hcju/isabelleps huggingface dataset:
mbisim_simulation1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and tl1 :: "'b \<times> ('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s1' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "\<not> \<tau>multithreaded.m\<tau>move \<tau>move1 s1 tl1 s1'"   and "multithreaded_base.redT final1 r1 convert_RA s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> multithreaded_base.redT final2 r2 convert_RA s2' tl2 s2'' \<and> \<not> \<tau>multithreaded.m\<tau>move \<tau>move2 s2' tl2 s2'' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1' s2'' \<and> FWbisimulation_base.mta_bisim bisim tl1 tl2"
A: fixes k :: "nat"   and n :: "nat" shows "offset k x xs \<noteq> None \<longrightarrow> offset (n + m) x xs = Some (the (offset n x xs) + m)"
suminf_PInfty: fixes f :: "nat \<Rightarrow> ereal"   and i :: "nat" assumes "\<And>i. 0 \<le> f i"   and "suminf f \<noteq> \<infinity>" shows "f i \<noteq> \<infinity>"
geq_my_U_imp_weak_preference: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf"   and q :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>"   and "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes"   and "some_best outcomes \<R> \<succeq>[\<R>] some_worst outcomes \<R> \<and> (some_worst outcomes \<R>, some_best outcomes \<R>) \<notin> \<R>"   and "my_U outcomes \<R> q \<le> my_U outcomes \<R> p" shows "p \<succeq>[\<R>] q"
evalUni_if: fixes cond :: "bool"   and x :: "real" shows "evalUni (if cond then TrueFUni else FalseFUni) x = cond"
stableD: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> stable A" shows "F \<in> A co A"
add_one_2_add_one: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_add_one m lm (b, Oc # list)" shows "(hd b = Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and> (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, Oc # Oc # list))) \<and> (hd b \<noteq> Oc \<longrightarrow> (b = [] \<longrightarrow> wprepare_add_one m lm ([], Bk # Oc # list)) \<and> (b \<noteq> [] \<longrightarrow> wprepare_add_one m lm (tl b, hd b # Oc # list)))"
inter: shows "Span F [v] \<inter> Span F Vs' = {\<zero>}"
has_derivative_at_withinI: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "(f has_derivative f') (at x)" shows "(f has_derivative f') (at x within s)"
plus_int_raw_rsp_aux: fixes a :: "nat \<times> nat"   and b :: "nat \<times> nat"   and c :: "nat \<times> nat"   and d :: "nat \<times> nat" assumes "a \<approx> b"   and "c \<approx> d" shows "plus_int_raw a c \<approx> plus_int_raw b d"
hoare_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
remove_min: shows "fset_of_list t = fset_of_list (h # t) |-| {|h|}"
inj_ab_group_add_hom_axioms: shows "inj_ab_group_add_hom hom"
V: shows "\<forall>x\<in>src.carrier. V x \<in> U \<and> x \<in> V x"
rank_eq_dim_image: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = vec.dim (range ((*v) A))"
pow_int_def: fixes x :: "'b"   and e :: "int" shows "pow_int x e = (if e < 0 then (1::'b) / x ^ nat (- e) else x ^ nat e)"
case_eq_if: fixes f1 :: "String.literal \<Rightarrow> trm list \<Rightarrow> 'a"   and f2 :: "String.literal \<Rightarrow> formula \<Rightarrow> formula \<Rightarrow> 'a"   and f3 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f4 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f5 :: "trm \<Rightarrow> trm \<Rightarrow> 'a"   and f6 :: "formula \<Rightarrow> 'a"   and f7 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f8 :: "formula \<Rightarrow> formula \<Rightarrow> 'a"   and f9 :: "formula list \<Rightarrow> 'a"   and f10 :: "formula \<Rightarrow> 'a"   and f11 :: "nat \<Rightarrow> agg_type \<times> event_data \<Rightarrow> nat \<Rightarrow> trm \<Rightarrow> formula \<Rightarrow> 'a"   and f12 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f13 :: "\<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f14 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f15 :: "formula \<Rightarrow> \<I> \<Rightarrow> formula \<Rightarrow> 'a"   and f16 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and f17 :: "\<I> \<Rightarrow> formula Regex.regex \<Rightarrow> 'a"   and formula :: "formula" shows "(case formula of Pred x xa \<Rightarrow> f1 x xa | formula.Let x xa xb \<Rightarrow> f2 x xa xb | formula.Eq x xa \<Rightarrow> f3 x xa | Less x xa \<Rightarrow> f4 x xa | LessEq x xa \<Rightarrow> f5 x xa | Neg x \<Rightarrow> f6 x | Or x xa \<Rightarrow> f7 x xa | And x xa \<Rightarrow> f8 x xa | Ands x \<Rightarrow> f9 x | Exists x \<Rightarrow> f10 x | Agg x xa xb xc xd \<Rightarrow> f11 x xa xb xc xd | Prev x xa \<Rightarrow> f12 x xa | Next x xa \<Rightarrow> f13 x xa | Since x xa xb \<Rightarrow> f14 x xa xb | Until x xa xb \<Rightarrow> f15 x xa xb | MatchF x xa \<Rightarrow> f16 x xa | MatchP x xa \<Rightarrow> f17 x xa) = (if is_Pred formula then f1 (un_Pred1 formula) (un_Pred2 formula) else if is_Let formula then f2 (un_Let1 formula) (un_Let2 formula) (un_Let3 formula) else if is_Eq formula then f3 (un_Eq1 formula) (un_Eq2 formula) else if is_Less formula then f4 (un_Less1 formula) (un_Less2 formula) else if is_LessEq formula then f5 (un_LessEq1 formula) (un_LessEq2 formula) else if is_Neg formula then f6 (un_Neg formula) else if is_Or formula then f7 (un_Or1 formula) (un_Or2 formula) else if is_And formula then f8 (un_And1 formula) (un_And2 formula) else if is_Ands formula then f9 (un_Ands formula) else if is_Exists formula then f10 (un_Exists formula) else if is_Agg formula then f11 (un_Agg1 formula) (un_Agg2 formula) (un_Agg3 formula) (un_Agg4 formula) (un_Agg5 formula) else if is_Prev formula then f12 (un_Prev1 formula) (un_Prev2 formula) else if is_Next formula then f13 (un_Next1 formula) (un_Next2 formula) else if is_Since formula then f14 (un_Since1 formula) (un_Since2 formula) (un_Since3 formula) else if is_Until formula then f15 (un_Until1 formula) (un_Until2 formula) (un_Until3 formula) else if is_MatchF formula then f16 (un_MatchF1 formula) (un_MatchF2 formula) else f17 (un_MatchP1 formula) (un_MatchP2 formula))"
der: fixes x :: "real" shows "(v has_real_derivative k) (at x)"
converse_relcomp: fixes r :: "('c \<times> 'd) set"   and s :: "('d \<times> 'b) set" shows "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
smult_power: fixes x :: "'a"   and n :: "nat"   and p :: "'a poly" shows "smult (x ^ n) (p ^ n) = smult x p ^ n"
inf_commute: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = y \<squnion> x"
intro: fixes D :: "('c, 'd) Order_scheme" assumes "\<forall>X. X \<subseteq> carrier D \<and> X \<noteq> {} \<longrightarrow> (\<exists>x. minimum_elem D X x)" shows "Worder_axioms D"
carrier_append_rows: fixes A :: "'b mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nr2 :: "nat" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc" shows "A @\<^sub>r B \<in> carrier_mat (nr1 + nr2) nc"
SUP_lessD: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'c"   and i :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less (Sup (f ` A)) y"   and "i \<in> A" shows "less (f i) y"
zero_nin_lin_indpt: fixes S :: "'c set" assumes "S \<subseteq> carrier V"   and "lin_indpt S"   and "carrier K \<noteq> {\<zero>\<^bsub>K\<^esub>}" shows "\<zero>\<^bsub>V\<^esub> \<notin> S"
project_constrains_mono: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and D :: "'f set"   and C :: "'f set"   and F :: "'f program"   and A :: "'d set"   and B :: "'d set" assumes "Extend h"   and "D \<subseteq> C"   and "project h C F \<in> A co B" shows "project h D F \<in> A co B"
sup_multiset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "sup A B = (if A < B then B else A)"
xy_le_xx_yy: fixes x :: "real"   and y :: "real" shows "x * y \<le> x\<^sup>2 / 2 + y\<^sup>2 / 2"
r_into_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
nonpos_Reals_one_I: shows "(1::'a) \<notin> \<real>\<^sub>\<le>\<^sub>0"
bisim: shows "(\<forall>P Q P'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q')) \<and> (\<forall>P Q Q'. P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q'))"
distrib_imp2: fixes 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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup"   and "\<And>x y z. sup x (inf y z) = inf (sup x y) (sup x z)" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
Dom_Cod: shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr> = (op_dg \<AA> \<times>\<^sub>D\<^sub>G op_dg \<BB>)\<lparr>Cod\<rparr>"
cod_char: fixes f :: "'a" shows "L.cod f = (if L.arr f then cod f else null)"
ltake_llist_of: fixes n :: "nat"   and xs :: "'c list" shows "ltake (enat n) (llist_of xs) = llist_of (take n xs)"
closure_finite_cspan: fixes T :: "'b set" assumes "finite T" shows "closure (cspan T) = cspan T"
simps_2: fixes x :: "char list"   and a :: "aexp" shows "time (x ::= a) = (\<lambda>s. Suc 0)"
1: fixes m :: "int" shows "poly_mod_type TYPE('b) m = (m = int CARD('b))"
irr_f: fixes m :: "int"   and f :: "'d mod_ring poly" assumes "poly_mod_type_irr m f" shows "irreducible\<^sub>d f"
MtransT_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com \<times> 'b"   and a2 :: "'b" assumes "PL.MtransT tval aval a1 a2" obtains cf :: "('a, 'c) com \<times> 'b" and cf' :: "('a, 'c) com \<times> 'b" and s'' :: "'b" where "a1 = cf" and "a2 = s''" and "PL.MtransC tval aval cf cf'" and "PL.transT tval aval cf' s''"
and_nat_def: fixes m :: "nat"   and n :: "nat" shows "m AND n = nat (int m AND int n)"
relation_of_Var: fixes v :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and A :: "('c, 'd) action" shows "action.relation_of (Var v A) = (R (true \<turnstile> \<lambda>(A, A'). \<exists>a. tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = increase v a (alpha_rp.more A)) ;; action.relation_of A ;; R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = decrease v (alpha_rp.more A)))"
l7_20_bis: fixes A :: "'p"   and B :: "'p"   and M :: "'p" assumes "A \<noteq> B"   and "Col A M B"   and "Cong M A M B" shows "M Midpoint A B"
even_minus: fixes a :: "'a" shows "even (- a) = even a"
elims_2: fixes x :: "('c, 'd) BinomialTree list" assumes "rank_invar x" obtains "x = []"   | t :: "('c, 'd) BinomialTree" where "x = [ta__]"   | t :: "('c, 'd) BinomialTree" and t' :: "('c, 'd) BinomialTree" and bq :: "('c, 'd) BinomialTree list" where "x = t # t' # bq" and "rank t < rank t' \<and> rank_invar (t' # bq)"
DFS_axioms: shows "DFS (rquot_succs A n) (dfa_is_node A) (rquot_invariant A) rquot_ins rquot_memb (rquot_empt A)"
CompleteLatticeI_simp: fixes po :: "'b potype"   and A :: "'b set" assumes "po \<in> PartialOrder"   and "\<And>S. S \<subseteq> A \<Longrightarrow> \<exists>L. isLub S po L"   and "A = pset po" shows "po \<in> CompleteLattice"
map2_map_map: fixes h :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and f :: "'f \<Rightarrow> 'd"   and xs :: "'f list"   and g :: "'f \<Rightarrow> 'e" shows "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
R1_idem2: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R1 (R1 x) = R1 x"
distinct_55: fixes x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "Await x71 x72 \<noteq> Stop"
assms: shows "timpls_transformable_to TI (Fun f T) t"
\<Gamma>\<^sub>v_TAtom''_1: fixes x :: "(('fun, 'a, 'sets) prot_fun, 'b prot_atom) Term.term \<times> 'c"   and a :: "'b" shows "(fst x = Var (prot_atom.Atom a)) = (\<Gamma>\<^sub>v x = Var (prot_atom.Atom a))"
R2P_app: fixes r :: "('c \<times> 'd) set"   and x :: "'c"   and y :: "'d" shows "R2P r x y = ((x, y) \<in> r)"
self_fin_trad_strat_mart: fixes N :: "'a measure"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "risk_neutral_prob N"   and "filt_equiv F M N"   and "trading_strategy pf"   and "self_financing Mkt pf"   and "stock_portfolio Mkt pf"   and "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"   and "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)" shows "martingale N F (discounted_value r (cls_val_process Mkt pf))"
assms: shows "\<Gamma> \<turnstile> Rcd fs : RcdT fTs"
sets: shows "{x \<in> space M. P x} \<in> sets M"
epsclo_increasing: fixes Q :: "hf set" shows "Q \<inter> nfa.states M \<subseteq> epsclo Q"
p: shows "p = (\<Sum>f\<in>F'. q f * f)"
not_less_iff_gr_or_eq: fixes y :: "'d"   and x :: "'d" shows "(\<not> y < x) = (x < y \<or> x = y)"
DIM_prod: shows "DIM('b \<times> 'c) = DIM('b) + DIM('c)"
assms: fixes x :: "'a" shows "(x \<in> fset A) = (x \<in> fset B)"
cut_same: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A"   and "insert A H \<turnstile> B" shows "H \<turnstile> B"
rot_diamond_cube_def: shows "rot_diamond_cube = prod.swap \<circ> diamond_cube \<circ> prod.swap"
vsubstInp_def: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" shows "vsubstInp ys y1 y2 \<equiv> substInp ys (Var ys y1) y2"
RF_g: shows "x \<notin> RF\<^sup>\<circ> (TER g)"
IH: shows "\<forall>y. (y, i', j') \<in> less_than <*lex*> less_than \<longrightarrow> (\<forall>x xa. y = (x, xa) \<longrightarrow> (0::'a) \<le> m k k \<longrightarrow> x \<le> n \<longrightarrow> xa \<le> n \<longrightarrow> k \<le> n \<longrightarrow> fwi m n k x xa x xa = min (m x xa) (m x k + m k xa))"
Fxy: shows "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
I2f: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s \<and> HInv2 s \<and> HInv2 s' \<and> HInv3 s \<and> HInv4 s \<and> HInv5 s \<and> HInv6 s" shows "HInv6 s'"
f: shows "\<forall>\<sigma>. dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k \<longrightarrow> (cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst (g2 \<sigma>) mem\<^sub>h\<rangle>"
mem_box_cart_2: fixes x :: "(real, 'c) vec"   and a :: "(real, 'c) vec"   and b :: "(real, 'c) vec" shows "(x \<in> cbox a b) = (\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i)"
self_financingE: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "self_financing Mkt p" shows "\<forall>n. val_process Mkt p n = cls_val_process Mkt p n"
q'_delta: shows "ego2.q' \<delta> = ego2.s' 0"
length_removeAll_less: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "length (removeAll x xs) < length xs"
I_2: fixes x :: "'a" assumes "x \<in> carrier" shows "open (I x)"
atU_fmt: fixes U :: "'a set"   and c :: "'a list multiset" assumes "atU U c" obtains ui :: "'a" and r :: "'a list" where "ui # r \<in># c" and "ui \<in> U"
simps_48: fixes x7 :: "'b ptrm"   and x6 :: "'b ptrm" shows "PSnd x7 \<noteq> PFst x6"
symp_symclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symp (symclp r)"
bind_cong: fixes x :: "'c option"   and y :: "'c option"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "Option.bind x f = Option.bind y g"
trapped_sol_right: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "{0..} \<subseteq> rev.existence_ivl0 x"
lm027: fixes f :: "('b \<times> 'a set) set" assumes "{} \<notin> Range f"   and "runiq f" shows "is_non_overlapping (omega ` f)"
assms_3: fixes a :: "'a" assumes "a \<noteq> \<zero>" shows "f a \<noteq> []"
pareto_losersI: fixes x :: "'a"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "x \<prec>[Pareto R] y" shows "x \<in> pareto_losers R"
triangular_toI: fixes k :: "nat"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>j < k; j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)" shows "local.triangular_to k A"
set_sel_1: fixes a :: "('g, 'h, 'i) generat" assumes "is_Pure a" shows "generat.result a \<in> generat_pures a"
degree_prod_le: fixes f :: "'c \<Rightarrow> 'b poly"   and S :: "'c set" shows "degree (prod f S) \<le> (\<Sum>i\<in>S. degree (f i))"
emeasure_mono_AE: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "AE x\<in>A in M. x \<in> B"   and "B \<in> sets M" shows "emeasure M A \<le> emeasure M B"
dagger_of_transpose_is_cnj: fixes M :: "complex Matrix.mat" shows "M\<^sup>t\<^sup>\<dagger> = M\<^sup>\<star>"
FstSnd: fixes x :: "ZF" assumes "isOpair x" shows "Opair (Fst x) (Snd x) = x"
d2: shows "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>"
objI_src: fixes a :: "'a" assumes "C.arr a"   and "src\<^sub>C a = a" shows "C.obj a"
arctan_ms_aux_inf: fixes xs :: "('a \<times> real) msllist"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "\<exists>e>0. ms_aux_hd_exp xs = Some e"   and "0 < fst (dominant_term_ms_aux xs)"   and "trimmed_ms_aux xs"   and "basis_wf basis"   and "is_expansion_aux xs f basis" shows "is_expansion_aux (minus_ms_aux (const_ms_aux (pi / 2)) (arctan_ms_aux (inverse_ms_aux xs))) (\<lambda>x. arctan (f x)) basis"
mod: fixes a :: "nat" shows "principal_dchar n (a mod n) = principal_dchar n a"
mem: shows "x \<in> r \<down>m P \<union> r \<down>m {#p#}"
elims_2: fixes x :: "'b extB"   and xa :: "'b extB" assumes "x \<le> xa" obtains uu_ :: "'b extB" where "xa__ = extB.Bot" and "xa = uu_"   | x :: "'b" and y :: "'b" where "xa__ = extB.Val x" and "xa = extB.Val y" and "x \<le> y"
inverse_arrows_def: fixes f :: "'a1"   and g :: "'a1" shows "C1.inverse_arrows f g \<equiv> C1.ide (g \<cdot>\<^sub>1 f) \<and> C1.ide (f \<cdot>\<^sub>1 g)"
ucast_scast_id: fixes x :: "'b signed word" shows "ucast (scast x) = x"
sorted_wrt_take: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (take n xs)"
list_slice_aux_length: fixes xs :: "'b list"   and k :: "nat"   and n :: "nat" shows "length (list_slice_aux xs k n) = n"
bernoulli_stream_pref_prob: fixes M :: "bool stream measure"   and p :: "real"   and n :: "nat"   and x :: "bool stream" assumes "M = bernoulli_stream p"   and "0 \<le> p"   and "p \<le> 1"   and "Suc 0 \<le> n" shows "emeasure M {w \<in> space M. stake n w = stake n x} = ennreal (prod (prob_component p x) {0..n - 1})"
in_abs_rangeI: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "(nat \<times> nat) list"   and v :: "nat"   and a :: "nat" assumes "abs_ast_prob problem"   and "ast_problem.wf_partial_state problem s"   and "(v, a) \<in> set s" shows "a \<in> set (the (ast_problem.abs_range_map problem v))"
notin_range_map_of: fixes y :: "'a"   and xys :: "('b \<times> 'a) list" assumes "y \<notin> snd ` set xys" shows "Some y \<notin> range (map_of xys)"
simps_2: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a list"   and xs :: "'a list list"   and ys :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.Tensor f (x # xs) ys = mult.vec_mat_Tensor f x ys @ mult.Tensor f xs ys"
PiE_E: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set"   and x :: "'f" assumes "f \<in> Pi\<^sub>E A B" obtains "x \<in> A" and "f x \<in> B x"   | "x \<notin> A" and "f x = undefined"
finsert_fminus: fixes a :: "'b"   and A :: "'b fset" assumes "a |\<in>| A" shows "finsert a (A |-| {|a|}) = A"
lift_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'m2" shows "m2.lift f x = bind2 x (\<lambda>x. return2 (f x))"
SeqRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e';; e\<^sub>2,s',b'\<rangle>"
finite_open_segment: fixes a :: "'a"   and b :: "'a" shows "finite (open_segment a b) = (a = b)"
count_roots_interval_iff: fixes p :: "real poly" assumes "square_free p" shows "count_roots_interval p = count_roots_interval_sf p"
extr_subset_dy_fake_msg: fixes bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set" shows "extr bad IK chan \<subseteq> dy_fake_msg bad IK chan"
D: shows "\<langle>n, Pwd n\<rangle> \<in> spied s"
product_commutativity: fixes x :: "i"   and y :: "i" assumes "CM"   and "O x y" shows "x \<otimes> y = y \<otimes> x"
complete_helper: fixes ts :: "'a set list"   and ss :: "'a set list" assumes "{} \<notin> set ts"   and "\<Union> (set ss) \<subseteq> \<Union> (set ts)" shows "\<Union> (set ts) = \<Union> (set (partitioning1 ss ts))"
closure_halfspace_lt: fixes a :: "'c"   and b :: "real" assumes "a \<noteq> (0::'c)" shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
element_tag_name: shows "h \<turnstile> get_tag_name element_ptr \<rightarrow>\<^sub>r element_tag_name"
gauss_sum_reduction: fixes k :: "nat" assumes "coprime k n" shows "gauss_sum k = cnj (\<chi> k) * gauss_sum 1"