hcju/isabelleps huggingface dataset:
assoc'_simps\<^sub>A\<^sub>W\<^sub>C_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "local.dom (local.inv \<a>[f, g, h]) = f \<star> g \<star> h"
rotate_downneg_kauff_mat: shows "kauff_mat (basic [cup, vert] \<circ> basic [vert, brick.under]) = kauff_mat (basic [vert, cup] \<circ> basic [over, vert])"
integrable_real_geometric_pmf: fixes p :: "real" assumes "p \<in> {0<..1}" shows "integrable (measure_pmf (geometric_pmf p)) real"
NATDistr: fixes N :: "((int \<times> int) set set, int) Combinators list"   and F :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent \<Rightarrow> unit decision option"   and NAT :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent \<Rightarrow> 'a decision option" assumes "N \<noteq> []"   and "F = C (list2policyR N)" shows "(\<lambda>(x, y). x) o\<^sub>f (NAT \<Otimes>\<^sub>2 F \<circ> (\<lambda>x. (x, x))) = list2policy ((NAT \<Otimes>\<^sub>L map C N) (\<Otimes>\<^sub>2) (\<lambda>(x, y). x) (\<lambda>x. (x, x)))"
vsv_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "vsv (VLambda A f)"
distinct_1: fixes x1 :: "'d"   and x2 :: "'d" shows "Normal x1 \<noteq> Abrupt x2"
friend_assm: fixes v :: "'v"   and u :: "'v" assumes "v \<in> V"   and "u \<in> V"   and "v \<noteq> u" shows "\<exists>!n. adjacent v n \<and> adjacent u n"
regular_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "regular f = (setmonotone f \<and> continuous f)"
distinct_ExcessTable_cont: fixes vs :: "nat list"   and g :: "graph" assumes "distinct vs" shows "distinct (map fst (ExcessTable_cont (ExcessAt g) vs))"
id2: shows "monom 1 CARD('a) \<circ>\<^sub>p v = v ^ CARD('a)"
intros_15: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> \<Rightarrow> Stuck"
empty_eff_drest_no_eff: fixes vs :: "'d set"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap" assumes "fmdom' (fmrestrict_set vs (snd a)) = {}" shows "fmrestrict_set vs (state_succ s (action_proj a vs)) = fmrestrict_set vs s"
prime_elem_iff_prime_abs: fixes k :: "int" shows "prime_elem k = prime \<bar>k\<bar>"
c1_P2: fixes xb :: "nat"   and xa :: "nat"   and xc :: "nat"   and x :: "nat"   and y :: "nat" assumes "xb < q"   and "xa < q"   and "xc < q"   and "x < q" shows "((y, xa, (xb * xa + q - xc) mod q, (x + xb) mod q), (x * ((y + q - xa) mod q) + q - xc) mod q, (x * y + q - (x * ((y + q - xa) mod q) + q - xc) mod q) mod q) = ((y, xa, (((x + xb) mod q + q - x) mod q * xa + q - xc) mod q, (x + xb) mod q), (x * ((y + q - xa) mod q) + q - xc) mod q, (x * y + q - (x * ((y + q - xa) mod q) + q - xc) mod q) mod q)"
induct: fixes P :: "'b \<Rightarrow> ('b \<times> bal) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b \<times> bal) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l a b r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x l; cmp x a = GT \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, (a, b), r\<rangle>" shows "P a0 a1"
uprodI: fixes M :: "('a, 'b) node set"   and A :: "('a, 'b) node set set"   and N :: "('a, 'b) node set"   and B :: "('a, 'b) node set set" assumes "M \<in> A"   and "N \<in> B" shows "Scons M N \<in> uprod A B"
lookup_operator_wf_2: fixes name :: "char list"   and \<pi> :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "lookup_operator name = Some \<pi>" shows "fst \<pi> = name"
cong_square_alt: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "[a * a = 1] (mod p)" shows "[a = 1] (mod p) \<or> [a = p - 1] (mod p)"
simps_1: fixes t :: "'b tree" shows "combine t Leaf = t"
zero_exp: fixes n :: "nat"   and r :: "'b list" assumes "n = 0" shows "r \<^sup>@ n = \<epsilon>"
infsetsum_nonneg_is_SUPREMUM: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "infsetsum f A = \<Squnion> (sum f ` {F. finite F \<and> F \<subseteq> A})"
img_set_submodule: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "submodule R N (f ` carrier M)"
sign_simps_29: fixes am :: "'b"   and bm :: "'b" shows "am + bm = bm + am"
complex_cnj_add: fixes x :: "complex"   and y :: "complex" shows "cnj (x + y) = cnj x + cnj y"
nonempty: fixes x :: "'a"   and y :: "'a" shows "{\<Sum>i = 0..<n. f (u i) (u (Suc i)) |u n. u 0 = x \<and> u n = y} \<noteq> {}"
eq2: shows "outer_measure_of M (K - {x \<in> space M. f x \<le> a}) = emeasure M K"
isCont_arccos: fixes x :: "real" assumes "- 1 < x"   and "x < 1" shows "isCont arccos x"
map_\<Gamma>_def: shows "map_\<Gamma> \<equiv> map_fun id (map_fun Rep_trace Abs_trace) (\<lambda>f. smap (\<lambda>(x, i). (f x, i)))"
frontier_complement: fixes S :: "'a set" shows "frontier (- S) = frontier S"
le_of_comp_def: fixes acomp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" shows "le_of_comp acomp x y = (case acomp x y of Gt \<Rightarrow> False | _ \<Rightarrow> True)"
exe_ars_conds_def: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_ars_conds arss = (alist_conds arss \<and> (\<forall>ars\<in>snd ` set arss. alist_conds ars))"
mprefix_chainpreserving: fixes Y :: "nat \<Rightarrow> 'b \<Rightarrow> 'b process"   and A :: "'b set" assumes "chain Y" shows "chain (\<lambda>i. Mprefix A (Y i))"
dehomogenize_zeroD: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "dehomogenize x p = 0"   and "homogeneous p" shows "p = 0"
H_QA: shows "Hermite_of_upt_row_i A k ass res = Q ** A"
ce_rel_lm_8: fixes r :: "(nat \<times> nat) set" shows "(r \<in> ce_rels) = (ce_rel_to_set r \<in> ce_sets)"
simps_1: fixes ts :: "nat"   and ts' :: "nat" shows "max_tstp (Inl ts) (Inl ts') = Inl (max ts ts')"
map_comp: fixes g1 :: "'j \<Rightarrow> 'h"   and g2 :: "'k \<Rightarrow> 'i"   and f1 :: "'l \<Rightarrow> 'j"   and f2 :: "'m \<Rightarrow> 'k"   and v :: "'l + 'm" shows "map_sum g1 g2 (map_sum f1 f2 v) = map_sum (g1 \<circ> f1) (g2 \<circ> f2) v"
igSubstIGAbsSTR_imp_igSubstIGAbs: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igSubstIGAbsSTR MOD" shows "FixSyn.igSubstIGAbs Delta MOD"
FOREACH_rule: fixes S :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b nres"   and P :: "'b \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "FOREACH S f \<sigma>0 \<le> SPEC P"
igWlsInp_def: fixes MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'i option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
univ_poly_units': fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (poly_ring R)" shows "(p \<in> Units (K [X]\<^bsub>poly_ring R\<^esub>)) = (p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>) \<and> p \<noteq> [] \<and> degree p = 0)"
is_concrete_set_category: shows "concrete_set_category (\<cdot>) \<AA> UNIV UP"
assms: fixes i :: "nat" assumes "i < length l" shows "valid_point l i"
map_graph_def: fixes f :: "('f \<times> 'e) set"   and G :: "('d, 'f) labeled_graph" shows "map_graph f G = LG (on_triple f `` edges G) (f `` vertices G)"
\<mu>_reach_conv: fixes f :: "'b \<Rightarrow> real"   and u :: "'a"   and v :: "'a" shows "(\<mu> f u v < \<infinity>) = (u \<rightarrow>\<^sup>* v)"
correctCompositionDiffLevelsA21: shows "correctCompositionDiffLevels sA21"
map_add_Some_iff: fixes m :: "'e \<Rightarrow> 'd option"   and n :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and x :: "'d" shows "(Map.map_add m n k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
max_li_is_basis: fixes A :: "'c set" assumes "maximal A (\<lambda>S. S \<subseteq> carrier V \<and> lin_indpt S)" shows "basis A"
older_seniors_monotonic: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x \<le> n" shows "older_seniors x (n + m) \<subseteq> older_seniors x n"
Min_eqI: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> x \<le>s y"   and "x \<in> A" shows "signed.Min A = x"
\<Delta>''_def: shows "\<Delta>'' = \<lparr>edge = edg, capacity = split (capacity \<Delta>), source = Vertex (source \<Delta>), sink = Vertex (sink \<Delta>)\<rparr>"
l_def: shows "l \<equiv> liminf (\<lambda>n. ereal (real n * (f n / f (Suc n) - 1)))"
3: shows "{xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> xs ! e} \<inter> {xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> \<not> xs ! e} = {}"
isEmpty_def: fixes t :: "('c, 'd) trie" shows "isEmpty t = is_empty_trie (impl_of t)"
borel_measurable_times: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
strict_mono_implies_mono: fixes m :: "('b, 'c) state" assumes "strict_mono_state m" shows "mono_state m"
lt_monom_mult: fixes c :: "'d"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and t :: "'a" assumes "c \<noteq> (0::'d)"   and "p \<noteq> 0" shows "lt (monom_mult c t p) = t \<oplus> lt p"
node_deliver_messages_Cons: fixes x :: "'msg event"   and xs :: "'msg event list" shows "node_deliver_messages (x # xs) = node_deliver_messages [x] @ node_deliver_messages xs"
conv_radius_gchoose: fixes a :: "'a" shows "conv_radius ((gchoose) a) = (if a \<in> \<nat> then \<infinity> else 1)"
enat_set_def: shows "enat_set \<equiv> {x. enat_setp x}"
SN: fixes S :: "('b \<times> 'b) set" assumes "SN_ars S" shows "SN S"
intros: fixes X :: "'b nmultiset"   and M :: "'b nmultiset multiset" assumes "X \<in># M" shows "(X, MSet M) \<in> sub_nmset"
FAddSA_RootEx_Root: fixes G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and F :: "('d, 'e, 'f) seqauto set"   and SA :: "('d, 'e, 'f) seqauto"   and S :: "'d" assumes "\<Union> (ran G) = F - {Root F G}"   and "\<Union> (States ` F) = dom G"   and "dom G \<inter> States SA = {}"   and "S \<in> dom G"   and "RootEx F G" shows "Root (insert SA F) (G [f+] (S, SA)) = Root F G"
le_of_comp_def: fixes acomp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" shows "le_of_comp acomp x y = (case acomp x y of Gt \<Rightarrow> False | _ \<Rightarrow> True)"
rep_eq: fixes x :: "'c set" shows "space_as_set (ccspan x) = closure (cspan x)"
io_targets_elim: fixes p :: "'d"   and M :: "('e, 'f, 'd) FSM"   and q :: "'d"   and io :: "('e \<times> 'f) list" assumes "p \<in> io_targets M q io" obtains tr :: "'d list" where "target (io || tr) q = p \<and> path M (io || tr) q \<and> length io = length tr"
collapse: fixes option :: "'a option" assumes "option \<noteq> None" shows "Some (the option) = option"
AB: shows "A \<or> B \<or> C \<or> D \<or> E"
induct: fixes x1 :: "'g list"   and x2 :: "'h list"   and r :: "('g \<times> 'h) set"   and P :: "'g list \<Rightarrow> 'h list \<Rightarrow> bool" assumes "(x1, x2) \<in> listrel r"   and "P [] []"   and "\<And>x y xs ys. \<lbrakk>(x, y) \<in> r; (xs, ys) \<in> listrel r; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
Id_refine: fixes S :: "'b nres" shows "S \<le> \<Down> Id S"
B: shows "\<forall>X\<in>A. (xs, X) \<in> failures P"
ereal_le_minus: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "(x \<le> z - y) = (x + y \<le> z)"
simps: fixes a :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "(a \<in> generate_field R H) = (a = \<one>\<^bsub>R\<^esub> \<or> (\<exists>h. a = h \<and> h \<in> H) \<or> (\<exists>h. a = \<ominus>\<^bsub>R\<^esub> h \<and> h \<in> generate_field R H) \<or> (\<exists>h. a = inv\<^bsub>R\<^esub> h \<and> h \<in> generate_field R H \<and> h \<noteq> \<zero>\<^bsub>R\<^esub>) \<or> (\<exists>h1 h2. a = h1 \<oplus>\<^bsub>R\<^esub> h2 \<and> h1 \<in> generate_field R H \<and> h2 \<in> generate_field R H) \<or> (\<exists>h1 h2. a = h1 \<otimes>\<^bsub>R\<^esub> h2 \<and> h1 \<in> generate_field R H \<and> h2 \<in> generate_field R H))"
normalize_def'_1: fixes p :: "'a list list" shows "p = replicate (length p - length (UP.normalize p)) \<zero>\<^bsub>K [X]\<^esub> @ drop (length p - length (UP.normalize p)) p"
irc_onI: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "\<And>B a. \<lbrakk>B \<subseteq> A; a \<in> A; a \<notin> f (insert a B)\<rbrakk> \<Longrightarrow> f (insert a B) = f B" shows "irc_on A f"
sp_rewrite: fixes n :: "nat"   and xs :: "('a, 'b, 'c, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr list"   and F :: "'c \<Rightarrow> 'f fundef option"   and x :: "('a, 'b, 'c, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr" assumes "n < length xs"   and "\<forall>\<Sigma>. Subx.sp_instr F (xs ! n) \<Sigma> = Subx.sp_instr F x \<Sigma>" shows "Subx.sp F (rewrite xs n x) = Subx.sp F xs"
ltakeWhile_eq_LNil_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(ltakeWhile P xs = LNil) = (xs \<noteq> LNil \<longrightarrow> \<not> P (lhd xs))"
wf_eq_check_append'': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t V (S @ [\<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
parent_opt: shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r parent_opt"
antisym: fixes x :: "('c, 'd) Term.term"   and y :: "('c, 'd) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
meval_intro: fixes cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and n :: "nat"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val"   and cm' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)"   and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "(cms ! n, mem) \<leadsto>\<^sub>C (cm', mem')"   and "n < length cms" shows "conc.mevalp cms mem n (cms[n := cm']) mem'"
iIN_iT_Div_Int_mod_0: fixes k :: "nat"   and n :: "nat"   and A :: "nat set"   and d :: "nat" assumes "0 < k"   and "n mod k = 0"   and "\<forall>x\<in>A. x div k \<le> (n + d) div k \<longrightarrow> x \<le> n + d" shows "A \<inter> [n\<dots>,d] \<oslash> k = (A \<oslash> k) \<inter> ([n\<dots>,d] \<oslash> k)"
assms: shows "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B"
flatten_block: fixes init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and result :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "flatten (block init bdy return result) = [block init bdy return result]"
power_Suc: fixes a :: "'b list"   and n :: "nat" shows "a \<^sup>@ Suc n = a \<cdot> a \<^sup>@ n"
aux_k3req: fixes k :: "nat"   and req :: "nat \<Rightarrow> reqType list"   and t :: "nat"   and m :: "nat" assumes "\<forall>m<k + 3. req (t + m) \<noteq> [send]"   and "m \<le> k" shows "req (Suc (Suc (t + m))) \<noteq> [send]"
xb: shows "real_of_int m \<le> x \<and> x < real_of_int (n + 1)"
at_least_at_most_member: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> carrier L"   and "a \<sqsubseteq> x"   and "x \<sqsubseteq> b" shows "x \<in> \<lbrace>a..b\<rbrace>"
f12_def_2: shows "f2 = Poly_Mapping.single k (lookup f k)"
\<phi>_\<psi>: fixes f :: "'a" assumes "arr f" shows "\<phi> (\<psi> f) = f"
invar_insert: fixes s :: "'a tree"   and x :: "'a" assumes "S2.invar s" shows "S2.invar (Tree_Set.insert x s)"
tab\<^sub>0_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg r) = src r"
cod_app_subst: fixes v :: "nat"   and S :: "nat \<Rightarrow> typ"   and n :: "nat" assumes "v \<in> free_tv (S n)"   and "v \<noteq> n" shows "v \<in> cod S"
assms_4: shows "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"
chine_hcomp_arr_ide: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "ide f"   and "src \<mu> = trg f" shows "chine_hcomp \<mu> f = \<langle>Chn \<mu> \<cdot> \<p>\<^sub>1[Leg0 (Dom \<mu>), Leg1 (Dom f)] \<lbrakk>Leg0 (Cod \<mu>), Leg1 (Cod f)\<rbrakk> \<p>\<^sub>0[Leg0 (Dom \<mu>), Leg1 (Dom f)]\<rangle>"
mark_guards_block: fixes f :: "'f"   and init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'g) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'g) com" shows "mark_guards f (block init bdy return c) = block init (mark_guards f bdy) return (\<lambda>s t. mark_guards f (c s t))"
member_RBT: fixes t :: "('b, unit) rbt"   and x :: "'b" assumes "ID ccompare \<noteq> None"   and "ord.is_rbt cless t" shows "member (Mapping_RBT t) x = (ord.rbt_lookup cless t x = Some ())"
Ex1_def: fixes P :: "'c \<Rightarrow> bool" shows "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"
fps_mult_var: fixes f :: "('d, 'c) fps"   and g :: "('d, 'c) fps"   and n :: "'d list" shows "(f \<cdot> g) $ n = \<Sum> {f $ fst p \<cdot> g $ snd p |p. p \<in> splitset n}"
OK_le_conv: fixes x :: "'a"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'a err" shows "(OK x \<sqsubseteq>\<^bsub>Err.le r\<^esub> e) = (e = Err \<or> (\<exists>y. e = OK y \<and> x \<sqsubseteq>\<^bsub>r\<^esub> y))"
pdevs_apply_zero_pdevs: fixes i :: "nat" shows "pdevs_apply zero_pdevs i = (0::'a)"
that_1: shows "\<I>_full,\<I> \<turnstile>\<^sub>C conv1 \<sim> conv1'"
S1_RNextdisabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S1 rmhist p \<longrightarrow> \<not> Enabled (<RNext memCh mm (resbar rmhist) p>_(rtrner memCh!p, resbar rmhist!p))"
assms_6: fixes xs :: "'b set list" assumes "xs \<in> set (map snd xss)" shows "is_finite_weak_ranking xs"
image_affinity_atLeastAtMost_diff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and m :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. minus (times m x) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (minus (times m a) c) (minus (times m b) c) else ord.atLeastAtMost less_eq (minus (times m b) c) (minus (times m a) c))"
quotient_eq: shows "{f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_permutation A B = {F \<in> (A \<rightarrow>\<^sub>E B) // domain_permutation A B. univ (\<lambda>f. f ` A = B) F}"
map_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and p :: "'c spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
crename_ImpR': fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = ImpR x.<c>.N d"   and "c \<sharp> (R, a, b)"   and "x \<sharp> R"   and "d \<sharp> a" shows "(\<exists>N'. R = ImpR x.<c>.N' d \<and> N'[a\<turnstile>c>b] = N) \<or> (\<exists>N'. R = ImpR x.<c>.N' a \<and> b = d \<and> N'[a\<turnstile>c>b] = N)"
targets_are_isomorphic: fixes b :: "'a"   and \<mu> :: "'a"   and b' :: "'a" assumes "b \<in> targets \<mu>"   and "b' \<in> targets \<mu>" shows "b \<cong> b'"
restrict_extend_indets: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "restrict_indets (extend_indets p) = p"
universe_character: fixes x :: "i" assumes "CMU" shows "P x u"
Pd_shuffle_to_top_simps_2: fixes d :: "'d"   and ds' :: "'d set"   and Y :: "'x set" assumes "d \<in> ds'" shows "Pd_shuffle_to_top ds' Y d = shuffle_to_top Y d"
Assign_starD: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and t :: "char list \<Rightarrow> int" assumes "(x ::= a, s) \<rightarrow>* (c, t)" shows "c \<in> {x ::= a, SKIP}"
spec_def: fixes x :: "'a"   and p :: "'a"   and q :: "'a" shows "(x \<le> Ref p q) = H p x q"
asm1: shows "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. P d"
refine: shows "(create_name_gba_impl, create_name_gba_aimpl) \<in> \<langle>Id\<rangle>ltlr_rel \<rightarrow> \<langle>gbav_impl_rel_ext unit_rel nat_rel (\<langle>Id\<rangle>fun_set_rel)\<rangle>nres_rel"
iSince_False: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "\<not> (P t'. t' \<S> t I. False)"
Sabs_2: shows "\<not> shadows_consts (Sabs cs)"
poly_Cons: fixes h :: "'a"   and t :: "'a list"   and x :: "'a" shows "poly (h # t) x = h + x * poly t x"
idealpow_1_self: fixes I :: "'a set" assumes "ideal R I" shows "I \<^bsup>\<diamondsuit>R Suc 0\<^esup> = I"
ttree_restr_nxt_subset2: fixes x :: "'a"   and S :: "'a set"   and t :: "'a ttree" assumes "x \<notin> S" shows "paths (ttree_restr S (nxt t x)) \<subseteq> paths (ttree_restr S t)"
ereal_less_eq_7: fixes r :: "real" shows "(1 \<le> ereal r) = (1 \<le> r)"
fun_upd_triv: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "f(x := f x) = f"
Abs_downset_inverse: fixes y :: "'a set" assumes "y \<in> range \<down>" shows "Rep_downset (Abs_downset y) = y"
c3: shows "pi \<bullet> a' \<sharp> pi \<bullet> h"
foldr_Cons: fixes xs :: "'a list" shows "foldr (#) xs [] = xs"
zunion: fixes x :: "'b"   and a :: "'b zet"   and b :: "'b zet" shows "zin x (zunion a b) = (zin x a \<or> zin x b)"
set_tries_def: fixes t :: "('d, 'c list) trie" shows "set_tries t = \<Union> {gs. \<exists>a. gs = set (lookup_tries t a)}"
finished_succ_impl_path_in_tree: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> E `` {v}"   and "\<delta> s v < \<delta> s w" shows "(v, w) \<in> (tree_edges s)\<^sup>+"
0: shows "map (TE.int \<xi> \<circ> tNN) Tl = map (Ik.int \<xi>) Tl"
sound: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "P \<sim> Q"
max2_equals1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(r.max2 a b = a) = ((b, a) \<in> r)"
mbisim_thrD2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x2"   and ln :: "'l \<Rightarrow>f nat" assumes "s1 \<approx>m s2"   and "thr s2 t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>x'. thr s1 t = \<lfloor>(x', ln)\<rfloor> \<and> t \<turnstile> (x', shr s1) \<approx> (x, shr s2) \<and> (wset s2 t = None \<or> True)"
targets_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "targets (\<nu> \<star> \<mu>) = targets \<nu>"