hcju/isabelleps huggingface dataset:
psimps: fixes C :: "acom"   and b :: "bexp"   and s :: "char list \<Rightarrow> int" assumes "postQs_dom (C, b, s)" shows "postQs C b s = (if bval b s then postQs C b (postQ C s) else s)"
invimage_vempty: fixes A :: "V" shows "0 -`\<^sub>\<circ> A = 0"
dma_\<C>_vars: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and y :: "'a"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "\<forall>x\<in>\<C>_vars y. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma mem\<^sub>1 y = dma mem\<^sub>2 y"
countably_compact_imp_compact_second_countable: fixes U :: "'b set" assumes "countably_compact U" shows "compact U"
induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'b set \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "'c"   and a2 :: "'b set" assumes "\<And>f x. P f x {}"   and "\<And>A f x. infinite A \<Longrightarrow> P f x A"   and "\<And>A f x. \<lbrakk>finite A; A \<noteq> {}; \<And>xa. xa \<in> set_pmf (pmf_of_set A) \<Longrightarrow> P f (f xa x) (A - {xa})\<rbrakk> \<Longrightarrow> P f x A" shows "P a0 a1 a2"
mC_C_eff_cont_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C_eff_cont theta c d s t P F \<equiv> \<forall>I i j. I \<in> P \<and> i \<in> I \<and> j \<in> F I \<longrightarrow> eff c s i \<approx> eff d t j \<and> (local.cont c s i, local.cont d t j) \<in> theta"
abs_eq: fixes xb :: "'b"   and xa :: "'c"   and x :: "'b \<Rightarrow> 'c option" shows "update xb xa (Mapping x) = Mapping (x(xb \<mapsto> xa))"
ret_modif: shows "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"
list_emb_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "list_emb P xs ys"   and "list_emb P ys zs" shows "list_emb P xs zs"
obtain_inverse_matrix: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "invertible_mat A" obtains B :: "'b mat" where "inverts_mat A B" and "inverts_mat B A" and "B \<in> carrier_mat n n"
pos_less_minus_divideR_eq: fixes 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 scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and a :: "'a"   and b :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less a (uminus (scaleR (inverse c) b)) = less (scaleR c a) (uminus b)"
assms: fixes G :: "'a set" assumes "G \<subseteq> F"   and "finite G" shows "real (card G) \<le> C"
List_app: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and as :: "'b list"   and bs :: "'b list" shows "List h x (as @ bs) = (\<exists>y. Path h x as y \<and> List h y bs)"
simps_3: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" shows "wf\<^sub>s\<^sub>t V (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t (V \<union> fv t) S"
atLeastatMost_empty_iff2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.preorder less_eq less" shows "({} = ord.atLeastAtMost less_eq a b) = (\<not> less_eq a b)"
eval_wfe: fixes \<Gamma> :: "char list set"   and e :: "expr"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "wfe \<Gamma> e"   and "match \<sigma> \<Gamma>" shows "eval \<sigma> e \<noteq> None"
run_owner_v_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_owner_v n s)"
normalize_quot_def: shows "normalize_quot = (\<lambda>(a, b). if b = (0::'b) then (0::'b, 1::'b) else let d = gcd a b * unit_factor b in (a div d, b div d))"
excess_non_negative': shows "\<forall>v\<in>V - {s}. (0::'a) \<le> excess f v"
add_power: fixes u :: "nat"   and x :: "nat" assumes "u < 2 ^ x" shows "nat_to_set (u + 2 ^ x) = nat_to_set u \<union> {x}"
S_intersection_coeff1_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection_coeff1 p q \<equiv> (- p \<bullet> (M *v q) + sqrt (quarter_discrim p q)) / (p \<bullet> (M *v p))"
Subring_minus_ring_minus: fixes S :: "('a, 'd) Ring_scheme"   and x :: "'a" assumes "Subring R S"   and "x \<in> carrier S" shows "-\<^sub>a\<^bsub>S\<^esub> x = -\<^sub>a x"
mult_less_cancel_left_pos: fixes 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 abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less (times c a) (times c b) = less a b"
assms_2: fixes x :: "nat" shows "eval r_phi [j, x] = eval g [x]"
unit_factor_idem: fixes a :: "'a" shows "unit_factor (unit_factor a) = unit_factor a"
termi: shows "\<Gamma>\<turnstile>c \<down> Normal s"
induct: fixes P :: "int \<times> int \<Rightarrow> real \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "real"   and a2 :: "(int \<times> int) list" assumes "\<And>uu_ uv_. P uu_ uv_ []"   and "\<And>uw_ ux_ p. P uw_ ux_ [p]"   and "\<And>p \<delta> p\<^sub>0 v va. (\<not> \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p) \<Longrightarrow> P p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (v # va)) \<Longrightarrow> P p \<delta> (p\<^sub>0 # v # va)" shows "P a0 a1 a2"
cont_D0: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "(lim_proc (range Y) [+] S) = lim_proc (range (\<lambda>i. Y i [+] S))"
is_word_terminals_take: fixes p :: "(symbol \<times> character list) list"   and n :: "nat" assumes "wellformed_tokens p" shows "is_word (terminals (take n p))"
grey_eq_imp: fixes r :: "'d" shows "eq_imp (\<lambda>_ s. r \<in> (\<Union>p. WL p s)) (grey r)"
hyps: shows "mset (remdups (a' @ b)) = mset (remdups a' @ remdups (list_diff b a'))"
is_liftingD_3: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and C :: "'a" assumes "is_lifting f basis basis'" shows "trimmed (f C) = trimmed C"
cov_invertible_real: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and T :: "(real, 'a) vec set"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> real"   and b :: "real" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h (g x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g (h y) = y"   and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g' (h y) = id" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) = b) = (f absolutely_integrable_on T \<and> integral T f = b)"
G_def: shows "(G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (;) (c::'a::refinement_lattice)"
sup_unit_def: fixes uu :: "unit"   and uv :: "unit" shows "sup uu uv = ()"
drinks_vend_sufficient: fixes r :: "nat \<Rightarrow>f value option"   and x1 :: "int" assumes "r $ 2 = Some (Num x1)"   and "100 \<le> x1" shows "possible_steps drinks 1 r STR ''vend'' [] = {|(2, vend)|}"
proots_count_def: fixes p :: "'a poly"   and s :: "'a set" shows "proots_count p s = (\<Sum>r\<in>proots_within p s. order r p)"
execute_parallel_operator_sas_plus_def: fixes s :: "'c \<Rightarrow> 'd option"   and ops :: "('c, 'd) sas_plus_operator list" shows "execute_parallel_operator_sas_plus s ops \<equiv> foldl (++) s (map (map_of \<circ> effect_of) ops)"
simps_11: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (NOT p) = (\<not> Ifm bs p)"
computable_complement_1: fixes A :: "nat set" assumes "computable A" shows "computable (- A)"
better_ex_eval_fmla: fixes e :: "(name, nat) finfun"   and k :: "name"   and A :: "fmla" shows "eval_fmla e (exi k A) = (\<exists>x. eval_fmla (finfun_update e k x) A)"
expand: fixes fundef :: "'a fundef"   and fundef' :: "'a fundef" assumes "body fundef = body fundef' \<and> arity fundef = arity fundef'" shows "fundef = fundef'"
fset_eq_iff: fixes A :: "'a fset"   and B :: "'a fset" shows "(A = B) = (\<forall>x. (x |\<in>| A) = (x |\<in>| B))"
completeness'': fixes s :: "form list" assumes "infinite (deriv (ns_of_s s))"   and "init (ns_of_s s)" shows "\<forall>A. A \<in> set s \<longrightarrow> \<not> FEval (model (ns_of_s s)) ntou A"
counit_in_hom_1: shows "\<guillemotleft>T.\<epsilon> : trg\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright>"
wf_matchF_invar_step: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and st :: "(nat \<times> event_data option list set list \<times> event_data option list set) list \<times> (mregex, event_data option list set) mapping"   and i :: "nat"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and mrs :: "mregex list"   and rels :: "event_data option list set list"   and rel :: "event_data option list set"   and entry :: "nat \<times> event_data option list set list \<times> event_data option list set"   and nt :: "nat" assumes "wf_matchF_invar \<sigma> V n R I r st (Suc i)"   and "safe_regex Futu Strict r"   and "to_mregex r = (mr, \<phi>s)"   and "mrs = sorted_list_of_set (LPDs mr)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels"   and "qtable n (fv_regex r) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < i + length (fst st) \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> Regex.match (Formula.sat \<sigma> V (map the v)) r i j) rel"   and "entry = (\<tau> \<sigma> i, rels, rel)"   and "nt = \<tau> \<sigma> (i + length (fst st))" shows "wf_matchF_invar \<sigma> V n R I r (update_matchF_step I mr mrs nt entry st) i"
collapse: fixes list :: "'c list" assumes "list \<noteq> []" shows "lead_coeff list # tl list = list"
simps_4: fixes b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and X :: "char list set" shows "L (IF b THEN c\<^sub>1 ELSE c\<^sub>2) X = vars b \<union> L c\<^sub>1 X \<union> L c\<^sub>2 X"
list_rel_split_left_iff: fixes l :: "'a list"   and y :: "'b"   and ys :: "'b list"   and R :: "('a \<times> 'b) set" shows "((l, y # ys) \<in> \<langle>R\<rangle>list_rel) = (\<exists>x xs. l = x # xs \<and> (x, y) \<in> R \<and> (xs, ys) \<in> \<langle>R\<rangle>list_rel)"
l_new_document_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
induct: fixes P :: "('d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'd list \<Rightarrow> 'e list \<Rightarrow> 'f \<Rightarrow> bool"   and a0 :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f"   and a1 :: "'f"   and a2 :: "'d list"   and a3 :: "'e list"   and a4 :: "'f" assumes "\<And>f err init. P f err [] [] init"   and "\<And>f err x xs y ys init. P f err xs ys (f x y init) \<Longrightarrow> P f err (x # xs) (y # ys) init"   and "\<And>uu_ err v va ux_. P uu_ err (v # va) [] ux_"   and "\<And>uu_ err v va ux_. P uu_ err [] (v # va) ux_" shows "P a0 a1 a2 a3 a4"
set_of_times: fixes X :: "'b interval"   and Y :: "'b interval" shows "set_of (X * Y) = {x * y |x y. x \<in>\<^sub>i X \<and> y \<in>\<^sub>i Y}"
fds_nth_fds': fixes f :: "nat \<Rightarrow> 'b" assumes "f 0 = (0::'b)" shows "($) (fds f) = f"
lsd_def: fixes x :: "'b \<times> 'c" shows "prod.lsd x = less_eq_prod (times_prod' top_prod x) x"
\<psi>_def: shows "\<psi> \<equiv> (1 - sqrt 5) / 2"
LTL_suffix_G: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and i :: "nat" assumes "w \<Turnstile> G \<phi>" shows "suffix i w \<Turnstile> G \<phi>"
r: shows "real_of_int \<lfloor>real_of_int m * 2 powr real_of_int e * 2 powr real_of_int p\<rfloor> = real_of_int m * 2 powr real_of_int e * 2 powr real_of_int p"
single_valued_as_brE: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" obtains \<alpha> :: "'a \<Rightarrow> 'b" and invar :: "'a \<Rightarrow> bool" where "R = br \<alpha> invar"
a_ge_0: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and a :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "a \<in> set as" shows "0 \<le> a"
A: fixes x :: "val"   and y :: "val" assumes "x \<in> space N'"   and "y \<in> space (stock_measure t1)" shows "indicator (snd' -` X \<inter> space N) <|y, x|> = indicator X x"
plus_up: fixes x :: "real"   and y :: "real"   and prec :: "nat" shows "x + y \<le> plus_up prec x y"
bind_eq_None_conv: fixes a :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "(Option.bind a f = None) = (a = None \<or> f (the a) = None)"
singular_chain_empty: fixes X :: "'c topology"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {}" shows "singular_chain p X c = (c = 0)"
gc_obj_fields_marked_invL_niE: fixes s :: "(('e, 'f, 'g) response, char list, 'h process_name, 'h process_name \<times> ('e, 'h, 'f, 'g) request_op, ('e, 'h, 'f, 'g) Model.local_state) system_state"   and s' :: "(('e, 'f, 'g) response, char list, 'h process_name, 'h process_name \<times> ('e, 'h, 'f, 'g) request_op, ('e, 'h, 'f, 'g) Model.local_state) system_state" assumes "obj_fields_marked_invL s"   and "AT s gc = AT s' gc"   and "s\<down> gc = s'\<down> gc"   and "\<forall>x. map_option obj_fields (sys_heap s\<down> x) = map_option obj_fields (sys_heap s'\<down> x)"   and "\<forall>x. map_option obj_mark (sys_heap s\<down> x) = map_option obj_mark (sys_heap s'\<down> x)"   and "sys_fM s\<down> = sys_fM s'\<down>"   and "sys_W s\<down> = sys_W s'\<down>"   and "filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers gc s\<down>) = filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers gc s'\<down>)" shows "obj_fields_marked_invL s'"
pwd_spy_enc_mult_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_enc_mult n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_enc_mult_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
nth_least_mono: fixes l :: "nat"   and A :: "'b set"   and k :: "nat" assumes "enat l < esize A"   and "k \<le> l" shows "nth_least A k \<le> nth_least A l"
null_eqI: fixes n :: "'c" assumes "\<And>f. n \<cdot> f = n \<and> f \<cdot> n = n" shows "n = C.null"
retraction_def: fixes g :: "'b" shows "B.retraction g \<equiv> \<exists>f. B.ide (g \<cdot>\<^sub>B f)"
cases: fixes x :: "'a word \<times> nat" obtains pre :: "'a word" and len :: "nat" where "x = (pre, len)"
nlists_def: fixes A :: "'a set"   and n :: "nat" shows "nlists A n \<equiv> {l. set l \<subseteq> A \<and> length l = n}"
inf_aci_3: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
e_nth_append_small: fixes n :: "nat"   and xs :: "nat"   and ys :: "nat" assumes "n < e_length xs" shows "e_nth (e_append xs ys) n = e_nth xs n"
simps_3: fixes a :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "\<langle>a: t \<doteq> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = \<langle>a: (t \<cdot> \<theta>) \<doteq> s \<cdot> \<theta>\<rangle>"
empty_le: fixes A :: "'c multiset" shows "{#} \<subseteq># A"
of_rat_lec_add: fixes c :: "rat le_constraint"   and d :: "rat le_constraint" shows "of_rat_lec (c + d) = of_rat_lec c + of_rat_lec d"
power2_norm_eq_inner: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(norm x)\<^sup>2 = inner x x"
axioms_1: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and inPorts :: "'d \<Rightarrow> 'e fset"   and outPorts :: "'d \<Rightarrow> 'f fset"   and nodeOf :: "'g \<Rightarrow> 'd"   and hyps :: "'d \<Rightarrow> 'f \<Rightarrow> 'e option"   and vertices :: "'g fset"   and vidx :: "'g \<Rightarrow> nat"   and inst :: "'g \<Rightarrow> 'c"   and edges :: "(('g \<times> 'f) \<times> 'g \<times> 'e) set"   and local_vars :: "'d \<Rightarrow> 'e \<Rightarrow> 'a set" assumes "Well_Scoped_Instantiation freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps vertices vidx inst edges local_vars" shows "Instantiation inPorts outPorts hyps vertices freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx"
Alpha_def: fixes x :: "'a"   and y :: "'a" shows "A x y = d (x \<cdot> y) \<cdot> ad y"
degree_MP_Rel: shows "(MP_Rel ===> (=)) degree_m degree"
WRP_unique: fixes x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{OrdP x, WRP x y, WRP x y'} \<turnstile> y' EQ y"
P'_add_Q: shows "P' + Q = matrix_sum d (\<lambda>l. tensor_P (proj_psi_l l) (proj_k l)) (R + 1)"
\<phi>l_def: shows "\<phi>l = asList \<phi>s"
id_4: shows "length (bef @ t # aft) = Suc (length bef + length aft)"
eq: shows "{0..} = {0..\<delta>} \<union> {\<delta>..}"
polytope_def: fixes S :: "'b set" shows "polytope S \<equiv> \<exists>v. finite v \<and> S = convex hull v"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "F f = C.null"
hfd: fixes x :: "complex"   and n :: "nat" assumes "x \<in> S" shows "(f n has_field_derivative f' n x) (at x)"
block_closed: fixes a :: "'a \<times> 'a"   and A :: "('a \<times> 'a) set" assumes "a \<in> A"   and "A \<in> rel.Partition" shows "a \<in> R \<times> S"
finrank_imp_findim: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V"   and "R_scalar_mult.R_finrank UNIV smult V" shows "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV smult ms = V"
wls_induct: fixes phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; \<And>Y. (X, Y) \<in> swapped \<Longrightarrow> phi s Y; \<And>ys y1 y2. phi s (X #[y1 // y2]_ys); \<And>Y. \<lbrakk>wls s Y; skel Y = skel X\<rbrakk> \<Longrightarrow> phi s Y\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
pred_spmf_def: fixes P :: "'b \<Rightarrow> bool"   and p :: "'b spmf" shows "pred_spmf P p = (\<forall>x\<in>set_spmf p. P x)"
transitive_closurep_trans'_4: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
neg_I0: fixes i :: "'a"   and A :: "fmla"   and H :: "fmla set" assumes "\<And>B. atom i \<sharp> B \<Longrightarrow> insert A H \<turnstile> B" shows "H \<turnstile> neg A"
gens_span_free_group: fixes gens :: "'e set" shows "\<langle>\<iota> ` gens\<rangle>\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> = carrier \<F>\<^bsub>gens\<^esub>"
in_chain_finite: fixes A :: "'a set" assumes "Complete_Partial_Order.chain (\<le>) A"   and "finite A"   and "A \<noteq> {}" shows "\<Squnion> A \<in> A"
simps: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" shows "mgu s t = (case unify [(s, t)] [] of None \<Rightarrow> None | Some res \<Rightarrow> Some (subst_of res))"
nnf_simps_8: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<le> y) = (y < x)"
simps_16: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Public of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f4"
typeof_lit_eq_Integer: fixes v :: "'a val" shows "(typeof v = \<lfloor>Integer\<rfloor>) = (\<exists>i. v = Intg i)"
at_1: shows "atoms F \<subseteq> atoms_mset (\<Gamma> + \<Delta>)"
simps_2: fixes p :: "'c \<times> 'd"   and ps :: "('c \<times> 'd) list" shows "clearjunk (p # ps) = p # clearjunk (delete (fst p) ps)"
cases: fixes A :: "'b \<Rightarrow> bool"   and a :: "'b llist" assumes "finlstsp A a" obtains "a = LNil"   | l :: "'b llist" and aa :: "'b" where "a = aa ## l" and "finlstsp A l" and "A aa"