hcju/isabelleps huggingface dataset:
Suc: shows "(sinpoly (Suc n) has_real_derivative cospoly n x) (at x)"
prems_2: shows "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc n1)"
UNION_OK_lift_I: fixes I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program" assumes "\<forall>i\<in>I. F i \<in> preserves snd"   and "\<forall>i\<in>I. \<Union> (Acts ` preserves fst) \<subseteq> AllowedActs (F i)" shows "OK I (\<lambda>i. lift i (F i))"
bij_lens_equiv: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "bij_lens X"   and "X \<approx>\<^sub>L Y" shows "bij_lens Y"
csqrt2p: shows "1 / csqrt (2 ^ Suc k) * 1 / csqrt 2 = 1 / csqrt (2 ^ Suc (Suc k))"
sconjunctive_simp: fixes x :: "'a"   and Q :: "'a set"   and S :: "'a \<Rightarrow> 'b" assumes "x \<in> Q"   and "sconjunctive S" shows "S (Inf Q) = Inf (S ` Q)"
ctn_def_1: shows "check \<C> es (Type t1s) = ctn"
comp_null_1: fixes f :: "'a" shows "S.null \<cdot>\<^sub>S f = S.null"
simps: fixes f :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue"   and v :: "C \<rightarrow> CValue" shows "CFn\<cdot>f \<down>CFn v = f\<cdot>v"
RID\<^sub>L_def: fixes s :: "('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr" shows "RID\<^sub>L s \<equiv> case s of (\<sigma>, \<tau>, e) \<Rightarrow> RID\<^sub>S \<sigma> \<union> RID\<^sub>S \<tau> \<union> RID\<^sub>E e"
ospec: fixes A :: "'b option"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>set_option A. P x"   and "A = Some x" shows "P x"
ennreal_minus_top: fixes a :: "real" shows "ennreal a - \<top> = 0"
deduct_if_hom_deduct: fixes M :: "('fun, 'var) Term.term set"   and A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and S :: "('fun, 'var) Term.term set"   and m :: "('fun, 'var) Term.term" assumes "\<langle>M;A;S\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m m" shows "M \<turnstile> m"
mod_exp_aux_code: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and m :: "'b"   and y :: "'b"   and x :: "'b"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "euclidean_semiring_cancel.mod_exp_aux times modulo m y x n = (if n = 0 then y else if n = 1 then modulo (times x y) m else if even n then euclidean_semiring_cancel.mod_exp_aux times modulo m y (modulo (times x x) m) (n div 2) else euclidean_semiring_cancel.mod_exp_aux times modulo m (modulo (times x y) m) (modulo (times x x) m) (n div 2))"
bot_least: fixes bot :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.order_bot bot less_eq less" shows "less_eq bot a"
less: shows "0 < Float.exponent x + (int (bias TYPE(('e, 'f) IEEE.float)) + int LENGTH('f))"
bind_returns_result_E3: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and y :: "'n"   and x :: "'o" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "pure f h" shows "h \<turnstile> g x \<rightarrow>\<^sub>r y"
pp_dist_star: fixes x :: "'a" shows "- - x\<^sup>\<star> = components x"
outer_prod_mult_vec: fixes a :: "'c vec"   and d1 :: "nat"   and b :: "'c vec"   and d2 :: "nat"   and c :: "'c vec" assumes "a \<in> carrier_vec d1"   and "b \<in> carrier_vec d2"   and "c \<in> carrier_vec d2" shows "outer_prod a b *\<^sub>v c = inner_prod b c \<cdot>\<^sub>v a"
insertion_var: fixes var :: "nat"   and L :: "'a list"   and x :: "'a" assumes "var < length L" shows "insertion (nth_default (0::'a) (L[var := x])) (Var var) = x"
monom_0: fixes a :: "'a" shows "monom a 0 = [:a:]"
S'_def: shows "S' = {(p, i). prime p \<and> 1 < i \<and> real (p ^ i) \<le> x}"
simps_2: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat" shows "expr_sem_rf \<sigma> (Var x) = \<sigma> x"
getFr_Fvars: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and \<phi> :: "'fmla" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s" shows "getFr xs ts \<phi>s \<notin> Fvars \<phi>"
P_def: fixes x :: "real" shows "P x = (\<Sum>n. D (real n + x))"
af_equals_af\<^sub>G_base_cases_4: fixes a :: "'a"   and w :: "'a set list" shows "af (np(a)) w = af\<^sub>G (np(a)) w"
regular_closed_star: fixes x :: "'b" assumes "regular x" shows "regular (x\<^sup>\<star>)"
height_gt_0_in_alphabet_imp_sibling_right: fixes t\<^sub>2 :: "'a tree"   and a :: "'a"   and t\<^sub>1 :: "'a tree"   and w :: "nat" assumes "0 < height t\<^sub>2"   and "a \<in> alphabet t\<^sub>1" shows "sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>1 a"
finite_image_iff: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
subset: shows "ideal_generated {a} \<subseteq> ideal_generated {b}"
std_inca_simulation_axioms: shows "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"
goal3_6: shows "Suc (Suc n) \<noteq> ni2"
agree: shows "Vagree \<nu> \<nu>' (FVDiff (Times t1 t2))"
clearjunk_updates: fixes ks :: "'c list"   and vs :: "'d list"   and al :: "('c \<times> 'd) list" shows "clearjunk (updates ks vs al) = updates ks vs (clearjunk al)"
Derives1_unique_dest: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list"   and w :: "symbol list" assumes "Derives1 u i r v"   and "Derives1 u i r w" shows "v = w"
assms_1: fixes bl :: "'b set" shows "2 \<le> card bl"
mult1_union: fixes B :: "'k multiset"   and D :: "'k multiset"   and r :: "('k \<times> 'k) set"   and C :: "'k multiset" assumes "(B, D) \<in> mult1 r" shows "(C + B, C + D) \<in> mult1 r"
htai_prod_def: fixes H1 :: "(nat, nat) hashedTa"   and H2 :: "(nat, nat) hashedTa" shows "htai_prod H1 H2 \<equiv> hta_reindex (hta_prod H1 H2)"
heap_of_B_def: fixes xs :: "'b list" shows "heap_of_B xs = fst (heapify (length xs) xs)"
inject_1: fixes x1 :: "fresh_t"   and y1 :: "fresh_t" shows "(sesK x1 = sesK y1) = (x1 = y1)"
greaterThan_subset_iff: fixes x :: "'b"   and y :: "'b" shows "({x<..} \<subseteq> {y<..}) = (y \<le> x)"
obj_ty_Arr: fixes T :: "ty"   and i :: "int"   and vs :: "vn \<Rightarrow> val option" shows "obj_ty \<lparr>tag = Arr T i, values = vs\<rparr> = T.[]"
valuation_for_operator_variables_def: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<tau> :: "('b \<Rightarrow> bool option) list" shows "valuation_for_operator_variables \<Pi> \<pi> \<tau> \<equiv> let ops = \<Pi>\<^sub>\<O> in Let {Operator k (index ops op) |k op. k \<in> {0..<length \<tau> - 1} \<and> op \<in> set ops} (override_on \<A>\<^sub>0 (foldr (\<lambda>(k, op) \<A>. \<A>(Operator k (index ops op) := True)) (concat (map (\<lambda>k. map (Pair k) (\<pi> ! k)) [0..<length \<tau> - 1])) \<A>\<^sub>0))"
eventually_frequently_conj: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "frequently P F"   and "eventually Q F" shows "\<exists>\<^sub>F x in F. P x \<and> Q x"
Lq_num2: shows "let LLq = \<lambda>t1 t2. psubst Lq [(t1, zz), (t2, yy)] in \<forall>p\<in>num. \<exists>P\<subseteq>num. finite P \<and> prv (dsj (sdsj {eql (Var yy) r |r. r \<in> P}) (LLq p (Var yy)))"
rel_spmf_mono: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h spmf"   and g :: "'i spmf"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
int: shows "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
p: fixes s :: "'a" assumes "guard s" shows "p \<le> weight_spmf (local.while s)"
simps_1: fixes r :: "'a rexp" shows "pnPlus Zero r = r"
isomorphicE: fixes a :: "'c"   and a' :: "'c" assumes "C.isomorphic a a'" obtains f :: "'c" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f"
prod_algebraI_finite: fixes I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite I"   and "\<forall>i\<in>I. E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> prod_algebra I M"
SgN: shows "S \<subseteq> g ` {0..<Suc c}"
scount_eq_card: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "ev (alw (\<lambda>xs. \<not> P xs)) \<omega>" shows "scount P \<omega> = enat (card {i. P (sdrop i \<omega>)})"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "A \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. f k * A) n = matrix_sum d f n * A"
continuous_map_image_subset_topspace: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
adjoint_free: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "local.adjoint I \<sigma> \<nu> = \<lparr>Functions = \<lambda>f. case SFunctions \<sigma> f of None \<Rightarrow> Functions I f | Some f' \<Rightarrow> \<lambda>R. sterm_sem (extendf I R) f' (fst \<nu>), Predicates = \<lambda>p. case SPredicates \<sigma> p of None \<Rightarrow> Predicates I p | Some p' \<Rightarrow> \<lambda>R. \<nu> \<in> fml_sem (extendf I R) p', Contexts = \<lambda>c. case SContexts \<sigma> c of None \<Rightarrow> Contexts I c | Some c' \<Rightarrow> \<lambda>R. fml_sem (extendc I R) c', Programs = \<lambda>a. case SPrograms \<sigma> a of None \<Rightarrow> Programs I a | Some a' \<Rightarrow> prog_sem I a', ODEs = \<lambda>ode. case SODEs \<sigma> ode of None \<Rightarrow> ODEs I ode | Some ode' \<Rightarrow> ODE_sem I ode', ODEBV = \<lambda>ode. case SODEs \<sigma> ode of None \<Rightarrow> ODEBV I ode | Some ode' \<Rightarrow> ODE_vars I ode'\<rparr>"
simps_17: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = sip"
split_int_le_real': fixes a :: "int"   and b :: "real" shows "(real_of_int a + b \<le> 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> \<le> 0 \<or> real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0)"
empty_trie_def: shows "empty_trie = Trie.trie.Trie None []"
assms_3: shows "s2 = (x * y + (q - s1)) mod q"
lossless_funct_OT_12: fixes msgs :: "'a \<times> 'a"   and \<sigma> :: "bool" shows "lossless_spmf (funct_OT_12 msgs \<sigma>)"
nn_integral_has_integral_lborel: fixes f :: "'c \<Rightarrow> real"   and I :: "real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "(f has_integral I) UNIV" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal I"
induced_automorphism_adjacent_half_chamber_system_image_fg: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "D \<in> g \<turnstile> folding_g.\<C>"   and "C \<sim> D" shows "\
 ` D = C"
imp_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longrightarrow>. q) = (A \<union> {p} \<turnstile> q)"
is_bisimulation_def: fixes as :: "'a list"   and R :: "('a rexp \<times> 'a rexp) set" shows "is_bisimulation as R = (\<forall>(r, s)\<in>R. atoms r \<union> atoms s \<subseteq> set as \<and> nullable r = nullable s \<and> (\<forall>a\<in>set as. (nderiv a r, nderiv a s) \<in> R))"
not_strongly_preferred_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier"   and "b \<in> carrier" shows "(\<not> a \<prec>[le] b) = le b a"
set_step: fixes s :: "'a list"   and r :: "'a"   and act :: "nat \<times> nat list" shows "set (step s r act) = set s"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "M-lipschitz_on (U i) f"
of_char_eqI: fixes c :: "char"   and d :: "char" assumes "of_char c = of_char d" shows "c = d"
ereal_Inf_cmult: assumes "0 < c" shows "\<Sqinter> {ereal c * x |x. \<exists>i\<in>I. x = f i} = ereal c * \<Sqinter> {x. \<exists>i\<in>I. x = f i}"
indspaceD_transform: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set"   and g :: "'g"   and x :: "('r, 'g) aezfun"   and a :: "'r" assumes "f \<in> indspace G FH V"   and "g \<in> G"   and "x \<in> FH" shows "f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x"
w: fixes w :: "'a" assumes "v \<rightarrow> w" shows "w \<in> winning_region p"
in_sccs_verts_conv_reachable: fixes S :: "'a set" shows "(S \<in> G.sccs_verts) = (S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u))"
relation_of_cases: fixes y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "y \<in> {p. is_CSP_process p}" obtains x :: "('c, 'd) action" where "y = action.relation_of x"
induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes " x1 \<longrightarrow>\<^sub>1\<^sup>* x2"   and "\<And>t. P t t"   and "\<And>t s.  t \<longrightarrow>\<^sub>1 s \<Longrightarrow> P t s"   and "\<And>t1 t2 t3. \<lbrakk> t1 \<longrightarrow>\<^sub>1\<^sup>* t2; P t1 t2;  t2 \<longrightarrow>\<^sub>1\<^sup>* t3; P t2 t3\<rbrakk> \<Longrightarrow> P t1 t3" shows "P x1 x2"
is_normal: shows "\<parallel>col v 0\<parallel> = 1"
length_remove_nth: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "length (remove_nth i xs) = length xs - 1"
iterate_additive': fixes t\<^sub>1 :: "nat"   and t\<^sub>2 :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "iterate (t\<^sub>1 + t\<^sub>2) f x = iterate t\<^sub>2 f (iterate t\<^sub>1 f x)"
inR2: shows "s \<otimes> g1 \<in> stabilizer G \<phi> x #> r2"
RECT_unfold: fixes B :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "trimono B" shows "REC\<^sub>T B = B (REC\<^sub>T B)"
hyps: fixes env :: "('a, 'b, 'c) env"   and e :: "('a, 'b, 'c) env" assumes "lookup env xs = Some e" shows "lookup (update (xs @ y # ys) opt env) xs = Some (update (y # ys) opt e)"
WHILE_def: shows "WHILE \<equiv> WHILEI (\<lambda>_. True)"
rep_eq: fixes x :: "'c set_dlist" shows "member x = equal_base.list_member ceq' (list_of_dlist x)"
thetaIfWT_WretrT: shows "thetaIfWT \<subseteq> WretrT (thetaIfWT \<union> WbisT)"
main: shows "[:resultant p q:] = p' * p + q' * q"
insert_3: assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F"   and "I \<noteq> {}" shows "((\<lambda>x. MIN i\<in>I. f i x) \<longlongrightarrow> Min (a ` I)) F"
awalk_decomp: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "G.awalk u p v"   and "w \<in> set (G.awalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> G.awalk u q w \<and> G.awalk w r v"
conjugate_scalar_prod_Re: fixes v :: "complex vec" shows "0 \<le> Re (inner_prod v v)"
bisim_simple_\<R>\<^sub>3: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'" shows "fst (fst lc) = fst (fst lc')"
sum_ideals_la2: fixes I1 :: "'a set"   and I2 :: "'a set" assumes "ideal R I1"   and "ideal R I2" shows "I2 \<subseteq> I1 \<minusplus> I2"
initial_arrow_to_functor_axioms: shows "initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y e"
scopeFresh: fixes x :: "name"   and P :: "ccs" assumes "x \<sharp> P" shows "\<lparr>\<nu>x\<rparr>P \<sim> P"
mk_mat_def: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'd" shows "HMA_Connect.mk_mat f = (\<chi>i j. f i j)"
sub_set_system_axioms_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "sub_set_system_axioms \<U> \<A> \<V> \<B> \<equiv> \<U> \<subseteq> \<V> \<and> \<A> \<subseteq># \<B>"
discr_coind: fixes phi :: "('test, 'atom, 'choice) cmd \<Rightarrow> bool"   and c :: "('test, 'atom, 'choice) cmd" assumes "phi c"   and "\<And>c s i. \<lbrakk>phi c; i < brn c\<rbrakk> \<Longrightarrow> s \<approx> eff c s i \<and> (phi (local.cont c s i) \<or> discr (local.cont c s i))" shows "discr c"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) R local.map \<rr>"
maxl_fun_maxl: fixes x :: "'a" assumes "x \<in> carrier D"   and "maxl_fun D x = x" shows "maximal x"
op_times_UNIV_coll_def: fixes X :: "'a set" shows "op_times_UNIV_coll X = X \<times> UNIV"
WT_resourceD: fixes \<I> :: "('a, 'b) \<I>"   and res :: "('a, 'b) resource"   and q :: "'a"   and r :: "'b"   and res' :: "('a, 'b) resource" assumes "\<I> \<turnstile>res res \<surd>"   and "q \<in> outs_\<I> \<I>"   and "(r, res') \<in> set_spmf (run_resource res q)" shows "r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu w cr s1)) \<and> t2 = snd (fst (write_cpu w cr s2))"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "arr f"
RN_derivI: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and N :: "'b measure" assumes "f \<in> borel_measurable M"   and "density M f = N" shows "density M (RN_deriv M N) = N"
trss_xchange_context_s: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and csp :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and cn :: "'a list multiset" assumes "((s, c), e, s', csp + c) \<in> trss fg"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "((s, cn), e, s', csp + cn) \<in> trss fg"
Re_csqrt: fixes z :: "complex" shows "0 \<le> Re (csqrt z)"
compl_bot_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus bot = top"
cycle_free: shows "\<forall>(a, w, b)\<in>E. \<not> nodes_connected (delete_edge a w b G) a b"
br_table: fixes c :: "i32"   and "is" :: "nat list"   and i :: "nat" assumes "nat_of_int c < length is" shows "\<lparr>[$C ConstInt32 c, $Br_table is i]\<rparr> \<leadsto> \<lparr>[$Br (is ! nat_of_int c)]\<rparr>"
wtC_map_gL: fixes c :: "('fsym, 'psym) lit list" assumes "Ik.wtC c" shows "GE.wtC (map gL c)"
not_initedD: fixes C :: "qtname"   and g :: "oref \<Rightarrow> obj option" assumes "\<not> inited C g" shows "g (Inr C) = None"
ifbd_Sup_pres: shows "Sup_pres bd\<^sup>-\<^sub>\<F>"
SAT_def: fixes \<Phi> :: "('a, 'b) lit list set" shows "SAT \<Phi> \<equiv> \<forall>\<xi>. wtE \<xi> \<longrightarrow> satPB \<xi> \<Phi>"
stake_Suc: fixes n :: "nat"   and s :: "'d stream" shows "stake (Suc n) s = stake n s @ [s !! n]"
map_def: fixes Fg :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" shows "local.map Fg \<equiv> if A_BxA.arr Fg then A_B.Map (fst Fg) (snd Fg) else B.null"
ent_ex_postI: fixes P :: "assn"   and Q :: "'b \<Rightarrow> assn"   and x :: "'b" assumes "P \<Longrightarrow>\<^sub>A Q x" shows "P \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x"
eq: fixes n :: "nat" shows "sum f {..<n + Suc (Max N)} = sum f N"
simps: fixes it :: "int node ref option"   and s :: "int" shows "os_sum' it s = os_it_has_next it \<bind> (\<lambda>b. if b then os_it_next it \<bind> (\<lambda>(x, it'). os_sum' it' (s + x)) else return s)"
Ana_fun_subterm: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and K :: "('fun, 'var) Term.term list"   and R :: "('fun, 'var) Term.term list" assumes "Ana (Fun f T) = (K, R)" shows "set R \<subseteq> set T"
set_integral_reflect: fixes S :: "real set"   and f :: "real \<Rightarrow> 'a" shows "set_lebesgue_integral lborel S f = LBINT x:{x. - x \<in> S}. f (- x)"
gate_def: fixes n :: "nat"   and A :: "complex mat" shows "gate n A \<equiv> dim_row A = 2 ^ n \<and> square_mat A \<and> unitary A"
while_rule: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c"   and b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and Q :: "'c \<Rightarrow> bool"   and r :: "('c \<times> 'c) set" assumes "P s"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. \<lbrakk>P s; \<not> b s\<rbrakk> \<Longrightarrow> Q s"   and "wf r"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "Q (while b c s)"
nc: shows "length (row (mat_multI ze pl ti nr m1 m2) i) = nc"
Nil_is_map_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "([] = map f xs) = (xs = [])"
surjective: fixes F :: "('f \<times> 'g) set"   and f :: "'f \<Rightarrow> 'g"   and A :: "'f" assumes "su_rel_fun F f" obtains B :: "'g" where "(A, B) \<in> F"
axioms_4: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "functor A B G"
f2: fixes h :: "hmultiset"   and ha :: "hmultiset"   and hb :: "hmultiset" shows "h + ha - (h - hb) = hb + ha - (hb - h)"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier \<G>"   and "y \<in> carrier \<G>"   and "z \<in> carrier \<G>" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
interval_def: fixes T :: "real set" shows "interval T \<equiv> is_interval T"
lm090: fixes P :: "('a \<times> 'b) set"   and Y :: "'b set" assumes "runiq P" shows "P\<inverse> `` (Range P - Y) \<inter> P\<inverse> `` Y = {}"
23: shows "bprv (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (cnj (PP \<langle>PP \<langle>\<phi>L \<phi>\<rangle>\<rangle>) (PP \<langle>imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>\<rangle>)))"
ex1I: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "\<exists>!x. P x"