hcju/isabelleps huggingface dataset:
piecewise_C1_differentiable_D2: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "g1 +++ g2 piecewise_C1_differentiable_on {0..1}"   and "pathfinish g1 = pathstart g2" shows "g2 piecewise_C1_differentiable_on {0..1}"
incLevel_def: shows "incLevel = (\<lambda>(n, a). (Suc n, a))"
if_assertion: fixes p :: "'b Assertion"   and x :: "'b"   and y :: "'b" shows "(If p then x else y) = {\<cdot> p } * x \<squnion> {\<cdot> - p } * y"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "PdXXX.stable_on ds X" shows "PdXXX.CD_on ds X = X"
STL5: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<and> G) = (\<box>F \<and> \<box>G)"
init_gc_def: shows "local.init_gc \<equiv> \<lparr>global_config.processes = mset (map local.init_pc (program.processes prog)), state = \<lparr>global_state.variables = init_valuation (global_vars prog)\<rparr>\<rparr>"
p_squarefree: fixes x :: "real" shows "\<not> (poly p' x = 0 \<and> poly (sturm_squarefree p ! 1) x = 0)"
fft: shows "(\<lambda>x. f x - f t) absolutely_integrable_on {- pi..pi}"
Gr_domain_1: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "Domain (BNF_Def.Gr A f) = A"
seqr_left_one_point_true: fixes x :: "bool \<Longrightarrow> 'b"   and P :: "('c, 'b) urel"   and Q :: "('b, 'd) urel" assumes "vwb_lens x" shows "(P \<and> $x\<acute>) ;; Q = P\<lbrakk>true/$x\<acute>\<rbrakk> ;; Q\<lbrakk>true/$x\<rbrakk>"
gauss_cnj_add: fixes a :: "gauss_int"   and b :: "gauss_int" shows "gauss_cnj (a + b) = gauss_cnj a + gauss_cnj b"
circline_intersection_symetry: fixes H1 :: "circline"   and H2 :: "circline" shows "circline_intersection H1 H2 = circline_intersection H2 H1"
PiE_eq_singleton: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'e set"   and f :: "'d \<Rightarrow> 'e" shows "(Pi\<^sub>E I S = {restrict f I}) = (\<forall>i\<in>I. S i = {f i})"
Un_closed: fixes x :: "'a set"   and y :: "'a set" assumes "x closed"   and "y closed" shows "x \<union> y closed"
bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst: fixes A :: "('c strand_label \<times> ('b, 'a) stateful_strand_step) list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
normF_xl: fixes \<epsilon> :: "real"   and n :: "nat"   and F :: "'a set" assumes "I \<epsilon> \<le> n"   and "0 < \<epsilon>"   and "finite F" shows "normF F (\<lambda>a. x n a - l a) \<le> \<epsilon>"
invertible_inverse_closed: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "inverse u \<in> M"
iso_iff_section_and_retraction: fixes f :: "'a" shows "local.iso f = (section f \<and> retraction f)"
consec_un_defined: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> Rep_nat_int (i \<squnion> j) \<in> {S. \<exists>m n. {m..n} = S}"
sel_2: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "'c"   and x4 :: "'b" shows "actOf (Trans x1 x2 x3 x4) = x2"
imp_ex: fixes P :: "'d \<Rightarrow> bool"   and Q :: "bool" shows "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)"
applicableBackjump_def: fixes state :: "State" shows "applicableBackjump state \<equiv> \<exists>state'. appliedBackjump state state'"
le_less_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
assms: shows "arg z = pi / 2 \<or> arg z = - pi / 2"
vector_ssub_ldistrib: fixes c :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "c *s (x - y) = c *s x - c *s y"
lcs_comm: fixes s :: "'a"   and t :: "'a" shows "lcs s t = lcs t s"
less_induct: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
f3: shows "rk {A, B, C, A', B', C', P} = 3"
ES_valid_def: fixes ES :: "'c ES_rec" shows "ES_valid ES \<equiv> es_inputs_are_events ES \<and> es_outputs_are_events ES \<and> es_inputs_outputs_disjoint ES \<and> traces_contain_events ES \<and> traces_prefixclosed ES"
return_time_pos: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x"   and "closed P" shows "0 < return_time P x"
foldr_aux_1_2: fixes us :: "'var list"   and g :: "'g"   and n :: "'node"   and u :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and m :: "'node" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}"   and "m \<noteq> n" shows "Mapping.lookup u' m = Mapping.lookup u m"
obt_5: shows "(\<Sum>v\<in>S. u v *\<^sub>R v) = (0::'a)"
Skip: fixes s :: "char list \<Rightarrow> int option" shows "(SKIP, s) \<Rightarrow>\<^sub>A Suc 0 \<Down> s"
assms: shows "\<sigma>' \<in> small_nstep P \<sigma> (Suc n)"
\<R>\<^sub>L_inv: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and ls' :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr"   and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<R>\<^sub>L (inv \<alpha>) (inv \<beta>) ls' = ls) = (\<R>\<^sub>L \<alpha> \<beta> ls = ls')"
functors_preserve_objects: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and A :: "'b" assumes "Functor F : AA \<longrightarrow> BB"   and "A \<in> Ob\<^bsub>AA\<^esub>" shows "F \<^bsub>\<o>\<^esub> A \<in> Ob\<^bsub>BB\<^esub>"
k_v_included: shows "(k, v) \<in> {..<t} \<times> set (\<Pi>\<^sub>\<V>)"
induct: fixes P :: "(event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> 'a) \<Rightarrow> event_data option list set list \<times> event_data option list set list \<Rightarrow> bool"   and a0 :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> 'a"   and a1 :: "event_data option list set list \<times> event_data option list set list" assumes "\<And>f x xs y ys. P f (xs, ys) \<Longrightarrow> P f (x # xs, y # ys)"   and "\<And>f ys. P f ([], ys)"   and "\<And>f xs. P f (xs, [])" shows "P a0 a1"
independent_distinct: fixes Us :: "'a list" assumes "local.independent K Us" shows "distinct Us"
b_e_type_br_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Br i" shows "i < length (label \<C>)"
dg_Set_Hom_vifunion_in_Vset: fixes X :: "V"   and Y :: "V" assumes "X \<in>\<^sub>\<circ> Vset \<alpha>"   and "Y \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>X. \<Union>\<^sub>\<circ> (VLambda Y (Hom (dg_Set \<alpha>) a) `\<^sub>\<circ> Y)) `\<^sub>\<circ> X) \<in>\<^sub>\<circ> Vset \<alpha>"
qiter_fun_exp: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "prod.qiter_fun x y z = sup_prod x (times_prod' y z)"
A: shows "char_matrix (jordan_matrix n_as) ev ^\<^sub>m k \<in> carrier_mat (sum_list (map fst n_as)) (sum_list (map fst n_as))"
empty_subsetI: fixes A :: "'a set" shows "{} \<subseteq> A"
sub_funty: fixes f1 :: "funty"   and f2 :: "funty" assumes "f1 <:: f2" shows "TFun f1 <: TFun f2"
uniform_complement: assumes "\<k> \<le> \<v> - 1" shows "block_design \<V> \<B>\<^sup>C (\<v> - \<k>)"
en: fixes a :: "'action"   and b :: "'action"   and p :: "'state" assumes "ind a b"   and "en a p" shows "en b p = en b (ex a p)"
assms_1: shows "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
g_def: shows "g \<equiv> \<lambda>j. if 1 \<le> j then f (j - 1) else b"
ideal_generated_pair: fixes a :: "'a"   and b :: "'a" shows "ideal_generated {a, b} = {p * a + q * b |p q. True}"
scale_left_diff_distrib: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and b :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "(a - b) \<odot> x = a \<odot> x - b \<odot> x"
column_def: fixes j :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" shows "column j A = (\<chi>i. A $ i $ j)"
Mh: shows "h \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
rename_closed: 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 f :: "'b"   and p :: "'a \<Rightarrow> 'a" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "lconsts f = {}" shows "renameLCs p f = f"
trimmed_imp_eventually_sgn: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "trimmed F" shows "\<forall>\<^sub>F x in at_top. sgn (eval F x) = sgn (fst (dominant_term F))"
real_alg_2_main_def: fixes ri :: "root_info"   and rai :: "int poly \<times> rat \<times> rat" shows "real_alg_2_main ri rai \<equiv> let p = poly_real_alg_1 rai in if degree p = 1 then Rational (Rat.Fract (- coeff p 0) (coeff p 1)) else case normalize_bounds_1 rai of (p', l, r) \<Rightarrow> Irrational (root_info.number_root ri r) (p', l, r)"
converse_rtrancl_into_rtrancl: fixes a :: "'x"   and b :: "'x"   and r :: "('x \<times> 'x) set"   and c :: "'x" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
cons_ap_list: fixes f :: "'b \<Rightarrow> 'a"   and fs :: "('b \<Rightarrow> 'a) list"   and xs :: "'b list" shows "(f # fs) \<diamondop> xs = map f xs @ fs \<diamondop> xs"
ideI: fixes a :: "'a" assumes "right (src \<mu>) a"   and "ide a" shows "Right.ide a"
lspasl_u_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.tern_rel plus sep_disj h zero h"
bij_betw_restrict0: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw (f \<down> A) A B"
ef: shows "intersect_segment_xline prec ((x0, y0), x1, y1) x = Some (e, f)"
exhaust: fixes y :: "'a delete_message" obtains x :: "'a" where "y = DeleteMessage x"
vsubset_vsingleton_left: fixes a :: "V"   and A :: "V" shows "(set {a} \<subseteq>\<^sub>\<circ> A) = (a \<in>\<^sub>\<circ> A)"
Rolle_deriv: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "a < b"   and "f a = f b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)" shows "\<exists>z>a. z < b \<and> f' z = (\<lambda>v. 0)"
exec_moves_ci_appD: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and Es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and fr' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_moves ci P t Es h (stk, loc, pc, None) ta h' fr'" shows "ci_app ci (compEs2 Es ! pc) (compP2 P) h stk loc undefined undefined pc []"
v: shows "(r \<cdot> s) \<^sup>@ Suc i = v"
unique_letter_word: fixes w :: "'b list"   and a :: "'b" assumes "\<forall>c. c \<in> set w \<longrightarrow> c = a" shows "\<exists>k. w = [a] \<^sup>@ k"
D: shows "offs_num (length ns) (x # xs) index key mi ma 0 = 0"
finsertI1: fixes a :: "'x"   and B :: "'x fset" shows "a |\<in>| finsert a B"
refl_on_def': fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
input2state_take: fixes x1 :: "nat"   and A :: "nat"   and i :: "value list"   and x :: "vname" assumes "x1 < A"   and "A \<le> length i"   and "x = I x1" shows "input2state i $ x1 = input2state (take A i) $ x1"
coprime_Suc_left_nat: fixes n :: "nat" shows "coprime (Suc n) n"
hoare\<^sub>3'_def: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c \<Rightarrow> ('d \<times> 'c) option" shows "\<lbrace>P\<rbrace> M\<dagger> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> True | Some a \<Rightarrow> False)"
tp_max: shows "max (te p (i + 1)) (te q (i + 1)) = te p (i + 1)"
Nil_is_append_conv: fixes xs :: "'e list"   and ys :: "'e list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
absorb2: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b" shows "sup a b = b"
focus_eq_zero_iff: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(focus X p = 0) = (p = 0)"
cases: fixes a :: "color list" assumes "valid' a" obtains "a = []"   | n :: "nat" where "a = replicate n__ R" and "3 \<le> n__"   | xs :: "color list" where "a = B # xs__" and "valid' xs__"   | xs :: "color list" and n :: "nat" where "a = replicate n R @ B # xs" and "valid' xs" and "3 \<le> n"
invar_NodeL: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree"   and l' :: "('b \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "invar l'"   and "lvl l' = lvl l" shows "invar \<langle>l', (x, n), r\<rangle>"
divide_out_primepow_ex: fixes n :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "n \<noteq> (0::'a)"   and "\<exists>p\<in>prime_factors n. P p" obtains p :: "'a" and k :: "nat" and n' :: "'a" where "P p" and "prime p" and "p dvd n" and "\<not> p dvd n'" and "0 < k" and "n = p ^ k * n'"
pred_of_set_def: shows "pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)"
linear: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" shows "x \<preceq> y \<or> y \<preceq> x"
simps_2: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and as :: "'c negation_type list" shows "NegPos_map f (Pos a # as) = Pos (f a) # NegPos_map f as"
with_optimal_paths_solves': fixes R_pre :: "'a list"   and z :: "'a"   and R_post :: "'a list" assumes "R = R_pre @ z # R_post"   and "Q.hitting_paths z"   and "\<And>z'. z' \<in> set R_pre \<Longrightarrow> \<not> Q.hitting_paths z'" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
simps: fixes a1 :: "'c"   and a2 :: "'c"   and r :: "('c \<times> 'c) set" shows "((a1, a2) \<in> r\<^sup>+) = ((\<exists>a b. a1 = a \<and> a2 = b \<and> (a, b) \<in> r) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>+ \<and> (b, c) \<in> r))"
corr_1: shows "\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub> \<sqsubseteq> \<N>\<lbrakk> z \<rbrakk>\<^bsub>\<N>\<lbrace>\<Delta>\<rbrace>\<^esub>"
cs_return: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "IFC prob"   and "\<pi> n = IFC_def.return prob" shows "IFC_def.cs prob \<pi> n = [\<pi> n]"
Collect_eqI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Collect P = Collect Q"
try_SE_def: fixes ioprog :: "'e \<Rightarrow> ('f \<times> 'e) option" shows "try\<^sub>S\<^sub>E ioprog = (\<lambda>\<sigma>. case ioprog \<sigma> of None \<Rightarrow> Some (None, \<sigma>) | Some (outs, \<sigma>') \<Rightarrow> Some (Some outs, \<sigma>'))"
subresultant'_zero_lt: fixes J :: "nat"   and f :: "'b poly"   and g :: "'b poly"   and k :: "nat" assumes "J \<le> degree f"   and "J \<le> degree g"   and "J < k"   and "k < degree f + (degree g - J)" shows "det (subresultant'_mat J k f g) = (0::'b)"
fds_const_abs_converges: fixes c :: "'b"   and s :: "'b" shows "fds_abs_converges (fds_const c) s"
split_sel: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and list :: "'d list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (list = list.hd list # list.tl list \<longrightarrow> P (f2 (list.hd list) (list.tl list))))"
assms_2: shows "B \<lesssim>\<^sub>\<circ> A"
pickInp_asInp: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp" shows "(pickInp (asInp qinp), qinp) \<in> alphaInp"
real_of_int_div2: fixes n :: "int"   and x :: "int" shows "0 \<le> real_of_int n / real_of_int x - real_of_int (n div x)"
add_strict_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" assumes "a \<subset># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
take_interpret_floatariths: fixes d :: "nat"   and fas :: "floatarith list"   and vs :: "real list" assumes "d < length fas" shows "take d (interpret_floatariths fas vs) = interpret_floatariths (take d fas) vs"
liberal_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set" shows "liberal scf A Is \<equiv> \<forall>i\<in>Is. \<exists>x\<in>A. \<exists>y\<in>A. x \<noteq> y \<and> decisive scf A Is {i} x y \<and> decisive scf A Is {i} y x"
delete_edges_chain: fixes G :: "'b list_graph"   and as :: "('b \<times> 'b) list"   and bs :: "('b \<times> 'b) list" shows "FiniteListGraph.delete_edges G (as @ bs) = FiniteListGraph.delete_edges (FiniteListGraph.delete_edges G as) bs"
shiftl1_rev: fixes w :: "'b word" shows "shiftl1 w = word_reverse (shiftr1 (word_reverse w))"
eq2: shows "M0 * proj_k R * M0 = proj_k R"
BigAnd_semantics: fixes A :: "'a \<Rightarrow> bool"   and F :: "'a formula list" shows "A \<Turnstile> \<^bold>\<And>F = (\<forall>f\<in>set F. A \<Turnstile> f)"
integral_by_parts_in_bounded_closed_convex_set: fixes f :: "'b \<Rightarrow> real"   and I :: "real"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'c"   and P :: "'c"   and G :: "'c set" assumes "(f has_integral I) (cbox a b)"   and "((\<lambda>x. f x *\<^sub>R g x) has_integral P) (cbox a b)"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> g x \<in> G"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> 0 \<le> f x"   and "bounded G"   and "closed G"   and "convex G"   and "cbox a b \<noteq> {}" shows "P \<in> (*\<^sub>R) I ` G"
SN_induct: fixes r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "SN r"   and "\<And>a. (\<And>b. (a, b) \<in> r \<Longrightarrow> P b) \<Longrightarrow> P a" shows "P a"
q_prop: shows "q \<in> set qs \<and> squash (rpoly q x1) \<noteq> squash (rpoly q x2)"
q: shows "q = A * MPoly_Type.monom (monomial (Suc 0) x') 1 + b"
rel_map_2: fixes S1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('e, 'a, 'c) sp\<^sub>\<mu>"   and g1 :: "'f \<Rightarrow> 'b"   and g2 :: "'g \<Rightarrow> 'd"   and y :: "('e, 'f, 'g) sp\<^sub>\<mu>" shows "rel_sp\<^sub>\<mu> S1a S2a x (map_sp\<^sub>\<mu> g1 g2 y) = rel_sp\<^sub>\<mu> (\<lambda>x y. S1a x (g1 y)) (\<lambda>x y. S2a x (g2 y)) x y"
qe_eps\<^sub>1_def: fixes \<phi> :: "atom fm" shows "qe_eps\<^sub>1 \<phi> = (let as = DLO.atoms\<^sub>0 \<phi>; lbs = lbounds as; ebs = ebounds as in list_disj (inf\<^sub>- \<phi> # map (subst\<^sub>+ \<phi>) lbs @ map (subst \<phi>) ebs))"
simps_16: fixes M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and slpl :: "nat" shows "slp_of_fa_rev Pi M slp slpl = slp_of_fa_rev_cnst Pi Pi M slp slpl"
locally_similar_refl: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g" shows "locally_similar \<sigma> \<sigma>"
infs_any: fixes P :: "'a \<Rightarrow> bool"   and w :: "'a stream" assumes "infs P w" shows "Bex (sset w) P"
comp_cong: fixes f :: "'c::type \<Rightarrow> 'b::type"   and g :: "'d::type \<Rightarrow> 'c::type"   and x :: "'d::type"   and f' :: "'e::type \<Rightarrow> 'b::type"   and g' :: "'f::type \<Rightarrow> 'e::type"   and x' :: "'f::type" assumes "(f::'c::type \<Rightarrow> 'b::type) ((g::'d::type \<Rightarrow> 'c::type) (x::'d::type)) = (f'::'e::type \<Rightarrow> 'b::type) ((g'::'f::type \<Rightarrow> 'e::type) (x'::'f::type))" shows "((f::'c::type \<Rightarrow> 'b::type) \<circ> (g::'d::type \<Rightarrow> 'c::type)) (x::'d::type) = ((f'::'e::type \<Rightarrow> 'b::type) \<circ> (g'::'f::type \<Rightarrow> 'e::type)) (x'::'f::type)"
rep_eq: fixes x :: "'a"   and xa :: "'a poly" shows "coeff (Polynomial.smult x xa) = (\<lambda>n. x * coeff xa n)"
code_free_basis: fixes C :: "'b list set" assumes "code C" shows "C = \<BB>\<^sub>F C"
induct: fixes P :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"   and a0 :: "'a poly"   and a1 :: "'a poly" assumes "\<And>p q. (q \<noteq> 0 \<Longrightarrow> P q (primitive_part (pseudo_mod p q))) \<Longrightarrow> P p q" shows "P a0 a1"
borel_0_1_law: fixes F :: "nat \<Rightarrow> 'a set" assumes "indep_events F UNIV" shows "prob (\<Inter>n. \<Union> (F ` {n..})) = 0 \<or> prob (\<Inter>n. \<Union> (F ` {n..})) = 1"
g: shows "g \<in> N \<rightarrow>\<^sub>M L"
uminus_assn_def: fixes P :: "assn" shows "- P \<equiv> Abs_assn (\<lambda>h. in_range h \<and> \<not> h \<Turnstile> P)"
Eq: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dfa_of_pf n (Eq ks l) = eq_dfa n ks l"
stc_minus: fixes y :: "complex star" assumes "y \<in> HFinite" shows "stc (- y) = - stc y"
hyps: fixes n :: "nat" shows "mset ` {ys \<in> set (subseqs xs). length ys = n} = {ws. ws \<subseteq># mset xs \<and> size ws = n}"
single_one: shows "Poly_Mapping.single (0::'a) (1::'b) = 1"
initial_def: fixes start :: "'a \<Rightarrow> 'b"   and cfg :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.initial start cfg \<equiv> (\<forall>p. \<exists>v. msgs cfg <p, inM v> = 1) \<and> (\<forall>p m1 m2. m1 \<in># msgs cfg \<and> m2 \<in># msgs cfg \<and> isReceiverOf p m1 \<and> isReceiverOf p m2 \<longrightarrow> m1 = m2) \<and> (\<forall>v. msgs cfg <\<bottom>, outM v> = 0) \<and> (\<forall>p v. msgs cfg <p, v> = 0) \<and> states cfg = start"
INTEG_def: shows "INTEG = \<lparr>carrier = UNIV, mult = (*), one = 1, zero = 0, add = (+)\<rparr>"
homeq_imp_rcos: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier R"   and "x \<in> carrier R"   and "h x = h a" shows "x \<in> a_kernel R S h +> a"