hcju/isabelleps huggingface dataset:
intro\<^sub>1: fixes \<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)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<R>\<^sub>1p \<Gamma> \<S> P x xa xaa xaaa xaaaa xaaaaa) xa) lc'" shows "\<R>p \<Gamma> \<S> P lc lc'"
ide_in_hom: fixes a :: "(bool, 'arr list) concrete_category.arr" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
spr_simAbs_fst_snd: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "toSet (snd ec) \<subseteq> toSet (fst ec)"
simple_function_borel_measurable: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "f \<in> borel_measurable M"   and "finite (f ` space M)" shows "simple_function M f"
invertible_lf_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "invertible_lf s1 s2 f \<equiv> linear s1 s2 f \<and> invertible_lf_axioms f"
h4: shows "\<forall>m\<in>monomials (p + q). lookup m var \<le> n"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
le: fixes x :: "'a" assumes "x \<noteq> a" shows "norm (g x - m) \<le> norm (f x - l)"
assms: shows "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"
inext_atLeast: fixes n :: "nat"   and t :: "nat" assumes "n \<le> t" shows "inext t {n..} = Suc t"
connex_axioms: shows "connex A (\<lambda>x y. y \<sqsubseteq> x)"
comp_in_homI: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and c :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
type_less_x_Boolean: fixes \<tau> :: "'a basic_type" assumes "\<tau> < Boolean" obtains "\<tau> = OclVoid"
sqrt_eq_floor_sqrt: fixes n :: "nat" shows "Discrete.sqrt n = nat \<lfloor>sqrt (real n)\<rfloor>"
adv_P2_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'e spmf"   and S2 :: "'b \<Rightarrow> 'f \<Rightarrow> 'e spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf"   and m1 :: "'a"   and m2 :: "'b"   and D :: "'e \<Rightarrow> bool spmf" assumes "sim_det_def R1 S1 R2 S2 funct" shows "sim_det_def.adv_P2 R2 S2 funct m1 m2 D = \<bar>spmf (R2 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S2 m2 o2 \<bind> D)) True\<bar>"
match_tcp_flags_conjunct_option_None: fixes f1 :: "ipt_tcp_flags"   and f2 :: "ipt_tcp_flags"   and pkt :: "tcp_flag set" assumes "match_tcp_flags_conjunct_option f1 f2 = None" shows "\<not> (match_tcp_flags f1 pkt \<and> match_tcp_flags f2 pkt)"
postdominate_trans: fixes n'' :: "'node"   and n :: "'node"   and n' :: "'node" assumes "n'' postdominates n"   and "n' postdominates n''" shows "n' postdominates n"
foundedSubs: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool"   and Gamma :: "'b"   and sigma :: "'b" assumes "founded subs P (tree subs Gamma)"   and "sigma \<in> subs Gamma" shows "founded subs P (tree subs sigma)"
sum_wt_ParT_notWtFT_notFinished: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and s :: "'state" assumes "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "\<not> finished (cl ! n)" shows "sum (wt (ParT cl) s) {brnL cl n..<brnL cl n + brn (cl ! n)} = 1 / real (length cl) / (1 - WtFT cl) * sum (wt (cl ! n) s) {..<brn (cl ! n)}"
big_subsetI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "L F f \<subseteq> L F g"
worker_wrapper_id: fixes wrap :: "'d \<rightarrow> 'c"   and unwrap :: "'c \<rightarrow> 'd"   and computation :: "'c"   and body :: "'c \<rightarrow> 'c" assumes "wrap oo unwrap = ID"   and "computation = fix\<cdot>body" shows "computation = wrap\<cdot>(fix\<cdot>(unwrap oo body oo wrap))"
simps_1: fixes D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr\<^sub>p\<^sub>c [] D = [[]]"
Congruence_def: shows "Congruence = {(a, b). a \<in> G \<and> b \<in> G \<and> inverse a \<cdot> b \<in> K}"
distinct_9: fixes x1 :: "'a" shows "prot_atom.Atom x1 \<noteq> OccursSecType"
triangle_equiv_form_2: shows "((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) = (\<r>[g] \<cdot> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] = g)"
simps_2: fixes x :: "bit"   and l :: "bit list"   and i :: "nat"   and n :: "nat" shows "select (x # l) i n = (if i \<le> 0 then selecthelp (x # l) n else select l (i - 1) (n - 1))"
fs': shows "\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p}"
strict_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "strict_prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
assms_4: shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b))"
reaches_append_iff: fixes fsm :: "'b fsm"   and st :: "hf"   and xs :: "'b list"   and ys :: "'b list"   and st' :: "hf" shows "reaches fsm st (xs @ ys) st' = (\<exists>st''. reaches fsm st xs st'' \<and> reaches fsm st'' ys st')"
weakBisimClosed: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<approx> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<approx> p \<bullet> Q"
weight_return_spmf: fixes x :: "'a" shows "weight_spmf (return_spmf x) = 1"
next'_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('a, 'c) Propagate.configuration"   and c1 :: "('a, 'c) Propagate.configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.next' summary results_in c0 c1 = ((\<exists>loc t. dataflow_topology.next_propagate' summary results_in c0 c1 loc t) \<or> (\<exists>loc t n. dataflow_topology.next_change_multiplicity' c0 c1 loc t n) \<or> c1 = c0)"
comp_null_2: fixes f :: "'a \<times> 'a" shows "VoV.comp f VoV.null = VoV.null"
assms_3: shows "(s, a, s') \<in> automaton.trans A"
log_base_root: fixes n :: "nat"   and b :: "real"   and x :: "real" assumes "0 < n"   and "0 < b" shows "log (root n b) x = real n * log b x"
enat_unfold_stop: fixes stop :: "'c \<Rightarrow> bool"   and a :: "'c"   and "next" :: "'c \<Rightarrow> 'c" assumes "stop a" shows "enat_unfold stop next a = 0"
AudioPlayer_CTRL_Init_def: shows "AudioPlayer_CTRL_Init = ''Off''"
density_density_eq: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "density (density M f) g = density M (\<lambda>x. f x * g x)"
lm032: fixes a :: "('d \<times> 'c set) set" assumes "{} \<notin> Range a"   and "\<forall>pair\<in>a. finite (snd pair)"   and "finite a"   and "runiq a" shows "card (pseudoAllocation a) = sum (card \<circ> snd) a"
no_0_in_smods: fixes p :: "real poly"   and q :: "real poly" shows "0 \<notin> set (smods p q)"
f'_nonpos: fixes x :: "real" assumes "x0 \<le> x" shows "f' x \<le> 0"
assms_2: shows "local.prefixes (Suc t) j \<up>"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "zero_hom hom"   and "semigroup_add_hom hom" shows "monoid_add_hom hom"
countable_Collect: fixes A :: "'a set"   and \<phi> :: "'a \<Rightarrow> bool" assumes "countable A" shows "countable {a \<in> A. \<phi> a}"
u1: fixes x :: "'a" shows "x - P *\<^sub>V x \<in> orthogonal_complement (space_as_set M)"
s: shows "(\<Sum>i\<in>{a, b}. f i * i) = d"
MP_holds_def: fixes \<phi> :: "('sf, 'sc, 'sz) formula"   and \<psi> :: "('sf, 'sc, 'sz) formula" shows "MP_holds \<phi> \<psi> \<equiv> valid (\<phi> \<rightarrow> \<psi>) \<longrightarrow> valid \<phi> \<longrightarrow> valid \<psi>"
ifte_thms_2: fixes e1 :: "'a"   and e2 :: "'a" shows "If FF then e1 else e2 = e2"
b: shows "(a, LENGTH('a)) \<in> set (map (Pair a) (pfxes TYPE('a)))"
finite_big_d': fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "finite (set (big_d' a b y i))"
undual_leq: fixes x' :: "'c dual"   and y' :: "'c dual" shows "(undual x' \<sqsubseteq> undual y') = (y' \<sqsubseteq> x')"
prod_encode_inverse: fixes x :: "nat \<times> nat" shows "prod_decode (prod_encode x) = x"
valid_path_join_eq: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "pathfinish g1 = pathstart g2" shows "valid_path (g1 +++ g2) = (valid_path g1 \<and> valid_path g2)"
zero_fun_def: shows "0 = (\<lambda>x. 0::'j)"
arc_top_edge: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "(x[[top]]) * x = x[[x]]"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "well_founded A less"   and "transitive A less" shows "well_founded_ordered_set A less"
transitionWeakTransition: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P) \<Psi>" shows "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
nth_eq_iff_index_eq: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "(xs ! i = xs ! j) = (i = j)"
schur_upper_triangular_3: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])" shows "similar_mat A (schur_upper_triangular A es)"
that_2: fixes a :: "hf"   and b :: "hf" assumes "y = b \<triangleleft> a"   and "a \<^bold>\<notin> b" shows "thesis"
inject_6: fixes x2 :: "pi"   and x1 :: "pi"   and y2 :: "pi"   and y1 :: "pi" shows "(x2 \<oplus> x1 = y2 \<oplus> y1) = (x2 = y2 \<and> x1 = y1)"
assm: shows "\<forall>x\<in>X. 0 \<le> v x"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" and xs :: "'a list" where "x = xa # xs"
Sound_def: fixes Rule :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" shows "Sound Rule \<equiv> \<forall>I P1 P2 C. Rule P1 P2 C \<longrightarrow> I \<Turnstile> P1 \<longrightarrow> I \<Turnstile> P2 \<longrightarrow> I \<Turnstile> C"
simps_10: fixes V :: "'a"   and T :: "ty"   and e :: "'a exp" shows "max_vars {V:T; e} = max_vars e + 1"
a_Inj1_inv_V: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, Inj1 eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2" obtains e :: "Syntax.term" and eP :: "Syntax.term" where "ea__ = Inj1 e" and "ePa__ = Inj1 eP" and "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>1"
stake_pos_minWait: fixes rs :: "'rule stream"   and s :: "'state"   and r :: "'rule" assumes "fair rs"   and "minWait rs s < pos rs r"   and "r \<in> R"   and "s \<in> S" shows "pos (stl (trim rs s)) r = pos rs r - Suc (minWait rs s)"
asym: fixes a :: "'d"   and b :: "'d" assumes "a < b"   and "b < a" shows "False"
le_iff_meet: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> y) = (x \<squnion>\<^bsub>L\<^esub> y = y)"
semiring_norm_140: fixes n :: "num" shows "(numeral n = (1::'a)) = (n = num.One)"
AnalBinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "FBinds (delete x \<Gamma>)\<cdot>ae = FBinds \<Gamma>\<cdot>ae"
Suc_1: fixes t :: "('a, 'b) tm" assumes "x = num_args t - 1"   and "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b chop t"
post_rule': fixes P :: "assn"   and f :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and R :: "'a \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<not> <P> f <R>" shows "\<exists>x. \<not> (Q x \<Longrightarrow>\<^sub>A R x)"
smallo': fixes c :: "real" assumes "exp 1 < c" shows "(\<lambda>x. real (Lcm {1..nat \<lfloor>x\<rfloor>})) \<in> o((powr) c)"
intros_10: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and c\<^sub>1 :: "('d, 'e, 'f) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
list_of_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lfinite xs"   and "lfinite ys" shows "list_of (lappend xs ys) = list_of xs @ list_of ys"
closed_Real_halfspace_Re_le: fixes x :: "real" shows "closed (\<real> \<inter> {w. Re w \<le> x})"
uverts_edge_ugraph: fixes p :: "real"   and n :: "nat"   and es :: "nat set set" assumes "edge_space p" shows "uverts (edge_space.edge_ugraph n es) = edge_space.S_verts n"
valid_arbitrary_strategy: fixes p :: "Player" shows "G'.strategy p G'.\<sigma>_arbitrary"
response_AND_def: fixes R :: "'rand0 \<times> 'rand1"   and W :: "'witness0 \<times> 'witness1"   and s :: "'bool" shows "response_AND R W s = (let (r0, r1) = R; (w0, w1) = W in response0 r0 w0 s \<bind> (\<lambda>z0. response1 r1 w1 s \<bind> (\<lambda>z1. return_spmf (z0, z1))))"
Un_infinite: fixes S :: "'c set"   and T :: "'c set" assumes "infinite S" shows "infinite (S \<union> T)"
epigraph_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" shows "epigraph S f = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
prems_11: shows "\<psi> \<noteq> false\<^sub>r"
assms: shows "(x, y) = split_i_j_x (dim_vec (lpoly_to_vec v)) l X"
geom_rand_walk_borel_measurable: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and n :: "nat" assumes "prob_grw p M geom_proc u d init" shows "geom_proc n \<in> borel_measurable M"
\<phi>'\<psi>': shows "meta_adjunction C D G.F G' \<phi>' \<psi>'"
arr_iff_in_hom: fixes f :: "'a1" shows "A1.arr f = \<guillemotleft>f : A1.dom f \<rightarrow>\<^sub>A\<^sub>1 A1.cod f\<guillemotright>"
inc_measure_def: fixes c :: "nat \<times> cell list \<times> cell list" shows "inc_measure c = (abc_inc_stage1 c, abc_inc_stage2 c, abc_inc_stage3 c)"
MM5: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<or> \<box>G \<longrightarrow> \<box>(F \<or> G)"
SUP_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b \<times> 'c"   and D :: "'a \<Rightarrow> 'b \<times> 'c" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Sup_prod (\<P> C A) = Sup_prod (\<P> D B)"
rb_aux_inv_D9_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat" assumes "rb_aux_inv (bs, ss, ps)"   and "j < length fs"   and "Inr j \<notin> set ps" shows "is_RB_in dgrad rword (set bs) (term_of_pair (0::'a, j))"
rightUniquePair: fixes f :: "('c \<times> 'd) set"   and x :: "'c"   and y :: "'d" assumes "runiq f"   and "(x, y) \<in> f" shows "y = f ,, x"
word_less_def: fixes a :: "'a word"   and b :: "'a word" shows "(a < b) = (uint a < uint b)"
INF_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` {}) = top"
results_in_mono_raw: fixes t1 :: "'t"   and t2 :: "'t"   and s1 :: "'sum"   and s2 :: "'sum" assumes "t1 \<le> t2"   and "s1 \<le> s2" shows "results_in t1 s1 \<le> results_in t2 s2"
regular_Compl: fixes S :: "'b list set" assumes "regular S" shows "regular (- S)"
aqgrp_carrier: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "aGroup A"   and "A +> H" shows "set_rcs (b_ag A) H = set_ar_cos A H"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "indRelLp a1 a2" obtains S :: "'procS" where "\<lbrakk>S\<rbrakk> \<in>T a1" and "S \<in>S a2"
simps_2: fixes x :: "'a" shows "fromSingleton [x] = Inr x"
dec_above_def: fixes b :: "nat"   and i :: "nat"   and x :: "nat" shows "dec_above b i x = (if x \<le> b then x else x - i)"
eq_\<I>_converter_sym: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv' :: "('a, 'b, 'c, 'd) converter"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"
DERIV_unique: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x)"   and "(f has_field_derivative E) (at x)" shows "D = E"
measure_Pi_pmf_Pi: fixes A :: "'a set"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A" shows "measure_pmf.prob (Pi_pmf A dflt p) (Pi A B) = (\<Prod>x\<in>A. measure_pmf.prob (p x) (B x))"
bspec: fixes A :: "'h set"   and P :: "'h \<Rightarrow> bool"   and x :: "'h" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
brI: fixes a :: "'a"   and \<alpha> :: "'c \<Rightarrow> 'a"   and c :: "'c"   and I :: "'c \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
QR_decomposition_mult: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "A = fst (QR_decomposition A) ** snd (QR_decomposition A)"
treeLess_induct: fixes phi :: "('c, 'd) tree \<Rightarrow> bool"   and T :: "('c, 'd) tree" assumes "\<And>T'. (\<And>T. (T, T') \<in> treeLess \<Longrightarrow> phi T) \<Longrightarrow> phi T'" shows "phi T"
le_ay: fixes y :: "'a" assumes "y \<in> S" shows "a \<bullet> x \<le> a \<bullet> y"
chanEqTrans: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a"   and L :: "'a" assumes "\<Psi> \<turnstile> M \<leftrightarrow> N"   and "\<Psi> \<turnstile> N \<leftrightarrow> L" shows "\<Psi> \<turnstile> M \<leftrightarrow> L"
ord_eq_le_trans: fixes a :: "'f"   and b :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'f" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
limitin_atin_self: fixes Y :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and X :: "'c topology" shows "limitin Y f (f a) (atin X a) = (f a \<in> topspace Y \<and> (a \<in> topspace X \<longrightarrow> (\<forall>V. openin Y V \<and> f a \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> a \<in> U \<and> f ` U \<subseteq> V))))"
defl_eq_om_trivial: fixes x :: "'a" shows "(x\<^sup>\<omega> = (0::'a)) = (\<forall>y. y \<le> x \<cdot> y \<longrightarrow> y = (0::'a))"
add_im_sheaf_def: shows "im0.add_im_sheaf \<equiv> \<lambda>V x y. 0::'e"
enum_all_UNIV: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "enum_all P = Ball UNIV P"
some_img_def: shows "some_img = (SOME img. set_category_given_img (\<cdot>) img \<AA>)"
vector_matrix_mult_def: fixes v :: "('b, 'd) Finite_Cartesian_Product.vec"   and m :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "v v* m \<equiv> \<chi>j. \<Sum>i\<in>UNIV. m $h i $h j * v $h i"
inverts_components: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "inverse_transformation A B F G \<tau>"   and "partial_magma.ide A a" shows "category.inverse_arrows B (\<tau> a) (inverse_transformation.map A B G \<tau> a)"
r_inv_ex: fixes x :: "'a" shows "\<exists>y\<in>UNIV. x + y = (0::'a)"
sorted_wrt_list_of_set_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "sorted_wrt_list_of_set R A = (if finite A then THE xs. set xs = A \<and> distinct xs \<and> Linorder_Relations.sorted_wrt R xs else [])"
Koszul_syz_sigs_def: fixes bs :: "('a \<Rightarrow>\<^sub>0 'b) list" shows "Koszul_syz_sigs bs = filter_min (adds\<^sub>t) (Koszul_syz_sigs_aux bs 0)"
matrix_inv_1: fixes M :: "(('b, 'c) vec, 'd) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'b)"
sumhr_def: shows "sumhr = (\<lambda>(M, N, f). (*f2* (\<lambda>m n. sum f {m..<n})) M N)"
cert_okD1: fixes c :: "'c list"   and n :: "nat"   and T :: "'c"   and B :: "'c"   and A :: "'c set"   and pc :: "nat" assumes "cert_ok c n T B A"   and "pc < n" shows "c ! pc \<in> A"
morphism_locally_ringed_spaces_axioms_def: fixes X :: "'c set"   and is_open\<^sub>X :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>X :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and add_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'f set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'f \<Rightarrow> 'd" shows "morphism_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> \<forall>x V. x \<in> X \<longrightarrow> is_open\<^sub>Y V \<longrightarrow> f x \<in> V \<longrightarrow> ind_mor_btw_stalks.is_local X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f x V \<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub>"
simps_4: fixes l1 :: "nat bdd"   and r1 :: "nat bdd"   and l2 :: "nat bdd"   and r2 :: "nat bdd"   and T :: "bool list list" shows "check_eq (Branch l1 r1) (Branch l2 r2) T = (check_eq l1 l2 T \<and> check_eq r1 r2 T)"