\<^sub>' {P} x ::= a {Q}"
sin_Arcsin: fixes z :: "complex" shows "sin (Arcsin z) = z"
imp_nfoldli_def: fixes l :: "'f list"   and c :: "'e \<Rightarrow> bool Heap"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e Heap"   and s :: "'e" shows "imp_nfoldli l c f s \<equiv> heap.fixp_fun (\<lambda>D (l, s). case l of [] \<Rightarrow> return s | x # ls \<Rightarrow> c s \<bind> (\<lambda>b. if b then f x s \<bind> (\<lambda>s'. D (ls, s')) else return s)) (l, s)"
lm: fixes j :: "nat" assumes "leftmost j a" shows "j \<le> i"
deg_pp_zero: shows "deg_pp 0 = (0::'a)"
multiplicative_function_axioms: shows "multiplicative_function (ind squarefree)"
simps_3: fixes n :: "nat"   and x :: "nat itrm"   and y :: "nat itrm" shows "unlift_vars n (x \<diamondop> y) = unlift_vars n x \<degree> unlift_vars n y"
GrD1: fixes x :: "'d"   and fx :: "'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "x \<in> A"
invar_shift: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" assumes "nf_invar' c src dst PRED C N d"   and "C = {}" shows "nf_invar c src dst PRED N {} (Suc d)"
tries_of_list_def: fixes key :: "'b \<Rightarrow> 'a list"   and vs :: "'b list" shows "tries_of_list key vs = inserts_tries key vs (Trie None [])"
FrameStatEqTrans: fixes F :: "'b frame"   and G :: "'b frame"   and H :: "'b frame" assumes "F \<simeq>\<^sub>F G"   and "G \<simeq>\<^sub>F H" shows "F \<simeq>\<^sub>F H"
direct_decomp_is_basis: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "is_basis A (\<Union> (some_basis ` set ss))"
right_uniqueD: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and z :: "'c" assumes "right_unique A"   and "A x y"   and "A x z" shows "y = z"
merge_sorted_wrt_partitions_between': fixes lo :: "nat"   and hi :: "nat"   and p :: "nat"   and xs :: "'c list"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "lo \<le> hi"   and "lo < p"   and "p < hi"   and "hi < length xs"   and "isPartition_wrt R xs lo hi p"   and "sorted_sublist_wrt R xs lo (p - 1)"   and "sorted_sublist_wrt R xs (p + 1) hi"   and "\<forall>i j. lo \<le> i \<and> i < p \<and> p < j \<and> j \<le> hi \<longrightarrow> R (xs ! i) (xs ! j)" shows "sorted_sublist_wrt R xs lo hi"
one_eq_divide_iff: fixes a :: "'a"   and b :: "'a" shows "((1::'a) = a / b) = (b \<noteq> (0::'a) \<and> a = b)"
bisimTransitive: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, Q, R) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, P, R) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
mojmir_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" shows "mojmir \<Sigma> \<delta> q\<^sub>0 w F \<equiv> semi_mojmir \<Sigma> \<delta> q\<^sub>0 w \<and> mojmir_axioms \<delta> F"
norm_eq_zero: 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 x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x = 0) = (x = zero)"
eucl_sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "sup x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (sup (inner x i) (inner y i)) i) Basis"
AR: shows "b_graph.is_acc_run (degeneralize_ext m) r"
hfsynth_ID_R_def: fixes uu :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and uv :: "'a" shows "hfsynth_ID_R uu uv \<equiv> True"
set_intros_6: fixes yk :: "'a"   and x22 :: "('b, 'a) tm"   and x21 :: "('b, 'a) tm" assumes "yk \<in> vars x22" shows "yk \<in> vars (App x21 x22)"
top_on_sup: fixes o1 :: "(char list \<Rightarrow> 'av) option"   and X :: "char list set"   and o2 :: "(char list \<Rightarrow> 'av) option" assumes "top_on\<^sub>o o1 X"   and "top_on\<^sub>o o2 X" shows "top_on\<^sub>o (o1 \<squnion> o2) X"
list_all_simps_2: fixes P :: "'a \<Rightarrow> bool" shows "list_all P [] = True"
induct: fixes P :: "assn \<Rightarrow> bool"   and a0 :: "assn" assumes "P ANil"   and "\<And>v e as. P as \<Longrightarrow> P (ACons v e as)" shows "P a0"
CFG_Construct_transfer: fixes G :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and \<alpha>e1 :: "'f \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>e2 :: "'g \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>n1 :: "'f \<Rightarrow> 'h list"   and \<alpha>n2 :: "'g \<Rightarrow> 'h list"   and invar1 :: "'f \<Rightarrow> bool"   and invar2 :: "'g \<Rightarrow> bool"   and inEdges1 :: "'f \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and inEdges2 :: "'g \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and Entry1 :: "'f \<Rightarrow> 'h"   and Entry2 :: "'g \<Rightarrow> 'h"   and defs1 :: "'f \<Rightarrow> 'h \<Rightarrow> 'j set"   and defs2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'j set"   and uses1 :: "'f \<Rightarrow> 'h \<Rightarrow> 'j set"   and uses2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'j set" assumes "right_total G"   and "(G ===> (=)) \<alpha>e1 \<alpha>e2"   and "(G ===> (=)) \<alpha>n1 \<alpha>n2"   and "(G ===> (=)) invar1 invar2"   and "(G ===> (=)) inEdges1 inEdges2"   and "(G ===> (=)) Entry1 Entry2"   and "(G ===> (=)) defs1 defs2"   and "(G ===> (=)) uses1 uses2" shows "CFG_Construct \<alpha>e1 \<alpha>n1 invar1 inEdges1 Entry1 defs1 uses1 \<longrightarrow> CFG_Construct \<alpha>e2 \<alpha>n2 invar2 inEdges2 Entry2 defs2 uses2"
mono_neutral_left: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'c)" shows "sum g S = sum g T"
inheritedPropagateEq: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and delta :: "'b" assumes "inherited subs P"   and "fans subs"   and "\<not> terminal subs delta" shows "P (tree subs delta) = (\<forall>sigma\<in>subs delta. P (tree subs sigma))"
isCont_minus: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "isCont f a" shows "isCont (\<lambda>x. - f x) a"
inferences_between_def: fixes CC :: "'a literal multiset set"   and C :: "'a literal multiset" shows "ord_FO_resolution.inferences_between CC C = {\<gamma> \<in> ord_FO_\<Gamma> S. infer_from (CC \<union> {C}) \<gamma> \<and> C \<in># prems_of \<gamma>}"
to_fun_monic_monom: fixes R :: "('a, 'b) ring_scheme"   and b :: "'a"   and n :: "nat" assumes "UP_cring R"   and "b \<in> carrier R" shows "UP_cring.to_fun R (monom (UP R) \<one>\<^bsub>R\<^esub> n) b = b [^]\<^bsub>R\<^esub> n"
image_mset_add_mset: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and M :: "'c multiset" shows "image_mset f (a, M) = f a, image_mset f M"
conv: fixes f :: "'a fds" assumes "fds_abs_converges f s"   and "fds_nth f (Suc 0) = (0::'a)" shows "fds_abs_converges (fds_exp f) s"
floor_le_one: fixes x :: "'b" shows "(\<lfloor>x\<rfloor> \<le> 1) = (x < (2::'b))"
assms_2: shows "\<exists>\<sigma>\<in>states c. \<sigma>' = \<sigma>(v := e \<sigma>)"
cExt: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', R, S) \<in> Rel" shows "(\<Psi>' \<otimes> \<Psi>'', R, S) \<in> Rel"
extract_src_dst_ips_length: fixes rs :: "'b simple_rule list"   and acc :: "('b word \<times> nat) list" shows "length (extract_src_dst_ips rs acc) = 2 * length rs + length acc"
ternary_to_bool_Some: fixes t :: "ternaryvalue"   and X :: "bool" shows "(ternary_to_bool t = Some X) = (t = TernaryTrue \<and> X = True \<or> t = TernaryFalse \<and> X = False)"
eq_reflection: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "x \<equiv> y"
rel2p_mset: fixes A :: "('a \<times> 'b) set" shows "rel2p (\<langle>A\<rangle>mset_rel) = rel_mset (rel2p A)"
f1: shows "\<not> (fst instr = shift_type SLL \<and> get_operand_w5 (snd instr ! 3) \<noteq> 0)"
sym_def: fixes r :: "('d \<times> 'd) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
mult_2_right: fixes z :: "'b hrel" shows "z ;; numeral.numeral II (\<sqinter>) (num.Bit0 num.One) = z \<sqinter> z"
id_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(A ===> A) id id"
ev: shows "\<forall>\<^sub>F p in sequentially. (\<forall>q\<in>P. \<bar>\<beta> q\<bar> < int p) \<and> cmod (\<Prod>\<alpha>\<in>Roots'. of_int (l ^ n) * prod ((-) \<alpha>) (Roots' - {\<alpha>})) < real p"
snd_Gauss_Jordan_in_ij_det_P_is_snd_Gauss_Jordan_in_ij_PA: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e"   and P :: "(('d, 'f) vec, 'f) vec" shows "snd (Gauss_Jordan_in_ij_det_P A i j) = snd (Gauss_Jordan_in_ij_PA (P, A) i j)"
axioms_1: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry" shows "graph_path \<alpha>e \<alpha>n invar inEdges'"
lens_indep_override_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y" shows "(X \<bowtie> Y) = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "find_index P (x # xs) = (if P x then 0 else find_index P xs + 1)"
subset: shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (r # sb) \<subseteq> A"
nth_slice: fixes n :: "nat"   and w :: "'c word"   and m :: "nat" shows "slice n w !! m = (w !! (m + n) \<and> m < LENGTH('b))"