hcju/isabelleps huggingface dataset:
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> M"   and "y \<in> M" shows "\<eta> (x \<cdot> y) = \<eta> x \<cdot>' \<eta> y"
flow_trans: fixes s :: "real"   and t0 :: "real"   and x0 :: "'a"   and t :: "real" assumes "s - t0 \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 (flow0 x0 (s - t0))" shows "flow0 x0 (s + t - t0) = flow0 (flow0 x0 (s - t0)) t"
gcd_mult_left: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (times c a) (times c b) = normalize (times c (gcd a b))"
intensional_rews_6: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and w :: "'b" shows "(\<forall>x. A x) w \<equiv> \<forall>x. A x w"
keys_mult_scalar_monomial_right: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and c :: "'f"   and p :: "'d \<Rightarrow>\<^sub>0 'f"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair"   and "c \<noteq> (0::'f)" shows "keys (term_powerprod.mult_scalar pair_of_term term_of_pair p (monomial c v)) = (\<lambda>t. term_powerprod.splus pair_of_term term_of_pair t v) ` keys p"
congruence_eye: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "congruence eye H = H"
Pi_split_insert_domain: fixes x :: "'g \<Rightarrow> 'h"   and i :: "'g"   and I :: "'g set"   and X :: "'g \<Rightarrow> 'h set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
assms_1: shows "fps_nth f 0 = (1::'a)"
meets_asym: fixes p :: "'a"   and q :: "'a" assumes "p \<parallel> q" shows "\<not> q \<parallel> p"
has_bochner_integral_I0i_power_exp_m: fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. x ^ k * exp (- x) * indicat_real {0<..} x) (fact k)"
CatchMatch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "'e"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> Abrupt s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
nn_integral_disjoint_family: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and B :: "nat \<Rightarrow> 'a set" assumes "f \<in> borel_measurable M"   and "\<And>n. B n \<in> sets M"   and "disjoint_family B" shows "set_nn_integral M (\<Union> (range B)) f = (\<Sum>n. set_nn_integral M (B n) f)"
vf\<^sub>2: shows "vertices f\<^sub>2 = [v] @ between (vertices f) v u @ u # vs"
assms_2: shows "adjoint_functors (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G'"
child_ix: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p" shows "ix (child p dir d) d = 2 * ix p d + Grid_Point.sgn dir"
invertibleI: fixes A :: "(('d, 'f) vec, 'e) vec"   and B :: "(('d, 'e) vec, 'f) vec" assumes "A ** B = mat (1::'d)"   and "B ** A = mat (1::'d)" shows "invertible A"
compatTst_def: fixes tst :: "test" shows "Example_PL.compatTst tst \<equiv> \<forall>s t. Example_PL.indisAbbrev s t \<longrightarrow> tval tst s = tval tst t"
assms_4: fixes es :: "('a \<times> 'a) list" assumes "fst e \<noteq> snd e"   and "P e es" shows "P e ((snd e, fst e) # es)"
inject_5: fixes x61 :: "'e set"   and x62 :: "('e, 'f, 'g) com"   and y61 :: "'e set"   and y62 :: "('e, 'f, 'g) com" shows "(While x61 x62 = While y61 y62) = (x61 = y61 \<and> x62 = y62)"
chop_empty2: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<and> \<parallel>t\<parallel> = 0 \<longrightarrow> r = s"
s_prod_iff: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "((a, A) \<in> R \<cdot> S) = (\<exists>B. (a, B) \<in> R \<and> (\<exists>f. (\<forall>b\<in>B. (b, f b) \<in> S) \<and> A = \<Union> (f ` B)))"
tendsto_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and g :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> \<bottom>"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
derives1_is_word_is_prefix_drop: fixes w :: "symbol list"   and a :: "symbol list"   and b :: "symbol list" assumes "is_word w"   and "is_prefix w a"   and "derives1 a b" shows "derives1 (drop (length w) a) (drop (length w) b)"
assms_2: shows "f \<in>\<^sub>\<circ> Vset \<alpha>"
essential_class_def: fixes C :: "('s \<times> 's option) set" shows "KN.essential_class C = (C \<in> UNIV // KN.communicating \<and> (SIGMA x:UNIV. set_pmf (KN.Kp x))\<^sup>* `` C \<subseteq> C)"
list_all_length: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "list_all P xs = (\<forall>n<length xs. P (xs ! n))"
D: shows "Crypt (SesK SK) (Num 0) \<in> used s"
set_less_eq_aux'_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>'' B"   and "B \<sqsubseteq>'' A" shows "A = B"
tLength_tZip: fixes xs :: "'a Trace"   and ys :: "'b Trace"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "tLength xs = tLength ys" shows "tLength (tZip f xs ys) = tLength xs"
IH_1: fixes ps :: "(real, 'a) vec list" assumes "set_kdt l \<union> set ps - set (nearest_nbors n ps p l) \<noteq> {}" shows "length (nearest_nbors n ps p l) = n"
relH_def: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" shows "relH as h h' \<equiv> in_range (h, as) \<and> in_range (h', as) \<and> (\<forall>t. \<forall>a\<in>as. refs h t a = refs h' t a \<and> arrays h t a = arrays h' t a)"
assms_1: shows "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
nsets_disjoint_2: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<inter> Y = {}" shows "[X \<union> Y]\<^bsup>2\<^esup> = [X]\<^bsup>2\<^esup> \<union> [Y]\<^bsup>2\<^esup> \<union> (\<Union>x\<in>X. \<Union>y\<in>Y. {{x, y}})"
A_splay_max_ub3: fixes t :: "'b tree" shows "A_splay_max t \<le> 3 * \<phi> t + 1"
unowned_shared_nth_update: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<O>' :: "nat set"   and p' :: "'p"   and is' :: "instr list"   and xs' :: "nat \<Rightarrow> nat option"   and sb' :: "unit"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"   and "\<O> \<subseteq> \<O>'" shows "unowned_shared \<S> (ts[i := (p', is', xs', sb', \<D>', \<O>', \<R>')])"
r_def: shows "r = R / (2 * sqrt (real DIM('a)))"
wf_map_prod_image: fixes r :: "('g \<times> 'g) set"   and f :: "'g \<Rightarrow> 'h" assumes "wf r"   and "inj f" shows "wf (map_prod f f ` r)"
Sigma_msetD1: fixes a :: "'a"   and b :: "'b"   and A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset" assumes "(a, b) \<in># Sigma_mset A B" shows "a \<in># A"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c" assumes "\<And>a. partial_magma.ide A a \<Longrightarrow> category.iso B (\<tau> a)" shows "natural_isomorphism_axioms A B \<tau>"
d: shows "\<eta> $$ Y maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ Y to NTCod \<eta> @@ Y"
vimage_Un: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" shows "f -` (A \<union> B) = f -` A \<union> f -` B"
map_of_list_def: fixes xs :: "'c list" shows "map_of_list xs = (\<lambda>i. if i < length xs then Some (xs ! i) else None)"
match_list_matches: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and ms :: "'c match_expr list"   and a :: "action"   and p :: "'d" shows "match_list \<gamma> ms a p = (\<exists>m\<in>set ms. matches \<gamma> m a p)"
decrpoly_normh: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "polybound0 p" shows "isnpolyh (decrpoly p) (n0 - 1)"
intros: fixes q :: "'d"   and W :: "'d set"   and Q' :: "'d set"   and Q :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and W' :: "'d set"   and rcm' :: "('d, 'e) ta_rule \<Rightarrow> nat option" assumes "q \<in> W"   and "Q' = Q \<union> {lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1}"   and "W' = W - {q} \<union> ({lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1} - Q)"   and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm' r = (if q \<in> set (rhsq r) then Some (the (rcm r) - 1) else rcm r)" shows "((Q, W, rcm), Q', W', rcm') \<in> br'_step \<delta>"
timesD2: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf" assumes "\<langle>a, b\<rangle> \<^bold>\<in> A * B" shows "b \<^bold>\<in> B"
project_leadsETo_I_lemma: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "extend h F \<squnion> G \<in> stable C"   and "extend h F \<squnion> G \<in> (C \<inter> A) leadsTo[(\<inter>) C ` givenBy f] B" shows "F \<squnion> project h C G \<in> project_set h C \<inter> project_set h (C \<inter> A) \<longmapsto> project_set h B"
exp_boundaries': fixes x :: "float"   and prec :: "nat" assumes "x \<le> 0" shows "exp (real_of_float x) \<in> {real_of_float (lb_exp prec x)..real_of_float (ub_exp prec x)}"
assms_2: shows "occurs_on ev \<noteq> occurs_on ev'"
subcocycle_lim_ereal_atmost_uN_invariants: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and N :: "nat" assumes "subcocycle u"   and "0 < N" shows "AE x in M. subcocycle_lim_ereal u x \<le> ereal (real_cond_exp M Invariants (\<lambda>x. u N x / real N) x)"
invertible: fixes u :: "('a \<times> 'a) set" assumes "u \<in> carrier_local_ring_at" shows "additive.invertible u"
lookup_csubst: fixes a :: "coname"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name"   and c :: "coname"   and x :: "name"   and P :: "trm" assumes "a \<sharp> (\<theta>_n, \<theta>_c)" shows "lookup y c \<theta>_n ((a, x, P) # \<theta>_c) = lookup y c \<theta>_n \<theta>_c{a:=x.P}"
inverse_transformations_axioms: shows "inverse_transformations A B F G \<sigma> \<sigma>'"
ubs_ex_n_max: fixes A :: "nat set"   and n :: "nat" assumes "A \<noteq> {}"   and "A \<subseteq> {i. i \<le> n}" shows "\<exists>!m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)"
inti: fixes i :: "'b" assumes "i \<in> I" shows "LINT x|M. indicat_real A x * f i x = LINT x|M. indicat_real A x * real_cond_exp M F (f i) x"
subst_tm_HPair: fixes i :: "name"   and x :: "tm"   and a :: "tm"   and b :: "tm" shows "subst i x (HPair a b) = HPair (subst i x a) (subst i x b)"
height_balR: fixes m :: "nat"   and l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "HBT m"   and "HBT.hbt m l"   and "HBT.hbt m r"   and "height r = height l + m + 1" shows "height (HBT.balR m l a r) \<in> {height l + m + 1, height l + m + 2}"
wivs_overflow_uint_def: shows "wivs_overflow_uint \<equiv> push_bit (dflt_size - 1) 1"
elims_2: fixes x :: "bool list"   and xa :: "tree" assumes "branch x xa" obtains "x = []" and "xa = Leaf"   | d :: "bool" and ds :: "bool list" and l :: "tree" and r :: "tree" where "x = d # ds" and "xa = Branching l r" and "if d then branch ds l else branch ds r"
ev_blocks_def: fixes n :: "nat" shows "??.Jordan_Normal_Form_Existence.ev_blocks n \<equiv> ev_blocks_part n"
h6: shows "mat_of_rows_list 2 [[1, 1], [1, - 1]] = M_mat [[1], [- 1]] [[], [0]]"
simple_integral_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>S M f = (\<Sum>x\<in>f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
assms: shows "(xs, y, ys) \<in> set (list_splits zs)"
simps: fixes a :: "('i \<Rightarrow> 'a) set" shows "(a \<in> generator) = (\<exists>J X. a = emb I J X \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^sub>M J M))"
index0: fixes s :: "(nat \<times> form) list"   and n :: "nat" assumes "init s" shows "\<forall>u m A. (n, u) \<in> deriv s \<longrightarrow> (m, A) \<in> set u \<longrightarrow> \<not> is_FEx A \<longrightarrow> m = 0"
quote_all_perm_eq: fixes V :: "name set"   and i :: "'a"   and p :: "perm"   and i' :: "'b" assumes "finite V"   and "atom i \<sharp> (p, V)"   and "atom i' \<sharp> (p, V)" shows "quote_all ((atom i \<rightleftharpoons> atom i') + p) V = quote_all p V"
FF_def: shows "FF = Geq (Const 0) (Const 1)"
surjective: fixes r :: "('a, 'b, 'c) sa_rec_scheme" shows "r = \<lparr>g_V = g_V r, g_E = g_E r, g_V0 = g_V0 r, sa_L = sa_L r, \<dots> = sa_rec.more r\<rparr>"
a_host_shadow_root_rel_def: fixes h :: "(_) heap" shows "a_host_shadow_root_rel h = (\<lambda>(x, y). (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y)) ` {(host, shadow_root). host |\<in>| element_ptr_kinds h \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root}"
le_add_same_cancel1: fixes a :: "'f multiset"   and b :: "'f multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
scf_list_def: fixes scf :: "nat \<Rightarrow> nat"   and xs :: "'b list" shows "scf_list scf xs = concat (map2 (\<lambda>x i. replicate (scf i) x) xs [0..<length xs])"
chain_subdiv_chain_def: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "chain_subdiv_chain one_chain1 subdiv \<equiv> \<exists>f. \<Union> (f ` one_chain1) = subdiv \<and> (\<forall>c\<in>one_chain1. chain_subdiv_path (coeff_cube_to_path c) (f c)) \<and> pairwise (\<lambda>p p'. f p \<inter> f p' = {}) one_chain1 \<and> (\<forall>x\<in>one_chain1. finite (f x))"
MkArr_Map: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
env_restr_join2: fixes S :: "'c set"   and m :: "'c \<Rightarrow> 'd"   and S' :: "'c set" shows "m f|` S \<squnion> m f|` S' = m f|` (S \<union> S')"
axioms_1: fixes G :: "('e, 'f) igb_graph_rec_scheme" assumes "igb_graph G" shows "graph G"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "trg\<^sub>B g = trg\<^sub>B f"
vector_def: fixes l :: "'a list" shows "vector l = (\<chi>i. foldr (\<lambda>x f n. (f (n + (1::'b)))(n := x)) l (\<lambda>n x. 0::'a) (1::'b) i)"
double_compl: fixes x :: "'d" shows "- (- x) = x"
simps_1: fixes left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b \<times> 'c) spmf"   and s :: "'c"   and a :: "'d" shows "(left \<oplus>\<^sub>O right) s (Inl a) = map_spmf (apfst Inl) (left s a)"
sum_with_empty: fixes pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and f :: "'c \<Rightarrow> 'b" shows "sum_with pls z f {} = z"
arc_set_balance_def: fixes w :: "'a"   and A :: "'b set" shows "arc_set_balance w A = int (card (in_arcs G w \<inter> A)) - int (card (out_arcs G w \<inter> A))"
natded_complete: fixes p :: "(nat, nat) form"   and ps :: "(nat, nat) form list" assumes "closed 0 p"   and "list_all (closed 0) ps"   and "\<forall>e f g. e,f,g,ps \<Turnstile> p" shows "ps \<turnstile> p"
intro: fixes x :: "'b \<Longrightarrow> 'a" assumes "\<And>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v" shows "pbij_lens_axioms x"
sorted_list_of_set_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq {} = []"
get_disconnected_document_locs_def: shows "get_disconnected_document_locs = \<Union> (range get_disconnected_nodes_locs) \<union> (\<Union>ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
vcomp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b arrow_of_spans_data"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "span_vertical_category.vcomp C \<nu> \<mu> \<equiv> if arrow_of_spans C \<mu> \<and> arrow_of_spans C \<nu> \<and> Dom \<nu> = Cod \<mu> then \<lparr>Chn = C (Chn \<nu>) (Chn \<mu>), Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else \<lparr>Chn = partial_magma.null C, Dom = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>, Cod = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>\<rparr>"
bisim_inv2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s2 -2-ta2\<rightarrow> s2'" obtains s1' :: "'x1 \<times> 'm1" where "t \<turnstile> s1' \<approx> s2'"
discI_2: fixes tllist :: "('a, 'b) tllist"   and x21 :: "'a"   and x22 :: "('a, 'b) tllist" assumes "tllist = TCons x21 x22" shows "\<not> is_TNil tllist"
validt_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A \<equiv> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<and> (\<forall>s\<in>Normal ` P. \<Gamma>\<turnstile>c \<down> s)"
sum: fixes g :: "'c \<Rightarrow> 'a"   and S :: "'c set" shows "h (sum g S) = (\<Sum>a\<in>S. h (g a))"
image_def: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
mem_is_interval_1_I: fixes S :: "real set"   and a :: "real"   and c :: "real"   and b :: "real" assumes "is_interval S"   and "a \<in> S"   and "c \<in> S"   and "a \<le> b"   and "b \<le> c" shows "b \<in> S"
bigo_lesseq1: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "\<forall>x. \<bar>g x\<bar> \<le> \<bar>f x\<bar>" shows "g \<in> O(h)"
psgt: 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 Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b" assumes "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk" shows "0 < p"
cong_ex: fixes f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "(f1 \<in> L F g1) = (f2 \<in> L F g2)"
run_fail_writer: fixes fail :: "'a" shows "run_writer (fail_writer fail) = fail"
put_twice: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) v = put\<^bsub>x\<^esub> \<sigma> v"
II: fixes it :: "'x set"   and \<sigma> :: "'a" assumes "it \<subseteq> \<alpha> s"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> s - it. k \<le> j" shows "P \<sigma>"
abs_fresh_list_star': fixes a :: "name"   and xvec :: "name list"   and P :: "'d" assumes "a \<sharp> xvec" shows "xvec \<sharp>* [a].P = xvec \<sharp>* P"
flatten_sigma_2_length: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "size \<phi>[M]\<^sub>\<Sigma>\<^sub>2 \<le> 2 ^ (size \<phi> + 1)"
cspan_range_ket_finite: shows "cspan (range ket) = UNIV"
paste_Domain: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "Domain (P +< Q) = Domain P \<union> Domain Q"
focus_in_Polys': fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and Y :: "'b set"   and X :: "'b set" assumes "p \<in> P[Y]" shows "focus X p \<in> P[Y \<inter> X]"
ennreal_inverse_power: fixes x :: "ennreal"   and n :: "nat" shows "inverse (x ^ n) = inverse x ^ n"
d: fixes l :: "'lbl" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l A) \<inter> subst_domain \<delta> = {}"
table_subset: fixes n :: "nat"   and V :: "nat set"   and T :: "'b option list set"   and S :: "'b option list set" assumes "table n V T"   and "S \<subseteq> T" shows "table n V S"
field_differentiable_add: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_field_derivative f') F"   and "(g has_field_derivative g') F" shows "((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
fun_left_comm: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'b"   and x :: "'b"   and z :: "'c" assumes "comp_fun_commute f" shows "f y (f x z) = f x (f y z)"
executionally_equivalent_remove_unreachable_state_arbitrary: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and x :: "(String.literal \<times> value list) list" assumes "obtainable s r e"   and "\<not> reachable s' e" shows "executionally_equivalent e s r (remove_state s' e) s r x"
ket_nonzero: fixes i :: "'c" shows "ket i \<noteq> 0"
usubst_cond_upd_1: fixes \<sigma> :: "'e \<Rightarrow> 'e"   and x :: "'f \<Longrightarrow> 'e"   and u :: "('f, 'e) uexpr"   and b :: "(bool, 'e) uexpr"   and \<rho> :: "'e \<Rightarrow> 'e"   and v :: "('f, 'e) uexpr" shows "\<sigma>(x \<mapsto>\<^sub>s u) \<triangleleft> b \<triangleright>\<^sub>s \<rho>(x \<mapsto>\<^sub>s v) = (\<sigma> \<triangleleft> b \<triangleright>\<^sub>s \<rho>)(x \<mapsto>\<^sub>s u \<triangleleft> b \<triangleright> v)"
simple_path_imp_path: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "path g"
diff_add_cancel: fixes a :: "'a"   and b :: "'a" shows "a - b + b = a"
degree_Bernstein_le: fixes j :: "nat"   and p :: "nat" shows "degree (Bernstein_Poly_01 j p) \<le> p"
inj_imp_bij_betw_inv: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a set" assumes "inj f" shows "bij_betw (inv f) (f ` M) M"
decompose_zmset_of2: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" obtains A :: "'b multiset" and B :: "'b multiset" and C :: "'b zmultiset" where "M = zmset_of A + C" and "N = zmset_of B + C"
Tpows_def: shows "Tpows = map (VectorSpaceEnd.endpow V T) [0..<Suc (dim_R V)]"
nres_more_simps_1: fixes X :: "'b set" shows "(SUCCEED = RES X) = (X = {})"
carrier_matD_1: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_row A = nr"
ipresIGCons_def: fixes h :: "'l \<Rightarrow> 'm"   and hA :: "'n \<Rightarrow> 'o"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'p, 'l, 'n, 'q) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'r, 'opSym, 'p, 'm, 'o, 's) model_scheme" shows "ipresIGCons h hA MOD MOD' \<equiv> ipresIGVar h MOD MOD' \<and> ipresIGAbs h hA MOD MOD' \<and> ipresIGOp h hA MOD MOD'"
distinct_9: fixes x1 :: "agent"   and x61 :: "msg"   and x62 :: "msg" shows "Agent x1 \<noteq> \<lbrace>x61, x62\<rbrace>"