hcju/isabelleps huggingface dataset:
Cons_step1_Cons: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and ys :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "step1 r (y # ys) (x # xs) = (r y x \<and> xs = ys \<or> x = y \<and> step1 r ys xs)"
Class: fixes a :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "Neg a # G \<turnstile> FF" shows "G \<turnstile> a"
rep_eq: fixes x :: "('c, 'd) fmap"   and xa :: "('c, 'd) fmap" shows "($$) (x ++\<^sub>f xa) = ($$) x ++ ($$) xa"
absorb2: fixes a :: "'b word"   and b :: "'b word" assumes "a \<le>s b" shows "signed.max a b = b"
wlp_measure_def: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and WS :: "(complex mat \<Rightarrow> complex mat) list"   and P :: "complex mat" shows "wlp_measure n M WS P = matrix_sum d (\<lambda>k. adjoint (M k) * (WS ! k) P * M k) n"
Beta_plus1_right: fixes y :: "'a"   and x :: "'a" assumes "y \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(x + y) * Beta x (y + (1::'a)) = y * Beta x y"
Red_lcl_add: fixes e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and l\<^sub>0 :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "P,E \<turnstile> \<langle>e,(h, l\<^sub>0 ++ l)\<rangle> \<rightarrow>* \<langle>e',(h', l\<^sub>0 ++ l')\<rangle>"
assms_2: shows "snd (cs (S t i) cid) = Recording"
ev: shows "\<forall>\<^sub>F i in F. f i \<le> a"
mI_gen_in_I: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "\<Sigma>\<^sub>e K (\<lambda>k. Zl_mI K P I k \<cdot>\<^sub>r mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m_zmax_pdsI K n P I\<^esup>) n \<in> I"
encode_problem_forall_step_decoded_plan_is_serializable_ii: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" shows "list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>)) (concat (\<Phi>\<inverse> \<Pi> \<A> t))"
seg_end_bound: fixes i :: "nat" assumes "i < length B" shows "seg_end i \<le> length S"
a_factorgroup_is_group: shows "Group.group (G A_Mod H)"
invertible_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b" shows "invertible_functor A B G \<equiv> (category A \<and> category B) \<and> functor A B G \<and> invertible_functor_axioms A B G"
discrete_topology_closure_of: fixes U :: "'a set"   and S :: "'a set" shows "discrete_topology U closure_of S = U \<inter> S"
vec_to_iarray_minus_nth: fixes i :: "nat"   and A :: "('a, 'b) vec"   and B :: "('a, 'c) vec" assumes "i < IArray.length (vec_to_iarray A)"   and "i < IArray.length (vec_to_iarray B)" shows "(vec_to_iarray A - vec_to_iarray B) !! i = vec_to_iarray A !! i - vec_to_iarray B !! i"
S_quadratic': fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec"   and k :: "real" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q" shows "(proj2_abs (k *\<^sub>R p + q) \<in> S) = (p \<bullet> (M *v p) * k\<^sup>2 + p \<bullet> (M *v q) * 2 * k + q \<bullet> (M *v q) = 0)"
simps_6: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and var :: "nat" shows "liftmap f (Neg \<phi>) var = Neg (liftmap f \<phi> var)"
pred_envirD: fixes p :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and x :: "'f" assumes "pred_envir p f" shows "p (f x)"
simps_2: fixes e :: "'a edge_kind"   and es :: "'a edge_kind list"   and s :: "'a" shows "preds (e # es) s = (pred e s \<and> preds es (transfer e s))"
lambda_system_def: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x\<in>M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
Icc_eq_Icc: fixes l :: "'a word"   and h :: "'a word"   and l' :: "'a word"   and h' :: "'a word" shows "(signed.atLeastAtMost l h = signed.atLeastAtMost l' h') = (l = l' \<and> h = h' \<or> \<not> l \<le>s h \<and> \<not> l' \<le>s h')"
Abinds_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and aexp1 :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and aexp2 :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> aexp1 e = aexp2 e"   and "heap1 = heap2" shows "ArityAnalysis.ABinds aexp1 heap1 = ArityAnalysis.ABinds aexp2 heap2"
o_assoc: fixes f :: "'m \<Rightarrow> 'l"   and g :: "'n \<Rightarrow> 'm"   and h :: "'k \<Rightarrow> 'n" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
is_Eq_def: fixes formula :: "formula" shows "is_Eq formula = (\<exists>x31 x32. formula = formula.Eq x31 x32)"
rbt_insert_def: fixes x :: "'a"   and v :: "'b"   and t :: "('a, 'b) rbt" shows "rbt_insert x v t = paint B (ins x v t)"
linear_injective_rank_eq_ncols: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "linear (*s) (*s) f" shows "inj f = (rank (matrix f) = ncols (matrix f))"
assms_2: shows "A \<in>\<^sub>\<circ> Vset \<alpha>"
set_val_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_val_locs character_data_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
env_map_loc: fixes w :: "'a list" shows "env (map LOC w) = []"
slp_indexl_def: fixes slpl :: "nat"   and i :: "nat" shows "slp_indexl slpl i = slpl - Suc i"
inf_formula_def: fixes x :: "'a formula"   and y :: "'a formula" shows "x \<sqinter> y = Abs_formula (Rep_formula x \<inter> Rep_formula y)"
asymptotics2: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>0 \<le> x"   and "i < k" shows "(1 + hb * inverse (bs ! i) * ln x powr - (1 + e)) powr p * (1 - ln (bs ! i * x + hb * x / ln x powr (1 + e)) powr (- e / 2)) \<le> 1 - ln x powr (- e / 2)"
isUbI: fixes S :: "'a set"   and x :: "'a"   and R :: "'a set" assumes "S *<= x"   and "x \<in> R" shows "isUb R S x"
simps_2: fixes x :: "'a"   and t :: "'a bro" shows "del x (N1 t) = N1 (del x t)"
finite_nonempty_ex_succ_imp_cyclic: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "finite A"   and "A \<noteq> {}"   and "\<forall>x\<in>A. \<exists>y\<in>A. (y, x) \<in> r" shows "\<not> acyclic r"
simps_4: fixes red :: "('f \<times> nat) rgraph"   and black :: "('f, 'd, 'e) lts"   and subs :: "(('f \<times> nat) \<times> 'f \<times> nat) set"   and init_conf :: "('d, 'e) conf"   and confs :: "'f \<times> nat \<Rightarrow> ('d, 'e) conf"   and marked :: "'f \<times> nat \<Rightarrow> bool"   and strengthenings :: "'f \<times> nat \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> bool"   and more :: "'g" shows "init_conf \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = init_conf"
assms_3: fixes C :: "'a set set" assumes "C \<subseteq> M"   and "disjoint C"   and "finite C"   and "\<Union> C \<in> M" shows "f (\<Union> C) = sum f C"
assms: shows "(Some r, sr') \<in> option_while' C B"
casts_ref: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list"   and a :: "nat" assumes "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P \<turnstile> Class C casts Ref (a, Cs) to Ref (a, Ds) "
dim_2: shows "dim_col (subresultant_mat J f g) = degree f - J + (degree g - J)"
lsc_lsc_hull: fixes f :: "'b \<Rightarrow> ereal" shows "lsc (lsc_hull f)"
simps_8: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter (G \<phi>) \<nu> = G \<phi> and af_letter \<phi> \<nu>"
isFinalState_def: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "isFinalState state F0 decisionVars \<equiv> \<nexists>state'. transition state state' F0 decisionVars"
butterfly_eq_proj: fixes x :: "'b" assumes "norm x = 1" shows "selfbutter x = proj x"
2: shows "monom (UP (R\<lparr>carrier := S\<rparr>)) a n x = (if n = x then a else \<zero>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub>)"
ys_def: shows "ys = filter (\<lambda>y. y \<le> x) xs"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'k list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
meet_connection2: fixes x :: "nat"   and y :: "nat" shows "(x \<le> y) = (nat.meet y x = x)"
SLC_commute: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s\<^sub>3 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>3' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r' :: "'a"   and s\<^sub>2 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and s\<^sub>2' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s\<^sub>3 = s\<^sub>3'"   and "substitution.revision_step subst r' s\<^sub>2 s\<^sub>3"   and "substitution.revision_step subst r s\<^sub>2' s\<^sub>3'" shows "s\<^sub>3 \<approx> s\<^sub>3' \<and> (substitution.revision_step subst r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (substitution.revision_step subst r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
subring_Inter: fixes S :: "'a set set" assumes "\<And>I. I \<in> S \<Longrightarrow> subring I R"   and "S \<noteq> {}" shows "subring (\<Inter> S) R"
NDVDJ_def: fixes i :: "int"   and n :: "int"   and s :: "MIR.num" shows "NDVDJ i n s = foldr MIR.conj (map (\<lambda>j. MIR.disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) T"
npowf_exp_1_add: fixes x :: "'a"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x\<^bsub>K\<^esub>\<^bsup>(1 + j)\<^esup> = x \<cdot>\<^sub>r x\<^bsub>K\<^esub>\<^bsup>j\<^esup>"
fresh_9: fixes a :: "atom"   and term1 :: "Syntax.term"   and term2 :: "Syntax.term" shows "a \<sharp> App term1 term2 = (a \<sharp> term1 \<and> a \<sharp> term2)"
qtable_Union: fixes I :: "'a set"   and n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Qi :: "'a \<Rightarrow> 'b option list \<Rightarrow> bool"   and Xi :: "'a \<Rightarrow> 'b option list set"   and Q :: "'b option list \<Rightarrow> bool" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> qtable n A P (Qi i) (Xi i)"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> Q x = (\<exists>i\<in>I. Qi i x)" shows "qtable n A P Q (\<Union> (Xi ` I))"
Dom_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Dom C (free_monoidal_category.norm C t) = monoidal_language.Dom C t"
pdevs_val_coord_pdevs: fixes e :: "nat \<Rightarrow> real"   and i :: "nat" shows "pdevs_val e (coord_pdevs i) = e i"
rjoin\<^sub>\<epsilon>_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and r' :: "'a"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = None" shows "substitution.revision_step subst r s s' = (s' = \<epsilon>)"
sqrt_approx_main_sound: fixes x :: "'a" assumes "(0::'a) < x"   and "\<bar>x\<bar> < x * x" shows "\<bar>x\<bar> < sqrt_approx_main x * sqrt_approx_main x \<and> sqrt_approx_main x * sqrt_approx_main x - \<bar>x\<bar> < \<epsilon>"
orthogonal_gso: shows "orthogonal (map gs.gso [0..<length (map of_int_hom.vec_hom fs)])"
db\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('c, 'd) stateful_strand_step list"   and B :: "('c, 'd) stateful_strand_step list"   and I :: "'d \<Rightarrow> ('c, 'd) Term.term"   and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (A @ B) I D = db'\<^sub>s\<^sub>s\<^sub>t B I (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
set_subset_Univ: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "Hom.set (b, a) \<subseteq> S.Univ"
dominant_term_expands_to: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "trimmed F" shows "f \<sim>[at_top] eval_monom (dominant_term F) basis"
rref_condition3: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. i < i + (1::'d) \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row (i + (1::'d)) A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'d)) $ n \<noteq> (0::'b))"
sat_scale_rat_ns: fixes v :: "nat \<Rightarrow> 'a"   and ns :: "'a ns_constraint"   and f :: "rat" assumes "v \<Turnstile>\<^sub>n\<^sub>s ns" shows "v \<Turnstile>\<^sub>n\<^sub>s (f *R ns)"
h: fixes i :: "nat" shows "(h i, h (Suc i)) \<in> r"
doesnt_have_mode\<^sub>C: fixes dma\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and dma\<^sub>C :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'a \<Rightarrow> 'a set"   and \<C>_vars\<^sub>C :: "'c \<Rightarrow> 'c set"   and eval\<^sub>A :: "((('d \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('d \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and eval\<^sub>C :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'b)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'b)) set"   and var\<^sub>C_of :: "'a \<Rightarrow> 'c"   and x :: "'a"   and mds\<^sub>C :: "Mode \<Rightarrow> 'c set"   and m :: "Mode" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "x \<notin> sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C m" shows "var\<^sub>C_of x \<notin> mds\<^sub>C m"
B: shows "(yp # yps, Y) \<in> futures (c_process step out s\<^sub>0) xps"
ebmovovissifsiddib: shows "(e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>) O b \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
sum_single: fixes A :: "'a set"   and i :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<And>j. \<lbrakk>j \<noteq> i; j \<in> A\<rbrakk> \<Longrightarrow> f j = (0::'b)" shows "sum f A = (if i \<in> A then f i else (0::'b))"
d_def: fixes R :: "('a \<times> 'a set) set" shows "Multirelations.d R \<equiv> {(a, {a}) |a. \<exists>B. (a, B) \<in> R}"
decs_clocked_total: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list" shows "\<exists>res. evaluate_decs True mn env s ds res"
least_le: fixes B :: "'d set"   and A :: "'d set" assumes "B \<noteq> {}"   and "\<And>i. \<lbrakk>i \<le> least A; i \<le> least B; i \<in> B\<rbrakk> \<Longrightarrow> i \<in> A" shows "least A \<le> least B"
right_associated_sum: fixes w :: "i"   and x :: "i"   and y :: "i"   and z :: "i" shows "O w (x \<oplus> (y \<oplus> z)) = (O w x \<or> O w y \<or> O w z)"
h1c1var: shows "Var x ^ i \<noteq> 0"
sdescending_sappend: fixes u :: "'b list"   and v :: "'b stream" assumes "sdescending (u @- v)" obtains "sdescending v"
ln_Gamma_real_strict_mono: fixes x :: "real"   and y :: "real" assumes "3 / 2 \<le> x"   and "x < y" shows "ln_Gamma x < ln_Gamma y"
simps_3: fixes x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list"   and y31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and y32 :: "'c"   and y33 :: "'d"   and y34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" shows "(x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 = y31:y32\<hookrightarrow>\<^bsub>y33\<^esub>y34) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34)"
of_circline_clmat_def: shows "of_circline_clmat \<equiv> map_fun Rep_circline_mat Abs_circline_mat of_circline_cmat"
terminates_concat_raw: fixes g :: "'e \<Rightarrow> ('f list, 'e) step" assumes "terminates g" shows "terminates (concat_raw g)"
intros_8: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A >>> A'" shows "Fst (SimplyTyped.Pair A B) >>> A'"
simps_19: fixes state_var' :: "sparc_state_var \<Rightarrow> sparc_state_var"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "state_var_update state_var' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var' state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
normalize_eq: fixes p :: "(real, 2) vec" shows "norm p *\<^sub>R normalize p = p"
Units_eq: shows "Units \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr> = carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
generate_eq_finprod_Pi_int_image: fixes gs :: "'a set" assumes "finite gs"   and "gs \<subseteq> carrier G" shows "generate G gs = (\<lambda>x. \<Otimes>g\<in>gs. g [^] x g) ` (gs \<rightarrow> UNIV)"
Seq: fixes c1 :: "com"   and s1 :: "char list \<Rightarrow> int"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int"   and c2 :: "com"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int"   and z :: "nat" assumes "(c1, s1) \<Rightarrow> x \<Down> s2"   and "(c2, s2) \<Rightarrow> y \<Down> s3"   and "z = x + y" shows "(c1;; c2, s1) \<Rightarrow> z \<Down> s3"
measure_subprob: fixes M :: "'c pmf" shows "measure_pmf M \<in> space (subprob_algebra (count_space UNIV))"
filtered_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.filtered less_eq X = (\<forall>Y. finite Y \<and> Y \<subseteq> X \<longrightarrow> (\<exists>x\<in>X. \<forall>y\<in>Y. less_eq x y))"
prob_proj_idempot: fixes vs :: "'d set"   and vs' :: "'d set"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "vs \<subseteq> vs'" shows "prob_proj PROB vs = prob_proj (prob_proj PROB vs') vs"
subM: fixes t :: "'a set"   and M :: "'a set set" assumes "t \<in> M"   and "M \<subseteq> T" shows "t open\<^bsub>T\<^esub>"
reachable_del_arcD: fixes G :: "('c, 'd) pre_digraph"   and u :: "'d"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "v \<rightarrow>\<^sup>*\<^bsub>pre_digraph.del_arc G u\<^esub> w" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
exhaust_sel: fixes stream :: "'d stream" obtains "stream = shd stream ## stl stream"
acts': fixes n :: "nat"   and t :: "'s Trace" assumes "{t \<in> jpTraces jp. tLength t = n} = jkbpC\<^bsub>n\<^esub>"   and "t \<in> jkbpC\<^bsub>n\<^esub>" shows "actJP jp t = jAction MC\<^bsub>n\<^esub> t"
pick_Var_qVar: fixes xs :: "'varSort"   and x :: "'var" shows "pick (Var xs x) #= qVar xs x"
measurable_vimage_algebra1: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and M :: "'c measure" assumes "f \<in> X \<rightarrow> space M" shows "f \<in> vimage_algebra X f M \<rightarrow>\<^sub>M M"
bind_tuple_ineq: fixes p :: "pat"   and U :: "ty"   and x :: "name"   and t :: "btrm"   and u :: "btrm" assumes "ty_size (pat_type p) < ty_size U" shows "Bind U x t \<noteq> (\<lambda>[p]. u)"
simps_2: fixes x :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "list_diff x (y # ys) = list_diff (removeAll y x) ys"
isometry_on_UNIV_iterates: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "isometry_on UNIV f" shows "isometry_on UNIV (f ^^ n)"
H2: fixes t :: "'a set" assumes "t open\<^bsub>T\<^esub>" shows "t \<inter> carrier open"
l2_inv7_trans: shows "{l2_inv7 \<inter> l2_inv1 \<inter> l2_inv2 \<inter> l2_inv4 \<inter> l2_inv5 \<inter> l2_inv6} TS.trans l2 {> l2_inv7}"
alw_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.alw \<phi> \<equiv> Shallow.neg (Shallow.ev (Shallow.neg \<phi>))"
eq: fixes p :: "nat" assumes "prime p" shows "(\<Sum>\<^sub>ak. - ((fds_nth f p / nat_power p s) ^ Suc k * of_real (ln (real p)))) = - (of_real (ln (real p)) * ((1::'a) / ((1::'a) - fds_nth f p / nat_power p s) - (1::'a)))"
HF_axioms_imp_Pf: fixes \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "\<alpha> \<in> HF_axioms" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
gcd_mult_unit2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "gcd b (c * a) = gcd b c"
ngba_g_ipath_run: fixes A :: "('d, 'c) ngba"   and r :: "nat \<Rightarrow> 'c" assumes "ipath (g_E (ngba_g A)) r" obtains w :: "'d stream" where "run A (w ||| smap (r \<circ> Suc) nats) (r 0)"
alw_nrec_pos: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds (\<lambda>c. \<forall>t. 0 \<le> zcount (c_records c) t)) s"
F_n_nn_integral_pos: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "\<forall>x. 0 \<le> f x"   and "f \<in> borel_measurable (F n)" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = (\<Sum>w\<in>range (pseudo_proj_True n). emeasure M (pseudo_proj_True n -` {w} \<inter> space M) * ennreal (f w))"
rec_accum2_def: fixes f :: "recf" shows "rec_accum2 f = PR (CN f [CN Z [recf.Id 2 0], recf.Id 2 0, recf.Id 2 1]) (CN rec_mult [recf.Id 4 1, CN f [CN S [recf.Id 4 0], recf.Id 4 2, recf.Id 4 3]])"
subseq_conv_nths: fixes xs :: "'a list"   and ys :: "'a list" shows "subseq xs ys = (\<exists>N. xs = nths ys N)"
properties_mod_lambda: fixes i :: "nat" assumes "mu \<le> i" shows "seq i = seq (mu + (i - mu) mod lambda)"