hcju/isabelleps huggingface dataset:
nonneg_Reals_minus_numeral_I: fixes w :: "num" shows "- numeral w \<notin> \<real>\<^sub>\<ge>\<^sub>0"
ereal_diff_le_mono_left: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" assumes "x \<le> z"   and "0 \<le> y" shows "x - y \<le> z"
toList_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.toList t \<equiv> FingerTreeStruc.toList (Rep_FingerTree t)"
card_le_dim_spanning: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "V \<subseteq> local.span B"   and "finite B"   and "card B \<le> local.dim V" shows "local.independent B"
funion_iff: fixes c :: "'x"   and A :: "'x fset"   and B :: "'x fset" shows "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)"
lcm_proj1_iff: fixes m :: "'a"   and n :: "'a" shows "(lcm m n = normalize m) = (n dvd m)"
subtopology_topspace: fixes U :: "'d topology" shows "subtopology U (topspace U) = U"
Abs_fnat_homs_5: shows "1 = word_of_nat (Suc 0)"
strip_eq_If: fixes c :: "'a acom"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "(strip c = IF b THEN c1 ELSE c2) = (\<exists>d1 d2 P. c = IF b THEN d1 ELSE d2 {P} \<and> strip d1 = c1 \<and> strip d2 = c2)"
left_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<tau> :: "'b" assumes "weak_composition V H" shows "weak_composition.left V H \<tau> \<equiv> \<lambda>\<mu>. H \<tau> \<mu> \<noteq> partial_magma.null V"
\<R>_divergent_divergent: fixes xs :: "('c \<Rightarrow> real) stream" assumes "pred_stream (\<lambda>u. u \<in> V) xs"   and "\<R>_div (smap (\<lambda>u. [u]\<^sub>\<R>) xs)"   and "stream_trans xs"   and "PTA_Reachability.pairwise trans' xs"   and "\<forall>u. (\<forall>c\<in>\<X>. real (k c) < u c) \<longrightarrow> \<not> ev (alw (\<lambda>xs. shd xs = u)) xs" shows "divergent xs"
simps: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" shows "r\<^sup>*\<^sup>* a1 a2 = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>*\<^sup>* a b \<and> r b c))"
simps: fixes n :: "real"   and l :: "real" shows "invar (n, l) = (l0 \<le> l \<and> (l0 \<le> l / c \<longrightarrow> f1 * l \<le> n) \<and> n \<le> f2 * l)"
bigtheta_trans1': fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "f \<in> L F g"   and "h \<in> \<Theta>[F](g)" shows "f \<in> L F h"
rep_eq: fixes x :: "'b mod_ring" shows "Rep_mod_ring (- x) = (if Rep_mod_ring x = 0 then 0 else int CARD('b) - Rep_mod_ring x)"
nxt_both_left: fixes t :: "'b ttree"   and x :: "'b"   and t' :: "'b ttree" assumes "possible t x" shows "nxt t x \<otimes>\<otimes> t' \<sqsubseteq> nxt (t \<otimes>\<otimes> t') x"
im_sheaf_def: fixes V :: "'a set" shows "id.im_sheaf V \<equiv> \<FF> (identity S \<^sup>\<inverse> S V)"
exhaust: fixes y :: "IMP" obtains "y = Skip"   | x21 :: "Var" and x22 :: "Expr" where "y = Assign x21 x22"   | x31 :: "IMP" and x32 :: "IMP" where "y = Comp x31 x32"   | x41 :: "BExpr" and x42 :: "IMP" where "y = While x41 x42"   | x51 :: "BExpr" and x52 :: "IMP" and x53 :: "IMP" where "y = Iff x51 x52 x53"   | "y = Call"
scaleR_mono: 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 a :: "real"   and b :: "real"   and x :: "'a"   and y :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq x y"   and "0 \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b y)"
impl_of_cases: fixes y :: "('a, 'b) ArrayHashMap_Impl.hashmap" assumes "y \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}" obtains x :: "('a, 'b) hashmap" where "y = impl_of x"
INF_constant: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and c :: "'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf ((\<lambda>y. c) ` A) = (if A = {} then top else c)"
le: fixes z :: "complex" assumes "cmod z < r" shows "cmod (deriv f z) \<le> 2 * cmod (deriv f 0)"
mult_nonneg_nonpos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times a b) zero"
monad_writer_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and tell :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_writer return bind tell" shows "monad_writer (return_state return) (bind_state bind) (tell_state tell)"
f: shows "f \<in> [elts \<beta>]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}"
One_hf_def: shows "1 = \<lbrace>0\<rbrace>"
path_nret_inf_icd_seq: fixes \<pi> :: "nat \<Rightarrow> 'n" assumes "is_path \<pi>"   and "\<forall>i. \<pi> i \<noteq> local.return" obtains f :: "nat \<Rightarrow> nat" where "\<forall>i. f (Suc i) icd\<^bsup>\<pi>\<^esup>\<rightarrow> f i" and "range f = {i. \<forall>j>i. j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i}" and "\<nexists>i. f 0 cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"
dbl_neg: fixes x :: "'a" shows "- (- x) = x"
convex_hull_list_def: fixes n :: "nat"   and Vs :: "'b vec list" shows "gram_schmidt.convex_hull_list n Vs = {x. \<exists>c. gram_schmidt.convex_lincomb_list n c Vs x}"
comp_converter_id_right: fixes conv :: "('i, 'j, 'k, 'l) converter" shows "conv \<odot> 1\<^sub>C = conv"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ag" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
ide_leg1: shows "ide f\<^sub>0gh\<^sub>1.p\<^sub>1"
mem_sphere: fixes y :: "'c"   and x :: "'c"   and e :: "real" shows "(y \<in> sphere x e) = (dist x y = e)"
image_f: shows "local.f ` local.S \<subseteq> local.S"
no_overlap_ConsI: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('a, 'c) flow_entry_match"   and xs :: "('a, 'c) flow_entry_match list" assumes "check_no_overlap2 \<gamma> (x # xs)" shows "check_no_overlap2 \<gamma> xs"
t1_space_def: fixes X :: "'a topology" shows "t1_space X \<equiv> \<forall>x\<in>topspace X. \<forall>y\<in>topspace X. x \<noteq> y \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> y \<notin> U)"
p_def: fixes t :: "real" shows "p t = s0 + v * t + 1 / 2 * a * t\<^sup>2"
simps_2: fixes x :: "floatarith"   and xs :: "floatarith list" shows "max_Var_floatariths (x # xs) = max (max_Var_floatarith x) (max_Var_floatariths xs)"
not_in_Ints_imp_not_in_nonpos_Ints: fixes z :: "'a" assumes "z \<notin> \<int>" shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
Union: fixes S :: "'c set set"   and B :: "'c set"   and F :: "'c program" assumes "\<forall>A\<in>S. (A, B) \<in> leads F" shows "(\<Union> S, B) \<in> leads F"
coeffs_smult': fixes a :: "'b"   and p :: "'b poly" shows "coeffs (smult a p) = (if a = (0::'b) then [] else strip_while ((=) (0::'b)) (map ((*) a) (coeffs p)))"
26864: fixes x :: "'a"   and y :: "'a" shows "- - x \<le> - (- x \<sqinter> y)"
assms_4: shows "w \<equiv> the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a))"
SF2: fixes N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and B :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and M :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c"   and P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "|~ \<langle>N \<and> B\<rangle>_f \<longrightarrow> \<langle>M\<rangle>_g"   and "|~ P \<and> \<circle>P \<and> \<langle>N \<and> A\<rangle>_f \<longrightarrow> B"   and "\<turnstile> P \<and> Enabled \<langle>M\<rangle>_g \<longrightarrow> Enabled \<langle>A\<rangle>_f"   and "\<turnstile> \<box>[N \<and> \<not> B]_f \<and> SF(A)_f \<and> \<box>F \<and> \<box>\<diamond>Enabled \<langle>M\<rangle>_g \<longrightarrow> \<diamond>\<box>P" shows "\<turnstile> \<box>[N]_f \<and> SF(A)_f \<and> \<box>F \<longrightarrow> SF(M)_g"
powr_mono: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "a \<le> b"   and "1 \<le> x" shows "x powr a \<le> x powr b"
State_ext_def: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "\<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons State_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple (Record.iso_tuple_cons Record.tuple_iso_tuple getF getM) (Record.iso_tuple_cons Record.tuple_iso_tuple getConflictFlag getC)) more"
countably_additive_additive: fixes M :: "'b measure"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive (sets M) f"   and "countably_additive (sets M) f" shows "Measure_Space.additive (sets M) f"
assms: fixes x :: "'a" assumes "x \<in> set_pmf p" shows "P x x"
mnum_def: shows "mnum = (\<lambda>q. card {g \<in> G. q g \<odot> g \<noteq> 0 \<and> lt (q g \<odot> g) = mlt q})"
compass_non_zero_6: shows "vector [1, 0, 0] \<noteq> 0"
flush_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (flush_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
zip_conv_bind_option: fixes x :: "'c option"   and y :: "'d option" shows "zip_option x y = x \<bind> (\<lambda>x. y \<bind> (\<lambda>y. Some (x, y)))"
len_le_iff: fixes x :: "nat"   and l :: "nat" shows "(len x \<le> l) = (x < 2 ^ l)"
intros_3: fixes b :: "expr"   and c' :: "cmd"   and l :: "nat"   and s :: "char list \<Rightarrow> val option" assumes "labels (while (b) c') l (Skip;; while (b) c')" shows "while (b) c' \<turnstile> \<langle>Skip;; while (b) c',s,l\<rangle> \<leadsto> \<langle>while (b) c',s,0\<rangle>"
fps_unit_dvd_left: fixes f :: "'b fps" assumes "f $ 0 \<noteq> (0::'b)" shows "\<exists>g. 1 = f * g"
\<beta>1: shows "\<beta>1 = \<beta>0 + \<beta>1a"
dom_one: shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> (1::'a) = (1::'a)"
discrCf_cont: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and i :: "nat" assumes "discrCf cf"   and "i < brn (fst cf)" shows "discrCf (cont_eff cf i)"
linear_span_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear (*a) (*b) f" shows "vs2.span (f ` S) = f ` vs1.span S"
invar_loc_def: fixes G :: "('a, 'b) graph_rec_scheme"   and v0 :: "'a"   and D0 :: "'a set"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set" shows "invar_loc G v0 D0 p D pE \<equiv> fr_graph G \<and> invar_loc_axioms G v0 D0 p D pE"
not_add_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not (plus a b) = minus (not a) b"
isomorphic_dags_eq_def: fixes st\<^sub>1 :: "dag"   and st\<^sub>2 :: "dag"   and var :: "ref \<Rightarrow> nat" shows "isomorphic_dags_eq st\<^sub>1 st\<^sub>2 var = (\<forall>bdt\<^sub>1 bdt\<^sub>2. bdt st\<^sub>1 var = Some bdt\<^sub>1 \<and> bdt st\<^sub>2 var = Some bdt\<^sub>2 \<and> bdt\<^sub>1 = bdt\<^sub>2 \<longrightarrow> st\<^sub>1 = st\<^sub>2)"
ge_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A BGe C D" shows "A BGe D C"
collapse: fixes option :: "'c option" assumes "option \<noteq> None" shows "Some (the option) = option"
tts_AB_C_transfer: fixes \<UU>A :: "'c set"   and \<UU>B :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and \<UU>C :: "'e set"   and A :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> \<UU>A; b \<in> \<UU>B\<rbrakk> \<Longrightarrow> f a b \<in> \<UU>C"   and "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B"   and "Domainp C = (\<lambda>x. x \<in> \<UU>C)"   and "bi_unique C"   and "right_total C" shows "\<exists>rcdt. (A ===> B ===> C) f rcdt"
pre_subdivFace_face_in_f: fixes f :: "face"   and v :: "nat"   and ls :: "nat option list"   and a :: "nat" assumes "pre_subdivFace_face f v ls"   and "Some a \<in> set ls" shows "a \<in> set (verticesFrom f v)"
rec_impl_def: shows "local.rec_impl \<equiv> gds_init gds \<bind> FOREACH\<^sub>C\<^bsup>\<lambda>it s. gen_rwof' s \<and> (\<not> gds_is_break gds s \<longrightarrow> gds_is_empty_stack gds s \<and> V0 - it \<subseteq> gen_discovered s)\<^esup> V0 ((Not \<circ>\<circ> gds_is_break) gds) (\<lambda>v0 s. let s0 = GHOST s in if gds_is_discovered gds v0 s then RETURN s else gds_new_root gds v0 s \<bind> (\<lambda>s. if gds_is_break gds s then RETURN s else REC_annot (\<lambda>(u, s). gen_rwof' s \<and> \<not> gds_is_break gds s \<and> (\<exists>stk. stack s = u # stk) \<and> E \<inter> {u} \<times> UNIV \<subseteq> pending s) (\<lambda>(u, s) s'. gen_rwof' s' \<and> (\<not> gds_is_break gds s' \<longrightarrow> stack s' = tl (stack s) \<and> pending s' = pending s - {u} \<times> UNIV \<and> gen_discovered s \<subseteq> gen_discovered s')) (\<lambda>D (u, s). FOREACH\<^sub>C\<^bsup>\<lambda>it s'. gen_rwof' s' \<and> (\<not> gds_is_break gds s' \<longrightarrow> stack s' = stack s \<and> pending s' = pending s - {u} \<times> (E `` {u} - it) \<and> gen_discovered s \<union> (E `` {u} - it) \<subseteq> gen_discovered s')\<^esup> (E `` {u}) (\<lambda>s. \<not> gds_is_break gds s) (\<lambda>v s. choose_pending u (Some v) s \<bind> (\<lambda>s. if gds_is_discovered gds v s then if gds_is_finished gds v s then gds_cross_edge gds u v s else gds_back_edge gds u v s else gds_discover gds u v s \<bind> (\<lambda>s. if gds_is_break gds s then RETURN s else D (v, s)))) s \<bind> (\<lambda>s. if gds_is_break gds s then RETURN s else choose_pending u None s \<bind> (\<lambda>s. gds_finish gds u s \<bind> RETURN))) (v0, s)))"
T_seq: fixes P :: "'c process"   and Q :: "'c process" shows "T (P `;` Q) = {t. \<exists>X. (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> T Q} \<union> {t1 @ t2 |t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q} \<union> {t1 @ t2 |t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
cut_less_Max_all: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I < t" shows "I \<down>< t = I"
A: fixes c :: "real" shows "\<forall>\<^sub>F x in at_top. (\<forall>b\<in>set bs. \<bar>hb * ln x powr - (1 + e)\<bar> < b - C) \<and> c < x"
gdpr_scenario''_def: shows "gdpr_scenario'' \<equiv> Infrastructure ex_graph'' local_policies"
less: fixes y :: "nat" assumes "y < k" shows "p y \<in> \<rat>"
cases: fixes a :: "('a \<Rightarrow> 'b) set"   and \<R> :: "('a \<Rightarrow> 'b) set set"   and R :: "('a \<Rightarrow> 'b) set" assumes "a \<in> Succ \<R> R" obtains u :: "'a \<Rightarrow> 'b" and R' :: "('a \<Rightarrow> 'b) set" and t :: "'b" where "a = R'" and "u \<in> R" and "R \<in> \<R>" and "R' \<in> \<R>" and "(0::'b) \<le> t" and "R' = [u \<oplus> t]\<^sub>\<R>"
divide_by_norm_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f" shows "divide_by_norm_row A a = (\<chi>b. ((1 / norm (column b A)) *\<^sub>R column b A) $ a)"
prime_n_pd_prime: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and j :: "nat" assumes "distinct_pds K n P"   and "j \<le> n" shows "prime_ideal (O\<^bsub>K P n\<^esub>) (P\<^bsub>K P n\<^esub> j)"
refine_cmpserTr0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and H :: "'a set" assumes "0 < n"   and "compseries G n f"   and "i \<le> n - 1"   and "G \<guillemotright> H "   and "f (Suc i) \<subseteq> H \<and> H \<subseteq> f i" shows "H = f (Suc i) \<or> H = f i"
dis_log_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "local.dis_log.dis_log \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>x. let h = \<^bold>g [^] x in \<A> h \<bind> (\<lambda>x'. return_spmf [x = x'] (mod order \<G>))) ELSE return_spmf False"
normal_form_exact_decompE: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" obtains qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" where "valid_decomp X qs" and "standard_decomp 0 qs" and "monomial_decomp qs" and "cone_decomp (normal_form F ` P[X]) qs" and "exact_decomp X 0 qs" and "\<And>g. \<lbrakk>\<And>f. f \<in> F \<Longrightarrow> homogeneous f; g \<in> punit.reduced_GB F\<rbrakk> \<Longrightarrow> poly_deg g \<le> \<b> qs 0"
eq_neutr_iff: fixes a :: "'b"   and b :: "'b" shows "(a \<sqinter> b = \<top>) = (a = \<top> \<and> b = \<top>)"
iso_def: fixes G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
WT_plus_oracleI: fixes \<I>l :: "('h, 'i) \<I>"   and left :: "'j \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'j) spmf"   and s :: "'j"   and \<I>r :: "('k, 'l) \<I>"   and right :: "'j \<Rightarrow> 'k \<Rightarrow> ('l \<times> 'j) spmf" assumes "\<I>l \<turnstile>c left s \<surd>"   and "\<I>r \<turnstile>c right s \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>"
assms_2: shows "\<forall>i\<in>set (map fst xs). i < oid"
wt_err_step_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b err \<Rightarrow> (nat \<times> 'b err) list"   and \<tau>s :: "'b err list" shows "wt_err_step r step \<tau>s = wt_step (Err.le r) Err step \<tau>s"
intersect'_nodes_finite: fixes A :: "('e, 'd) nba"   and B :: "('e, 'f) nba" assumes "finite (NBA.nodes A)"   and "finite (NBA.nodes B)" shows "finite (NGBA.nodes (intersect' A B))"
m: shows "wfactors G (as @ bs) (a \<otimes> b)"
eq: shows "map (pairself (mirror_point (fst X))) (half_segments_of_aform X) ! n = (2 *\<^sub>R fst X - fst (half_segments_of_aform X ! n), 2 *\<^sub>R fst X - snd (half_segments_of_aform X ! n))"
finite_fg: shows "finite {x. (f x = 0 \<or> g x = 0) \<and> a \<le> x \<and> x \<le> b}"
lex_comp'_valE: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat_pp :: "'b \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'b"   and lex_comp' :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and deg' :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'"   and "lex_comp' s t \<noteq> Eq" obtains x :: "nat" where "x \<in> keys_pp (rep_nat_pp s) \<union> keys_pp (rep_nat_pp t)" and "comparator_of (lookup_pp (rep_nat_pp s) x) (lookup_pp (rep_nat_pp t) x) = lex_comp' s t" and "\<And>y. y < x \<Longrightarrow> lookup_pp (rep_nat_pp s) y = lookup_pp (rep_nat_pp t) y"
ext_funcset_to_sing_iff: fixes A :: "'d set"   and a :: "'e" shows "A \<rightarrow>\<^sub>E {a} = {\<lambda>x\<in>A. a}"
intros_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile>c \<down> s"   and "\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t" shows "\<Gamma>\<turnstile>c # cs,css \<Down> s"
comp_converter_parametric: shows "rel_fun (rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>) (rel_fun (rel_converter (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse> (BNF_Def.Grp UNIV id) (BNF_Def.Grp UNIV id)\<inverse>\<inverse>) (rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV id) (BNF_Def.Grp UNIV id)\<inverse>\<inverse>)) comp_converter comp_converter"
nummul_nb: fixes t :: "Ferrack.num"   and i :: "int" assumes "numbound0 t" shows "numbound0 (nummul t i)"
override_D_def: fixes m1 :: "'a \<Rightarrow> 'b decision option"   and m2 :: "'a \<Rightarrow> 'b decision option" shows "m1 \<Oplus>\<^sub>D m2 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>allow a\<rfloor> \<Rightarrow> case m1 x of \<bottom> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>allow \<alpha>\<rfloor> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>deny b\<rfloor> \<Rightarrow> \<lfloor>deny b\<rfloor> | \<lfloor>deny a\<rfloor> \<Rightarrow> \<lfloor>deny a\<rfloor>)"
assms_2: shows "g \<in> stabilizer G \<phi> x"
converse_rtrancl_on_cases: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> rtrancl_on F r" obtains "a = b" and "b \<in> F"   | c :: "'c" where "(a, c) \<in> r" and "(c, b) \<in> rtrancl_on F r"
bdia_export1: fixes x :: "'a"   and p :: "'a"   and t :: "'a" shows "\<langle>x| (r p \<cdot> r t) = \<langle>r t \<cdot> x| p"
simps_2: fixes a :: "Arity"   and as :: "Arity list"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "AEstack (a # as) (Alts e1 e2 # S) = \<A>\<^bsub>a\<^esub> e1 \<squnion> \<A>\<^bsub>a\<^esub> e2 \<squnion> AEstack as S"
tmi_le_tp: shows "te m i \<le> te p (i + 1)"
rep_eq: fixes x :: "'k multiset"   and xa :: "'k multiset" shows "count (x - xa) = (\<lambda>a. count x a - count xa a)"
card_bind_singleton: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite S"   and "inj_on f S" shows "card (S \<bind> (\<lambda>x. {f x})) = card S"
dom_compose: fixes xs :: "('d \<times> 'f) list"   and ys :: "('f \<times> 'e) list" shows "fst ` set (compose xs ys) \<subseteq> fst ` set xs"