hcju/isabelleps huggingface dataset:
elims_1: fixes x :: "Formula.formula"   and y :: "bool" assumes "Formula.future_bounded x = y" obtains uu_ :: "String.literal" and uv_ :: "Formula.trm list" where "x = formula.Pred uu_ uv_" and "y"   | p :: "String.literal" and \<phi> :: "Formula.formula" and \<psi> :: "Formula.formula" where "x = formula.Let p \<phi>__ \<psi>__" and "y = (Formula.future_bounded \<phi>__ \<and> Formula.future_bounded \<psi>__)"   | uw_ :: "Formula.trm" and ux_ :: "Formula.trm" where "x = formula.Eq uw_ ux_" and "y"   | uy_ :: "Formula.trm" and uz_ :: "Formula.trm" where "x = formula.Less uy_ uz_" and "y"   | va_ :: "Formula.trm" and vb_ :: "Formula.trm" where "x = formula.LessEq va_ vb_" and "y"   | \<phi> :: "Formula.formula" where "x = formula.Neg \<phi>'__" and "y = Formula.future_bounded \<phi>'__"   | \<phi> :: "Formula.formula" and \<psi> :: "Formula.formula" where "x = formula.Or \<phi>''__ \<psi>'__" and "y = (Formula.future_bounded \<phi>''__ \<and> Formula.future_bounded \<psi>'__)"   | \<phi> :: "Formula.formula" and \<psi> :: "Formula.formula" where "x = formula.And \<phi>'''__ \<psi>''__" and "y = (Formula.future_bounded \<phi>'''__ \<and> Formula.future_bounded \<psi>''__)"   | l :: "Formula.formula list" where "x = formula.Ands l" and "y = list_all Formula.future_bounded l"   | \<phi> :: "Formula.formula" where "x = formula.Exists \<phi>''''__" and "y = Formula.future_bounded \<phi>''''__"   | ya :: "nat" and \<omega> :: "Formula.agg_type \<times> event_data" and b :: "nat" and f :: "Formula.trm" and \<phi> :: "Formula.formula" where "x = formula.Agg ya \<omega> b f \<phi>'''''__" and "y = Formula.future_bounded \<phi>'''''__"   | I :: "\<I>" and \<phi> :: "Formula.formula" where "x = formula.Prev I__ \<phi>''''''__" and "y = Formula.future_bounded \<phi>''''''__"   | I :: "\<I>" and \<phi> :: "Formula.formula" where "x = formula.Next Ia__ \<phi>'''''''__" and "y = Formula.future_bounded \<phi>'''''''__"   | \<phi> :: "Formula.formula" and I :: "\<I>" and \<psi> :: "Formula.formula" where "x = formula.Since \<phi>''''''''__ Ib__ \<psi>'''__" and "y = (Formula.future_bounded \<phi>''''''''__ \<and> Formula.future_bounded \<psi>'''__)"   | \<phi> :: "Formula.formula" and I :: "\<I>" and \<psi> :: "Formula.formula" where "x = formula.Until \<phi> Ic__ \<psi>" and "y = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi> \<and> right Ic__ \<noteq> \<infinity>)"   | I :: "\<I>" and r :: "Formula.formula Regex.regex" where "x = formula.MatchP Id__ r__" and "y = regex.pred_regex Formula.future_bounded r__"   | I :: "\<I>" and r :: "Formula.formula Regex.regex" where "x = formula.MatchF I r" and "y = (regex.pred_regex Formula.future_bounded r \<and> right I \<noteq> \<infinity>)"
transversal_segment_pos: fixes x :: "'a"   and y :: "'a"   and w :: "'a" assumes "transversal_segment x y"   and "w \<in> {x--y}"   and "0 < f w \<bullet> rot (x - y)" shows "\<forall>z\<in>{x--y}. 0 < f z \<bullet> rot (x - y)"
prv_cnj_dsj_distrib2: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (dsj (cnj \<phi> \<chi>1) (cnj \<phi> \<chi>2)) (cnj \<phi> (dsj \<chi>1 \<chi>2)))"
elims: fixes x :: "ordinal"   and y :: "(ordinal \<times> nat) list" assumes "to_cnf2 x = y" obtains x :: "ordinal" where "xa__ = x" and "y = (if x = 0 then [] else (oLog \<omega> x, inv ordinal_of_nat (x div \<omega> ** oLog \<omega> x)) # to_cnf2 (x mod \<omega> ** oLog \<omega> x))"
b_e_type_nop: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Nop" shows "ts = ts'"
update_unalive_loc: fixes l :: "Location"   and s :: "Store"   and x :: "Value" assumes "\<not> alive (ref l) s" shows "s\<langle>l := x\<rangle> = s"
par_comp_prot_stateful_def: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set"   and Sec :: "('fun, 'var) Term.term set" shows "par_comp_prot_stateful \<P> Sec \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l1 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l1 \<A>)) (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l2 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l2 \<A>)) Sec) \<and> ground Sec \<and> (\<forall>s\<in>Sec. \<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Sec) \<and> (\<forall>(i, p)\<in>\<Union> (setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t ` \<P>). \<forall>(j, q)\<in>\<Union> (setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t ` \<P>). (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j) \<and> typing_cond_prot_stateful \<P>"
elims: fixes x :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "balance x = y" obtains xs :: "'a list" and ys :: "'a list" where "x = (xs, ys)" and "y = (if length xs \<le> length ys then (xs, ys) else ([], ys @ rev xs))"
legs_form_cospan_1: shows "C.cospan \<mu>.dom.leg0 \<nu>.dom.leg1"
p_def: fixes i :: "nat" shows "p i = snd (f i)"
simps_4: fixes h :: "heap"   and i :: "int" shows "typeof\<^bsub>h\<^esub> (Intg i) = \<lfloor>Integer\<rfloor>"
decr_ev: fixes m :: "'ord" assumes "(shd js, m) \<in> Ord" shows "ev (\<lambda>js. (shd js, m) \<in> Ord \<and> shd js \<noteq> m) js"
P: fixes i :: "'a" assumes "is_glb L i (insert x A)" shows "P i"
elims_1: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b stream"   and y :: "bool" assumes "holds x xa = y" obtains P :: "'b \<Rightarrow> bool" and xs :: "'b stream" where "x = P" and "xa = xs" and "y = P (shd xs)"
compactin_imp_Bolzano_Weierstrass: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "compactin X S"   and "infinite T \<and> T \<subseteq> S" shows "S \<inter> X derived_set_of T \<noteq> {}"
right_total_alt_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "right_total R = ((=) \<le> R\<inverse>\<inverse> OO R)"
adjoint_pair_def: fixes f :: "'a"   and g :: "'a" shows "C.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
wts'_wts: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:'] Ts" shows "P,E,h \<turnstile> es [:] Ts"
rel_fun_eq_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(=) ===> R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
list_of_dlist_inverse: fixes x :: "'c set_dlist" shows "Abs_dlist' (list_of_dlist x) = x"
\<L>\<^sub>P_derives: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and a :: "symbol list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "a \<in> CFG.\<L>\<^sub>P \<NN> \<TT> \<RR> \<SS>" shows "\<exists>b. CFG.derives \<NN> \<TT> \<RR> [\<SS>] (a @ b)"
adopt_node_locs_def: shows "Shadow_DOM.adopt_node_locs \<equiv> l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs Shadow_DOM.remove_child_locs set_disconnected_nodes_locs"
assms_1: shows "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
fst'_def: shows "fst' \<equiv> op_sem Fst"
Lyndon_pref_suf_less: fixes w :: "'a list"   and p :: "'a list"   and s :: "'a list" assumes "Lyndon w"   and "p \<le>p w"   and "s \<le>ns w"   and "s \<noteq> w" shows "p <lex s"
strip_while_def: fixes P :: "'a \<Rightarrow> bool" shows "strip_while P = rev \<circ> dropWhile P \<circ> rev"
ereal_strict_mono2: fixes x :: "real"   and y :: "real" assumes "x < y" shows "ereal x < ereal y"
unit_factor_mult_unit_right: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "unit_factor (b * a) = unit_factor b * a"
eSuc_le_iff: fixes x :: "enat"   and y :: "enat" shows "(eSuc x \<le> y) = (\<exists>y'. y = eSuc y' \<and> x \<le> y')"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "llam"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> llam \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<turnstile> x2 : x3 "   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (lPar x) T"   and "\<And>\<Gamma> t1 T1 T2 t2 y. \<lbrakk> \<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2 ; \<And>z. P z \<Gamma> t1 (T1 \<rightarrow> T2);  \<Gamma> \<turnstile> t2 : T1 ; \<And>z. P z \<Gamma> t2 T1\<rbrakk> \<Longrightarrow> P y \<Gamma> (lApp t1 t2) T2"   and "\<And>x t T1 \<Gamma> T2 y. \<lbrakk>x \<sharp> y; x \<sharp> t;  (x, T1) # \<Gamma> \<turnstile> freshen t x : T2 ; \<And>z. P z ((x, T1) # \<Gamma>) (freshen t x) T2\<rbrakk> \<Longrightarrow> P y \<Gamma> (lLam t) (T1 \<rightarrow> T2)" shows "P z x1 x2 x3"
CALL: fixes B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and R :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and S :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "({B} \<union> X) \<rhd> body : Sec A R S (\<phi> (FIX \<phi>))"   and "Monotone \<phi>"   and "B = Sec A R S (FIX \<phi>)" shows "X \<rhd> Call : B"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "w \<le> x"   and "z \<parallel> y \<le> y \<parallel> w" shows "z \<parallel> y \<le> y \<parallel> x"
steps_r_alt: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set"   and l'' :: "'d"   and R'' :: "('e \<Rightarrow> real) set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"   and "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle>" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"
IH: assumes "P \<turnstile> C sees_methods Mm" shows "\<exists>Mm' Mm\<^sub>2. P \<turnstile> C' sees_methods Mm' \<and> Mm' = Mm ++ Mm\<^sub>2 \<and> (\<forall>M m D. Mm\<^sub>2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C)"
image_mset_Diff: fixes B :: "'b multiset"   and A :: "'b multiset"   and f :: "'b \<Rightarrow> 'c" assumes "B \<subseteq># A" shows "image_mset f (A - B) = image_mset f A - image_mset f B"
that: fixes a :: "'a"   and b :: "'a" assumes "x = {a, b}"   and "a \<noteq> b" shows "thesis"
nb\<^sub>2: shows "inj_on (\<lambda>x. State (fst x) (index vs (snd x))) (set (List.product ks vs))"
3: shows "rawpsubstT (rawpsubstT (pls r1 r2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = pls (rawpsubstT (rawpsubstT r1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubstT (rawpsubstT r2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
pcr_blinfun_def: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_blinfun T Ta \<equiv> rel_fun T Ta OO cr_blinfun"
is_cparser_scan_upto: fixes e :: "char"   and "end" :: "char list" shows "is_cparser (scan_upto (e # end))"
le_sub_flip: fixes v :: "val"   and v' :: "val" assumes "v \<sqsubseteq> v'" shows "T v' <: T v"
simps_1: fixes x5 :: "String.literal"   and y5 :: "String.literal" shows "(Class x5 = Class y5) = (x5 = y5)"
CP_t_K: fixes k :: "'key"   and obs :: "'observation list" assumes "k \<in> keys"   and "obs \<in> OB ` msgs" shows "prob ((\<lambda>x. ((t \<circ> OB) x, fst x)) -` {(t obs, k)} \<inter> msgs) / prob (fst -` snd ` {(t obs, k)} \<inter> msgs) = real (card (t -` {t obs} \<inter> OB ` msgs)) * (prob ((\<lambda>x. (OB x, fst x)) -` {(obs, k)} \<inter> msgs) / prob (fst -` snd ` {(obs, k)} \<inter> msgs))"
iTILL_trans: fixes y :: "nat"   and x :: "nat"   and z :: "nat" assumes "y \<in> [\<dots>x]"   and "z \<in> [\<dots>y]" shows "z \<in> [\<dots>x]"
inpcp: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H \<bottom>" shows "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (F::'b formula)"
bound_on_all_plans_bounds_PLS_s: fixes P :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> bool"   and f :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> ('c, 'd) fmap \<Rightarrow> nat" assumes "\<forall>PROB as s. finite PROB \<and> P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB s)" shows "\<forall>PROB s as. finite PROB \<and> P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>x. x \<in> PLS s as \<and> x < f PROB s)"
finite_image_absD: fixes S :: "'b set" assumes "finite (abs ` S)" shows "finite S"
Jacobi_mod_cong: fixes a :: "int"   and b :: "int"   and n :: "int" assumes "[a = b] (mod n)" shows "Jacobi a n = Jacobi b n"
fminus_fempty: fixes A :: "'c fset" shows "A |-| {||} = A"
Rep_cfun_inverse: fixes x :: "'b \<rightarrow> 'c" shows "Abs_cfun (Rep_cfun x) = x"
dghm_is_arr_HomCod_1: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
multiset_nonemptyE: fixes A :: "'c multiset" assumes "A \<noteq> {#}" obtains x :: "'c" where "x \<in># A"
R_def: shows "R = EM_remainder 1 (\<lambda>x. - (s * complex_of_real (x + a) powr (- 1 - s))) 0"
match_ex_entry: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and start :: "nat"   and "end" :: "nat"   and catch_type :: "String.literal option"   and handler :: "nat \<times> nat" shows "matches_ex_entry P C pc (start, end, catch_type, handler) = (start \<le> pc \<and> pc < end \<and> (case catch_type of None \<Rightarrow> True | \<lfloor>C'\<rfloor> \<Rightarrow> P \<turnstile> C \<preceq>\<^sup>* C'))"
cOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "x \<sharp> M"   and "x \<sharp> \<Psi>"   and "distinct xvec"   and "distinct yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* yvec"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"
simple_path_cases: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "arc g \<or> pathfinish g = pathstart g"
simps_2: fixes x :: "'a"   and c :: "color"   and a :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c"   and b :: "('a, 'c) rbt" shows "rbt_del x (Branch c a y s b) = (if x < y then rbt_del_from_left x a y s b else if y < x then rbt_del_from_right x a y s b else combine a b)"
IH: assumes "(v, w) \<notin> E\<^sup>+" shows "\<exists>x y. (v, x) \<in> (E \<union> U)\<^sup>* \<and> (x, y) \<in> U \<and> (y, w) \<in> (E \<union> U)\<^sup>*"
i_set_Plus_closed: fixes I :: "nat set"   and k :: "nat" assumes "I \<in> i_set" shows "I \<oplus> k \<in> i_set"
induct: fixes P :: "nat \<Rightarrow> C \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "C" assumes "\<And>c. P c (C [])"   and "\<And>c ns. P (c + 1) (C ns) \<Longrightarrow> P c (C (C [] # ns))"   and "\<And>c n ns ms. P c (C (funC (C (n # ns)) (c + 2) @ ms)) \<Longrightarrow> P c (C (C (n # ns) # ms))" shows "P a0 a1"
unat_pow_le_intro: fixes n :: "nat"   and x :: "'b word" assumes "LENGTH('b) \<le> n" shows "unat x < 2 ^ n"
c: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "f x (f y z) = f y (f x z)"
at_hchop_dist2: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> (\<forall>v'. va = c > v' \<longrightarrow> \<phi> ts v') \<and> (\<forall>v'. u = c > v' \<longrightarrow> \<psi> ts v')) \<longrightarrow> (\<forall>v'. v = c > v' \<longrightarrow> (\<exists>v u. v'=v\<parallel>u \<and> \<phi> ts v \<and> \<psi> ts u))"
path_independent_on_consistency_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "path_independent_on A f" shows "consistency_on A f"
drop_pref: fixes u :: "'b list"   and w :: "'b list" shows "drop \<^bold>|u\<^bold>| (u \<cdot> w) = w"
intros: fixes X :: "'a set"   and u :: "'a \<Rightarrow> real"   and I :: "'a \<Rightarrow> Regions.intv"   and X\<^sub>0 :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<forall>x\<in>X. 0 \<le> u x"   and "\<forall>x\<in>X. Regions.intv_elem x u (I x)"   and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d}"   and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))" shows "u \<in> Regions.region X I r"
induced_subgraph_prob: fixes p :: "real"   and H :: "nat set \<times> nat set set"   and V :: "nat set"   and n :: "nat" assumes "edge_space p"   and "uverts H \<subseteq> V"   and "uwellformed H"   and "V \<subseteq> edge_space.S_verts n" shows "\<P>(es in edge_space.P n p. subgraph H (induced_subgraph V (edge_space.edge_ugraph n es))) = p ^ card (uedges H)"
cases: fixes a :: "nat \<times> nat" assumes "zerozero' a" obtains x :: "nat" and y :: "nat" where "a = (x, y)" and "equals (x, y) (0, 0)"
pick_mono_le: fixes n :: "nat"   and S :: "nat set"   and m :: "nat" assumes "n < card S"   and "m < n" shows "pick S m < pick S n"
dim'_2: shows "dim_col (subresultant_mat J f g * mat (degree f - J + (degree g - J)) (degree f - J + (degree g - J)) (\<lambda>(i, j). if i = j then if i < degree g - J then [:c:] else 1 else 0)) = degree f - J + (degree g - J)"
one_time_pad: fixes f :: "nat \<Rightarrow> nat"   and q :: "nat" assumes "inj_on f {..<q}"   and "f ` {..<q} = {..<q}" shows "map_spmf f (sample_uniform q) = sample_uniform q"
Inf_pres_radj_aux: fixes g :: "'c \<Rightarrow> 'd" assumes "Inf_pres g" shows "ladj g \<stileturn> g"
disj: shows "affine hull T \<inter> convex hull (S - T) = {}"
rel_G_witness2: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok :: "('z \<times> ('z \<times> 'aa) \<times> 'aa \<times> 'ab \<times> ('ab \<times> 'ac) \<times> 'ac \<times> 'ad) itself"   and L1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and x :: "('z, 'ab, 'n, 'q, 't, 'w, 'ad) G"   and y :: "('aa, 'ac, 'p, 's, 'v, 'y, 'ad) G" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_G L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_G (\<lambda>(x, y') y. y' = y \<and> L1 x y) (\<lambda>(x, y') y. y' = y \<and> L2 x y) Co1' Co2' Contra1' Contra2' (rel_G_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y)) y"
rreq_rrep_sn_simps_2: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and oip :: "nat"   and sip :: "nat" shows "rreq_rrep_sn (Rrep hops dip dsn oip sip) = (1 \<le> dsn)"
expectation_pair_pmf_fst: fixes p :: "'b pmf"   and q :: "'c pmf"   and f :: "'b \<Rightarrow> 'a" shows "measure_pmf.expectation (pair_pmf p q) (\<lambda>x. f (fst x)) = measure_pmf.expectation p f"
comp_arr_ide: fixes a :: "'j"   and f :: "'j" assumes "J.ide a"   and "J.seq f a" shows "f \<cdot>\<^sub>J a = f"
stack_not_finished: fixes x :: "'v" assumes "x \<in> set (stack s)" shows "x \<notin> dom (finished s)"
sc_minus_am: fixes a :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M" shows "-\<^sub>a a \<cdot>\<^sub>s m = a \<cdot>\<^sub>s (-\<^sub>a m)"
root_summation: fixes k :: "nat"   and n :: "nat" assumes "0 < k"   and "k < n" shows "sum ((^) (FFT.root n ^ k)) {0..<n} = 0"
simps_1: shows "subtrees_mset \<langle>\<rangle> = {#\<langle>\<rangle>#}"
hyps: assumes "is_tangle_diagram zs \<and> codomain_wall zs = domain_wall y" shows "is_tangle_diagram (zs \<circ> y)"
finite_chainsD: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'c set" assumes "finite_chains ord"   and "Complete_Partial_Order.chain ord Y" shows "finite Y"
cod_eval: fixes f :: "'c local.term" assumes "Arr f" shows "D.cod \<lbrace>f\<rbrace> = \<lbrace>Cod f\<rbrace>"
locate_pred_ex: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "\<exists>n_j. locate_pred f al i n_j"
l3_defs_6: fixes implem :: "chan \<Rightarrow> msg" assumes "dhlvl3 implem" shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
InfinitesimalD2: fixes x :: "'c star"   and r :: "real" assumes "x \<in> Infinitesimal"   and "0 < r" shows "hnorm x < hypreal_of_real r"
LstSeqP_cong: fixes H :: "fm set"   and s :: "tm"   and s' :: "tm"   and k :: "tm"   and k' :: "tm"   and y :: "tm"   and y' :: "tm" assumes "H \<turnstile> s EQ s'"   and "H \<turnstile> k EQ k'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> LstSeqP s k y IFF LstSeqP s' k' y'"
double_negation: fixes P :: "'b upred" shows "(\<not> \<not> P) = P"
eq_zero_iff: fixes x :: "nat" shows "(\<Phi> x = 0) = (\<not> coprime x c)"
stoch_proc_filt_sets: fixes n :: "nat"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "\<And>i. i \<le> n \<Longrightarrow> X i \<in> M \<rightarrow>\<^sub>M N" shows "sets (stoch_proc_filt M X N n) = sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets N})"
Minr_Suc_simp: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" shows "Minr Rr xs (Suc w) = (if Minr Rr xs w \<le> w then Minr Rr xs w else if Rr (xs @ [Suc w]) then Suc w else Suc (Suc w))"
that: fixes f' :: "'a \<Rightarrow> 'e" assumes "diff_fun k charts f'"   and "\<And>x. x \<in> A \<Longrightarrow> f' x = f x"   and "csupport_on carrier f' \<inter> carrier \<subseteq> U" shows "thesis"
cross_edge_def: fixes u :: "'v"   and v :: "'v"   and s :: "('v, 'es) state_scheme" shows "cross_edge u v s \<equiv> let c = insert (u, v) (cross_edges s) in s\<lparr>cross_edges := c\<rparr>"
subset_not_subset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subset> B) = strict (\<subseteq>) A B"