hcju/isabelleps huggingface dataset:
read_only_reads_unowned_nth_update: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and sb' :: "'p memref list"   and \<O>' :: "nat set"   and p' :: "'p"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<O>') (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<subseteq> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"   and "\<O>' \<union> all_acquired sb' \<subseteq> \<O> \<union> all_acquired sb" shows "read_only_reads_unowned (ts[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')])"
propnot\<^sub>0_def: shows "propnot \<equiv> \<lambda>p. \<^bold>\<lambda>\<^sup>0 (\<^bold>\<not>p)"
strict_implies_order: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" assumes "a \<rhd> b" shows "a \<unrhd> b"
final: shows "abc_steps_l (0, lm) A n = (length A, lm')"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (f \<otimes>\<^sub>C \<I>\<^sub>C)"
ltl_const_equiv_implies_ltl_prop_equiv: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "\<phi> \<sim>\<^sub>P \<psi>"
frontier_of_disjoint_eq_alt: fixes S :: "'a set"   and X :: "'a topology" shows "(S \<subseteq> topspace X - X frontier_of S) = openin X S"
c: shows "rawpsubst (dsj \<phi>1 \<phi>2) (zip (map Var us) (map snd txs)) = dsj (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (rawpsubst \<phi>2 (zip (map Var us) (map snd txs)))"
distinct_1: fixes x1 :: "('c, 'd) atm"   and x2 :: "('c, 'd) atm" shows "Pos x1 \<noteq> Neg x2"
rel_Grp: fixes A1 :: "'d set"   and f1 :: "'d \<Rightarrow> 'f"   and A2 :: "'e set"   and f2 :: "'e \<Rightarrow> 'g" shows "rel_prod (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> A1 \<and> Basic_BNFs.snds x \<subseteq> A2} (map_prod f1 f2)"
cr_ivl_def: shows "cr_ivl \<equiv> \<lambda>x. (=) (abs_ivl x)"
step: fixes A :: "('a \<times> ('c, real) cconstraint \<times> 'b \<times> 'c list \<times> 'a) set \<times> ('a \<Rightarrow> ('c, real) cconstraint)"   and l :: "'a"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'a"   and Z' :: "('c \<Rightarrow> real) set"   and l'' :: "'a"   and Z'' :: "('c \<Rightarrow> real) set" assumes "alpha_interp.steps_z_alpha A l Z l' Z'"   and "alpha_interp.step_z_alpha A l' Z' l'' Z''" shows "alpha_interp.steps_z_alpha A l Z l'' Z''"
supp_at_base: fixes a :: "'a" shows "supp a = {atom a}"
assms: shows "\<lbrace>!x \<cdot> !y\<rbrace>z\<lbrace>!x\<rbrace>"
dg_prod_is_arrD_7: fixes a :: "V"   and b :: "V"   and f :: "V"   and i :: "V" assumes "f : a \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>\<^esub> b"   and "i \<in>\<^sub>\<circ> ZFC_in_HOL.set {j}" shows "f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<CC>\<^esub> b\<lparr>i\<rparr>"
lcm_1_left: fixes a :: "'a" shows "lcm (1::'a) a = normalize a"
emeasure_extend_measure: fixes M :: "'c measure"   and \<Omega> :: "'c set"   and I :: "'d set"   and G :: "'d \<Rightarrow> 'c set"   and \<mu> :: "'d \<Rightarrow> ennreal"   and \<mu>' :: "'c set \<Rightarrow> ennreal"   and i :: "'d" assumes "M = extend_measure \<Omega> I G \<mu>"   and "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"   and "G ` I \<subseteq> Pow \<Omega>"   and "positive (sets M) \<mu>'"   and "countably_additive (sets M) \<mu>'"   and "i \<in> I" shows "emeasure M (G i) = \<mu> i"
wf_lang_wf_word: fixes n :: "nat"   and r :: "'a atom rexp" assumes "local.wf n r" shows "\<forall>w\<in>lang n r. wf_word n w"
nv_def: shows "nv = vec n (\<lambda>j. (- (1::'a)) ^ (length ws + j) * det (B j))"
topological_tendstoD: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter"   and S :: "'c set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
Nil_is_append_conv: fixes xs :: "'d list"   and ys :: "'d list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
G_3: shows "G = (pair (s, t), pair d) # G'"
assms: shows "RETURN (f x) \<le> \<Down> R M'"
simps_13: fixes b :: "expr"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "fv (if (b) e\<^sub>1 else e\<^sub>2) = fv b \<union> fv e\<^sub>1 \<union> fv e\<^sub>2"
maxsimpchain_def: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X" shows "SimplicialComplex.maxsimpchain X xs \<equiv> (\<forall>x\<in>set xs. SimplicialComplex.maxsimp X x) \<and> adjacentchain xs"
extensional_set: fixes a :: "'s"   and b :: "'s" assumes "ide a"   and "ide b"   and "local.set a = local.set b" shows "a = b"
map_props_def: fixes f :: "'c \<Rightarrow> 'b option"   and APs :: "'c set" shows "map_props f APs \<equiv> {i. \<exists>p\<in>APs. f p = Some i}"
events_distinct_paths: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "a \<in> \<E>"   and "b \<in> \<E>"   and "a \<noteq> b" shows "\<exists>R\<in>\<P>. \<exists>S\<in>\<P>. a \<in> R \<and> b \<in> S \<and> (R \<noteq> S \<longrightarrow> (\<exists>!c. c \<in> \<E> \<and> R \<inter> S = {c}))"
ideal_axioms: shows "ideal I R (+) (\<cdot>) \<zero> \<one>"
transD: fixes r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
map_fst_zip_take: fixes xs :: "'e list"   and ys :: "'f list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
slide_def: fixes t :: "'a tree"   and w :: "nat \<times> nat" shows "local.slide t w = (let ts = local.atomics (max (fst w) (Suc (r t))) (snd w); ts' = reusables t w in fold combine (ts @ ts') Leaf)"
Try_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and C :: "String.literal"   and V :: "nat"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (try e catch(C V) e') h s s'"
is_nth_power_0_iff: fixes n :: "nat" shows "is_nth_power n (0::'b) = (0 < n)"
eq_entails: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "P = Q" shows "P \<tturnstile> Q"
remove_lemma: fixes x :: "nat"   and u :: "nat" assumes "x \<in> nat_to_set u" shows "set_to_nat (nat_to_set u - {x}) = u - 2 ^ x"
converse_converse: fixes r :: "('d \<times> 'e) set" shows "(r\<inverse>)\<inverse> = r"
Collect_minI_cut: fixes k :: "'a"   and I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "k \<in> I"   and "P k" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I \<down>< x. \<not> P y)"
exhaust: fixes y :: "'a hterm" obtains x1 :: "'a" and x2 :: "'a hterm list" where "y = HApp x1 x2"
emp_hcomp: fixes P :: "interface" shows "(\<epsilon> \<otimes> P) = P"
rScope: fixes x :: "name"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "cP = \<lparr>\<nu>x\<rparr>P"   and "cRs = \<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"   and "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "x \<sharp> C"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop"
PQ: fixes i :: "nat" assumes "i \<in> pdevs_domain d" shows "P i (pdevs_apply b i) = Q i (pdevs_apply d i)"
intros_3: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P"
dom_simp: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.VVV.arr f" shows "C.VVV.dom f = C.VxVxV.dom f"
succ_range: fixes n :: "nat" shows "\<forall>v\<in>succs n. v < length (g_V G)"
IH: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" assumes "no_effectless_act as"   and "as \<in> valid_plans PROB"   and "prob_dom PROB \<subseteq> vs" shows "as_proj as vs = as"
spmf_fast_dice_roll_ub: fixes v :: "nat"   and x :: "nat" assumes "0 < v" shows "spmf (pmf_of_set {..<v} \<bind> local.fast_dice_roll v) x \<le> (if x < n then 1 / real n else 0)"
segment_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" shows "segment D a = (if a \<notin> carrier D then carrier D else {x. x \<prec>\<^bsub>D\<^esub> a \<and> x \<in> carrier D})"
sum_list_const_mult: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.semiring_0 plus zero times" shows "monoid_add.sum_list plus zero (map (\<lambda>x. times c (f x)) xs) = times c (monoid_add.sum_list plus zero (map f xs))"
zero_less_iff_neq_zero: fixes n :: "'d multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
rt_fresh_as_refl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "rt \<approx>\<^bsub>dip\<^esub> rt"
subst_support_if_mgt_subst_idem: fixes \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<theta> \<preceq>\<^sub>\<circ> \<delta>"   and "subst_idem \<theta>" shows "\<theta> supports \<delta>"
kronecker_factorization_main_complete: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and bnd :: "nat"   and p :: "int poly" assumes "divisors_fun df"   and "divisors_pos_fun dp"   and "kronecker_factorization_main df dp bnd p = None"   and "2 \<le> degree p" shows "\<nexists>q. 1 \<le> degree q \<and> degree q \<le> bnd \<and> q dvd p"
BinOpRedsThrow1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map B.map B.map"
less_imp_le: fixes x :: "'a"   and y :: "'a" assumes "x \<prec> y" shows "x \<preceq> y"
qinc: shows "quality_increases (\<sigma> nhip) (\<sigma>' nhip)"
good_ruleset_def: fixes rs :: "'c rule list" shows "good_ruleset rs \<equiv> \<forall>r\<in>set rs. (\<nexists>chain. get_action r = Call chain) \<and> get_action r \<noteq> Return \<and> (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
collapse_2: fixes event :: "('a, 'b, 'c) event" assumes "isSend event" shows "Send (getId event) (occurs_on event) (partner event) (un_Send4 event) (un_Send5 event) (getMsg event) = event"
A: shows "\<forall>x. \<exists>y. (x, y) \<in> R"
\<alpha>_inv: shows "\<alpha> i * inverse (\<alpha> i) = 1"
Sup_eq_Sup: fixes B :: "V set"   and A :: "V set" assumes "B \<subseteq> A"   and "small A"   and "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. x \<le> y" shows "\<Squnion> A = \<Squnion> B"
ord_isom_mem: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "f a \<in> carrier E"
INF_emeasure_decseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "decseq A"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter> (range A))"
bindWT_fmapWT: fixes f :: "'i \<rightarrow> 'h"   and m :: "'i\<cdot>('f, 'g) writerT"   and k :: "'h \<rightarrow> 'e\<cdot>('f, 'g) writerT" shows "bindWT\<cdot>(fmapWT\<cdot>f\<cdot>m)\<cdot>k = bindWT\<cdot>m\<cdot>(\<Lambda> x. k\<cdot>(f\<cdot>x))"
remdups_append: fixes y :: "'b"   and ys :: "'b list"   and ws :: "'b list" shows "y \<in> set ys \<longrightarrow> remdups (ws @ y # ys) = remdups (ws @ ys)"
Bernstein_changes_01_test: fixes P :: "real poly"   and p :: "nat" assumes "degree P \<le> p"   and "P \<noteq> 0" shows "int (proots_count P {x. 0 < x \<and> x < 1}) \<le> Bernstein_changes_01 p P \<and> even (Bernstein_changes_01 p P - int (proots_count P {x. 0 < x \<and> x < 1}))"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat" shows "fwi m n k (Suc i) 0 = fw_upd (fwi m n k i n) k (Suc i) 0"
fresh_ty: fixes x :: "name"   and T :: "ty" shows "x \<sharp> T"
hom_imp_img_monoid: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "Group.monoid (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>)"
bisimulation_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool" assumes "bisimulation trsys1 trsys2 bisim tlsim" shows "bisimulation trsys2 trsys1 (flip bisim) (flip tlsim)"
B_F_inv_rewrite: fixes \<alpha> :: "'index"   and \<tau> :: "'trap"   and y\<^sub>\<sigma>' :: "'range"   and m1 :: "bool" shows "(B \<alpha> (F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>') = (B \<alpha> (F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>') = m1)) = m1"
assms: shows "es @ [Deliver (i, Insert e m)] prefix of j"
d_OUT_add: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_OUT (\<lambda>e. f e + g e) x = d_OUT f x + d_OUT g x"
Map_comp: fixes t' :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a' :: "'a"   and a :: "'a" assumes "A_B.seq t' t"   and "A.seq a' a" shows "A_B.Map (t' \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] t) (a' \<cdot>\<^sub>A a) = A_B.Map t' a' \<cdot>\<^sub>B A_B.Map t a"
bangInduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and Prop :: "'e \<Rightarrow> 'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) residual \<Rightarrow> bool"   and C :: "'e" assumes "\<Psi> \<rhd> !P \<longmapsto> Rs"   and "\<And>\<alpha> P' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>)\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"   and "\<And>\<alpha> P' C. \<lbrakk>\<Psi> \<rhd> !P \<longmapsto> \<alpha> \<prec> P'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>); \<And>C. Prop C \<Psi> (!P) (\<alpha> \<prec> P')\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"   and "\<And>M N P' K xvec P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''; \<And>C. Prop C \<Psi> (!P) (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''); \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C; distinct xvec\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>M xvec N P' K P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''; \<And>C. Prop C \<Psi> (!P) (K\<lparr>N\<rparr> \<prec> P''); \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C; distinct xvec\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>Rs C. \<lbrakk>\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs; \<And>C. Prop C \<Psi> (P \<parallel> !P) Rs; guarded P\<rbrakk> \<Longrightarrow> Prop C \<Psi> (!P) Rs" shows "Prop C \<Psi> (!P) Rs"
elims_2: fixes x :: "('b \<times> 'c) tree"   and xa :: "'b" assumes "isin x xa" obtains l :: "('b \<times> 'c) tree" and a :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" and x :: "'b" where "xa__ = \<langle>l, (a, uu_), r\<rangle>" and "xa = x" and "case cmp x a of LT \<Rightarrow> isin l x | EQ \<Rightarrow> True | GT \<Rightarrow> isin r x"
assms_3: shows "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
map_FGf_def: fixes l1 :: "'k \<Rightarrow> 'u"   and l2 :: "'l \<Rightarrow> 'v"   and co1 :: "'n \<Rightarrow> 'w"   and co2 :: "'o \<Rightarrow> 'x"   and contra1 :: "'y \<Rightarrow> 'q"   and contra2 :: "'z \<Rightarrow> 'r" shows "map_FGf l1 l2 co1 co2 contra1 contra2 = map_F l1 l2 id co1 co2 id contra1 contra2 id"
rep_eq: fixes x :: "integer"   and xa :: "nat"   and xb :: "bool" shows "int_of_integer (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (int_of_integer x) xa xb"
abs_eq: fixes x :: "'b \<Rightarrow> 'c" shows "sum_cfc x = Abs_comp_fun_commute ((+) \<circ> x)"
in_Gr: fixes x :: "'b"   and y :: "'c"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "((x, y) \<in> BNF_Def.Gr A f) = (x \<in> A \<and> f x = y)"
PO_a0i_inv1_iagree_trans: shows "{a0i_inv1_iagree} TS.trans a0i {> a0i_inv1_iagree}"
enc: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and enc :: "'c \<Rightarrow> 'b"   and \<phi> :: "'c" assumes "Encode var trm fmla Var FvarsT substT Fvars subst num enc"   and "\<phi> \<in> fmla" shows "enc \<phi> \<in> num"
P2_ideal_model_def: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "P2_ideal_model M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. let l = b0 \<otimes> inv (h0 [^] r); \<sigma>' = if l = \<one> then False else True in mal_def.trusted_party (x0, x1) \<sigma>' \<bind> (\<lambda>(uu_, x\<sigma>). sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> (if \<sigma>' then \<one> else x\<sigma>); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> (if \<sigma>' then x\<sigma> else \<one>); e0 = (w0, z0); e1 = (w1, z1) in \<A>3 e0 e1 s' \<bind> (\<lambda>out. return_spmf ((), out)))))))))))))"
outside: fixes x :: "'a"   and y :: "'b" assumes "x \<notin> set_pmf p \<or> y \<notin> set_pmf q \<or> \<not> R x y" shows "h x y = 0"
cblinfun_id_cblinfun_apply: shows "(*\<^sub>V) id_cblinfun = (\<lambda>x. x)"
assms_1: fixes v :: "i" shows "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi> in v]"
SeqSubstFormP_imp_OrdP: fixes v :: "tm"   and u :: "tm"   and x :: "tm"   and x' :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqSubstFormP v u x x' s k} \<turnstile> OrdP k"
abs_ok_2: shows "term_ok \<Theta> (Abs_fv x \<tau>' t)"
tdropn_Suc: fixes n :: "nat"   and xs :: "('a, 'b) tllist" shows "tdropn (Suc n) xs = (case xs of TNil b \<Rightarrow> TNil b | TCons x xs' \<Rightarrow> tdropn n xs')"
spec_of_invariant: shows "\<forall>s a. invariant s \<longrightarrow> invariant (kstep s a)"
sin_ms_aux': fixes xs :: "('a \<times> real) msllist"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "ms_exp_gt 0 (ms_aux_hd_exp xs)"   and "basis_wf basis"   and "is_expansion_aux xs f basis" shows "is_expansion_aux (sin_ms_aux' xs) (\<lambda>x. sin (f x)) basis"
cr2: shows "cross_ratio b i1 a i2 = of_complex 2"
parallel_resource_def: shows "parallel_resource \<equiv> \<lambda>uu uua. corec_resource (\<lambda>(res1, res2) ac. case ac of Inl a \<Rightarrow> map_spmf (map_prod Inl (\<lambda>uu. Inr (uu, res2))) (run_resource res1 a) | Inr c \<Rightarrow> map_spmf (map_prod Inr (\<lambda>uu. Inr (res1, uu))) (run_resource res2 c)) (uu, uua)"
Axiom_of_Extensionality_3: fixes \<alpha> :: "V"   and a :: "V"   and x :: "V"   and y :: "V" assumes "\<Z> \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "x = y"   and "x \<in>\<^sub>\<circ> a" shows "y \<in>\<^sub>\<circ> Vset \<alpha>"
trcl_mono: fixes A :: "('c \<times> 'd \<times> 'c) set"   and B :: "('c \<times> 'd \<times> 'c) set" assumes "A \<subseteq> B" shows "trcl A \<subseteq> trcl B"
fconverse_fconverse: fixes r :: "V" assumes "fbrelation r" shows "(r\<inverse>\<^sub>\<bullet>)\<inverse>\<^sub>\<bullet> = r"
is_variable_imp_incr_boundvars_unchganged: fixes inc :: "nat"   and v :: "variable"   and T :: "typ" shows "incr_boundvars inc (Fv v T) = Fv v T"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (tVAR list)"   and "P tBOOL"   and "P tINT"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P ty1 \<rightarrow> ty2" shows "P ty"
inf_principal: fixes A :: "'a set"   and B :: "'a set" shows "inf (principal A) (principal B) = principal (A \<inter> B)"
T_def: shows "T = Min (insert t (d2 ` C'))"
smap_singleton_conv_W_image: fixes w :: "'w" assumes "w \<in> W" shows "dual_order.poset_simplex_map \<P> {w} = (+o) w ` dual_order.poset_simplex_map \<P> 0"