hcju/isabelleps huggingface dataset:
linear_order_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> Relation.total_on A r"
numeral_power_eq_of_nat_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num"   and n :: "nat"   and y :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (numeral.numeral one plus x) n = semiring_1.of_nat one plus zero y) = (numeral x ^ n = y)"
hyperpow_ge_zero: fixes r :: "'b star"   and n :: "nat star" assumes "0 \<le> r" shows "0 \<le> r pow n"
assms_3: fixes es :: "('a \<times> 'a) list" assumes "P e es" shows "P e (e # es)"
is_tm_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
prems: shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> T'' # Ts''"
r_neg: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "x + \<ominus>\<^bsub>V\<^esub> x = 0\<^sub>v n"
times_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a * b \<equiv> Abs_N (Rep_N a + Rep_N b)"
hoare_triple_success: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P" shows "success c h"
valid_cube_boundary: shows "\<forall>(k, \<gamma>)\<in>boundary C. valid_path \<gamma>"
collapse: fixes kripke :: "('b, 'c) kripke" shows "Kripke (\<pi> kripke) (\<K> kripke) = kripke"
memFreshSimp: fixes y :: "name"   and yvec :: "name list" shows "(y \<notin> set yvec) = y \<sharp> yvec"
valuation_aux_2: fixes M :: "object atom formula set" assumes "wm_basic M" shows "\<forall>G\<in>close_world M. valuation M \<Turnstile> G"
less_imp_le: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a \<le> b"
bexI: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e"   and A :: "'e set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
limit_conc: fixes w :: "'b list"   and x :: "nat \<Rightarrow> 'b" shows "limit (w \<frown> x) = limit x"
transient_strengthen: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> transient A"   and "B \<subseteq> A" shows "F \<in> transient B"
at1: assumes "n \<noteq> 0" shows "eval at1 [f \<triangleright> n] \<down>= the (f 1)"
distinct_4: fixes x3 :: "'a formula"   and x1 :: "'a" shows "\<^bold>\<not> x3 \<noteq> Atom x1"
SXcpt_mdecls_def: shows "SXcpt_mdecls \<equiv> []"
reflp_def: fixes r :: "'l \<Rightarrow> 'l \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
uc: shows "uniformly_continuous_on (cbox (a, c) (b, d)) f"
closed_halfspace_component_ge: fixes a :: "real"   and i :: "'b" shows "closed {x. a \<le> x \<bullet> i}"
Heap_cases: fixes f :: "'g Heap"   and h :: "heap" obtains x :: "'g" and h' :: "heap" where "execute f h = Some (x, h')"   | "execute f h = None"
strict_runit: fixes a :: "'d" assumes "D.ide a" shows "\<r>\<^sub>D[a] = a"
listset_closed_map2: fixes ys1 :: "'a list"   and xs :: "'a set list"   and ys2 :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "ys1 \<in> listset xs"   and "ys2 \<in> listset xs"   and "\<And>x y1 y2. \<lbrakk>x \<in> set xs; y1 \<in> x; y2 \<in> x\<rbrakk> \<Longrightarrow> f y1 y2 \<in> x" shows "map2 f ys1 ys2 \<in> listset xs"
conc_trans_additional_5: fixes A :: "'a nres"   and B :: "'a nres"   and C :: "'a nres" assumes "A \<le> \<Down> Id B"   and "B \<le> C" shows "A \<le> C"
is_Zp_continuous_def: fixes p :: "int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.is_Zp_continuous p f = (f \<in> carrier (function_ring (carrier (padic_int p)) (padic_int p)) \<and> (\<forall>s. padic_integers.is_Zp_cauchy p s \<longrightarrow> padic_integers.is_Zp_cauchy p (f \<circ> s)))"
gen_gbg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Rf :: "('e \<times> 'g set set) set" shows "\<langle>Rm, Rf\<rangle>gen_gbg_impl_rel_eext = {(\<lparr>gbgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>gbg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
Abs_CGS_cases: fixes x :: "CGS" obtains y :: "unit" where "x = Abs_CGS y" and "y \<in> UNIV"
inv_char_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b"   and \<phi> :: "'b" assumes "left_hom V H \<omega>"   and "partial_magma.arr (subcategory.comp V (weak_composition.left V H \<omega>)) \<phi>"   and "category.iso V \<phi>" shows "category.inverse_arrows (subcategory.comp V (weak_composition.left V H \<omega>)) \<phi> (category.inv V \<phi>)"
eq_free_eq_subst_te: fixes t :: "typ"   and S1 :: "nat \<Rightarrow> typ"   and S2 :: "nat \<Rightarrow> typ" assumes "\<forall>n. n \<in> free_tv t \<longrightarrow> S1 n = S2 n" shows "$ S1 t = $ S2 t"
check_allm_gen_index_def: fixes g :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b \<Rightarrow> nat \<Rightarrow> 'a + unit"   and n :: "nat"   and xs :: "'b list" shows "check_allm_gen_index g f n xs = snd (foldl (\<lambda>(i, m) x. (g x i, m \<bind> (\<lambda>_. f x i))) (n, Inr ()) xs)"
wf_measure: fixes f :: "'e \<Rightarrow> nat" shows "wf (measure f)"
assms_1: fixes i :: "'b" assumes "i \<in> S" shows "{x \<in> \<Omega>. P i x} \<in> M"
convex_Inter: fixes f :: "'b set set" assumes "\<And>s. s \<in> f \<Longrightarrow> convex s" shows "convex (\<Inter> f)"
simps_5: fixes f1 :: "fdecl \<Rightarrow> 'b"   and f2 :: "mdecl \<Rightarrow> 'b"   and x1 :: "fdecl" shows "(case fdecl x1 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f1 x1"
path_is_step: fixes c :: "'c"   and p :: "'a list"   and c' :: "'c" assumes "lts.path c p c'" shows "(c, c') \<in> step\<^sup>*"
less_sets_imp_disjnt: fixes A :: "'a set"   and B :: "'a set" assumes "A \<lless> B" shows "disjnt A B"
fin_make_untimed_inf_truncate_Nonempty_all0: fixes z :: "nat \<Rightarrow> 'd list" assumes "z 0 \<noteq> []" shows "\<forall>i. fin_make_untimed (inf_truncate z i) \<noteq> []"
that: fixes y :: "('idx, 'pred, 'act) formula" assumes "weak_formula y"   and "supp y \<subseteq> supp P"   and "y distinguishes P from Q" shows "thesis"
real_sq: fixes a :: "real" shows "a * a = a\<^sup>2"
selectlen: fixes l :: "bit list"   and i :: "nat"   and j :: "nat" shows "length (select l i j) = j - i + 1"
conv_involutive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x\<^sup>t\<^sup>t = x"
vars_empty_zero: fixes lp :: "linear_poly" shows "(vars lp = {}) = (lp = 0)"
XD_XH_gfp_F_2: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "snd (gfp_F ds) \<subseteq> snd XD_XH"
map_nth_default: fixes x :: "'f"   and xs :: "'f list" shows "map (nth_default x xs) [0..<length xs] = xs"
fun_rep_map2_rep: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ps1 :: "(char list \<times> 'a) list"   and ps2 :: "(char list \<times> 'a) list" assumes "f \<top> \<top> = \<top>" shows "fun_rep (map2_st_rep f ps1 ps2) = (\<lambda>x. f (fun_rep ps1 x) (fun_rep ps2 x))"
strong_int_gr_induct: fixes k :: "int"   and i :: "int"   and P :: "int \<Rightarrow> bool" assumes "k < i"   and "\<And>i. \<lbrakk>k < i; \<And>j. \<lbrakk>k < j; j < i\<rbrakk> \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i" shows "P i"
ide_compE: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.ide (g \<bullet> f)" obtains "V.seq g f" and "V.seq f g" and "g \<bullet> f = V.dom f" and "g \<bullet> f = V.cod g"
primitive_part_normalize: fixes f :: "'a poly" shows "primitive_part (normalize f) = normalize (primitive_part f)"
seq_R: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is R healthy"   and "Q is R healthy" shows "P ;; Q is R healthy"
select_convs_9: fixes igWls :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'c \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'h"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'c"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'h option) \<Rightarrow> ('e \<Rightarrow> 'c option) \<Rightarrow> 'h"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubst :: "'a \<Rightarrow> 'h \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubstAbs :: "'a \<Rightarrow> 'h \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'i" shows "igSwapAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igSwapAbs"
atm_of_atms_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and As :: "'b list"   and \<sigma> :: "'c"   and Bs :: "'b list" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "(subst_atm (atm_of_atms As) \<sigma> = atm_of_atms Bs) = (map (\<lambda>A. subst_atm A \<sigma>) As = Bs)"
assms_3: assumes "\<not> a" shows "Case_Labeling.CTXT (Suc outp') (ct' outp' ''else'') outp c"
simps_2: fixes uu :: "color"   and l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "entries (Branch uu l k v r) = entries l @ (k, v) # entries r"
hol_pal_lem2: fixes S :: "complex set"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and k :: "real"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "open S"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "d \<noteq> 0"   and "d \<bullet> a \<le> k"   and "d \<bullet> b \<le> k"   and "f holomorphic_on {z \<in> S. d \<bullet> z < k}"   and "f holomorphic_on {z \<in> S. k < d \<bullet> z}"   and "continuous_on S f" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
word_suicide_lang_l: fixes X :: "'a list set"   and Y :: "'a list set" assumes "\<not> l_ewp X"   and "Y \<noteq> {}" shows "\<exists>n. \<not> Y \<subseteq> X\<^bsup>Suc n\<^esup> \<cdot> Y"
interp_ins_maybe_grow2: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" assumes "insert_ops (xs @ [x])" shows "set (interp_ins (xs @ [x])) = set (interp_ins xs) \<or> set (interp_ins (xs @ [x])) = set (interp_ins xs) \<union> {fst x}"
Cons_1: fixes as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and lss :: "('a, bool) fmap list" assumes "ss = vars_change as vs s"   and "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "no_effectless_act as"   and "sat_precond_as s as" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> S vs lss PROB (fmrestrict_set vs s)"
inf_sup_aci_5: fixes x :: "'d"   and y :: "'d" shows "x \<sqinter> y = y \<sqinter> x"
ent_pure_pre_iff_sng: fixes b :: "bool"   and Q :: "assn" shows "(\<up> b \<Longrightarrow>\<^sub>A Q) = (b \<longrightarrow> (emp \<Longrightarrow>\<^sub>A Q))"
orthogonal_basis_exists: fixes V :: "(real, 'a) vec list" assumes "is_basis (set V)"   and "distinct V" shows "vec.independent (set (Gram_Schmidt V)) \<and> set V \<subseteq> vec.span (set (Gram_Schmidt V)) \<and> card (set (Gram_Schmidt V)) = vec.dim (set V) \<and> pairwise orthogonal (set (Gram_Schmidt V))"
det_def: fixes A :: "'a mat" shows "Determinant.det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'a))"
steps_smaller_card_Max_fun': fixes A :: "'a set"   and l :: "nat"   and f :: "'a \<Rightarrow> nat"   and up :: "nat"   and left :: "nat" assumes "finite A"   and "\<exists>x\<in>A. l \<le> f x" shows "steps A f l up left + l * up \<le> left * card A + up * Max (f ` A \<union> {0})"
mem_succ_iff: fixes j :: "hf"   and i :: "hf" assumes "Ord j" shows "(succ i \<^bold>\<in> succ j) = (i \<^bold>\<in> j)"
SFAssRedsVal: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sfs :: "sfields" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,(h\<^sub>2, l\<^sub>2, sh\<^sub>2(D \<mapsto> (sfs(F \<mapsto> v), Done))),False\<rangle>"
ord_isom_segment_mem: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a"   and x :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "x \<in> segment D a" shows "f x \<in> segment E (f a)"
sat_models_Abs: fixes \<A> :: "'a ltln set"   and \<phi> :: "'a ltln" shows "(\<A> \<in> sat_models (abs_ltln\<^sub>P \<phi>)) = \<A> \<Turnstile>\<^sub>P \<phi>"
le_rexp_trans: fixes r :: "'a rexp"   and s :: "'a rexp"   and t :: "'a rexp" assumes "local.le_rexp r s"   and "local.le_rexp s t" shows "local.le_rexp r t"
simps_3: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a"   and vb :: "'a"   and vc :: "'a list" shows "msort f (v # vb # vc) = PAC_Checker_Init.merge f (msort f (take (length (v # vb # vc) div 2) (v # vb # vc))) (msort f (drop (length (v # vb # vc) div 2) (v # vb # vc)))"
sorted_aux': shows "sorted_wrt (\<lambda>x y. fst y < fst x) aux'"
b_e_type_checker_weaken: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and ts :: "t list" assumes "b_e_type_checker \<C> es (t1s _> t2s)" shows "b_e_type_checker \<C> es (ts @ t1s _> ts @ t2s)"
BinOpRedsThrow2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(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 e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>"
interval_integral_Icc_approx_integrable: fixes a :: "ereal"   and b :: "ereal"   and l :: "nat \<Rightarrow> real"   and u :: "nat \<Rightarrow> real"   and f :: "real \<Rightarrow> 'a" assumes "a < b"   and "einterval a b = (\<Union>i. {l i..u i})"   and "incseq u"   and "decseq l"   and "\<And>i. l i < u i"   and "\<And>i. a < ereal (l i)"   and "\<And>i. ereal (u i) < b"   and "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"   and "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"   and "set_integrable lborel (einterval a b) f" shows "(\<lambda>i. interval_lebesgue_integral lborel (ereal (l i)) (ereal (u i)) f) \<longlonglongrightarrow> interval_lebesgue_integral lborel a b f"
If_transT_simp: fixes b :: "'test"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" shows "\<not> (if b then c1 else c2, s) \<rightarrow>t s'"
subst_eqvt: fixes pi :: "(tyvrs \<times> tyvrs) list"   and T :: "ty"   and X :: "tyvrs"   and T' :: "ty" shows "pi \<bullet> T[X \<mapsto> T']\<^sub>\<tau> = (pi \<bullet> T)[pi \<bullet> X \<mapsto> pi \<bullet> T']\<^sub>\<tau>"
s\<^sub>2': shows "s\<^sub>2' = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r'')]), r'' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
extend_inverse: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h" shows "project h UNIV (extend h F) = F"
mod_sub_if_z: fixes x :: "int"   and z :: "int"   and y :: "int" assumes "x < z"   and "y < z"   and "0 \<le> y"   and "0 \<le> x"   and "0 \<le> z" shows "(x - y) mod z = (if y \<le> x then x - y else x - y + z)"
Rep_data: fixes x :: "'b data" shows "Rep_data x \<in> data"
assoc: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" shows "gcp (gcp a b) c = gcp a (gcp b c)"
one_aux: fixes f :: "nat \<Rightarrow> 'a" assumes "completely_multiplicative_function f" shows "f (Suc 0) = (1::'a)"
set_Collect_arr_Par: fixes T :: "V"   and \<alpha> :: "V" shows "(T \<in>\<^sub>\<circ> ZFC_in_HOL.set (Collect (arr_Par \<alpha>))) = arr_Par \<alpha> T"
attractor_set_induction: fixes W :: "'a set"   and P :: "'a set \<Rightarrow> bool"   and p :: "Player" assumes "W \<subseteq> V"   and "\<And>S. \<lbrakk>S \<subseteq> V; P S\<rbrakk> \<Longrightarrow> P (W \<union> S \<union> directly_attracted p S)"   and "\<And>M. \<forall>S\<in>M. S \<subseteq> V \<and> P S \<Longrightarrow> P (\<Union> M)" shows "P (attractor p W)"
complex_Im_numeral: fixes v :: "num" shows "Im (numeral v) = 0"
exi_rename2: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y = x \<or> y \<notin> Fvars \<phi>" shows "exi x \<phi> = exi y (subst \<phi> (Var y) x)"
Sup_le_iff: fixes b :: "'b"   and A :: "'b set" shows "(b \<le> \<Sqinter> A) = (\<forall>a\<in>A. b \<le> a)"
r_cons_else_prim: shows "prim_recfn 2 r_cons_else"
gSubstPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSubstPresGWls MOD \<equiv> \<forall>ys Y' Y y s X' X. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gWls MOD s (gSubst MOD ys Y' Y y X' X)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : g \<star> w \<Rightarrow> g \<star> w'\<guillemotright>"   and "(r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w) = ((r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = g \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (f \<star> \<gamma>)"
HomFtor_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "Hom\<^bsub>C\<^esub>[X,\<midarrow>] \<equiv> MakeFtor (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"
P_lnull_ltl_deadend_v0: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0"   and "lnull (ltl P)" shows "\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v0 \<rightarrow>\<^bsub>G\<^esub> w)"
max_min_set_dual_var: fixes X :: "'a set" shows "\<partial> ` max_set X = min_set (\<partial> ` X)"
iso_iff_group_isomorphisms: fixes G :: "('d, 'e) monoid_scheme"   and f :: "'d \<Rightarrow> 'f"   and H :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "(f \<in> Group.iso G H) = (\<exists>g. group_isomorphisms G H f g)"
assms_4: shows "h \<turnstile> local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr () \<rightarrow>\<^sub>r owner_document"
known_ptrs_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "object_ptr_kinds h = object_ptr_kinds h'" shows "DocumentClass.known_ptrs h = DocumentClass.known_ptrs h'"
sort_set_unique_h: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" assumes "irreflp r \<and> transp r"   and "set (x # xs) = set (y # ys)"   and "\<forall>z\<in>set xs. r x z"   and "\<forall>z\<in>set ys. r y z" shows "x = y \<and> set xs = set ys"
OUTfromVCorrect1_data4: shows "OUTfromVCorrect1 data4"
HVZK_unfold2: assumes "\<Sigma>0.\<Sigma>_protocol" shows "\<forall>h e out. e \<in> carrier L \<longrightarrow> h \<in> valid_pub0 \<longrightarrow> out \<in> set_spmf (S0_raw h e) \<longrightarrow> check0 h (fst out) e (snd out)"
sup_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
invariant_def: fixes A :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and P :: "'e \<Rightarrow> bool" shows "invariant A P \<equiv> \<forall>s. reachable A s \<longrightarrow> P s"
bij_betw_primepows: shows "bij_betw (\<lambda>(p, k). p ^ Suc k) (Collect prime \<times> UNIV) (Collect primepow)"
WHILE_nofail_imp_rwof_nofail: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and s :: "'a" assumes "nofail (m0 \<bind> WHILE cond step)"   and "rwof m0 cond step s"   and "cond s" shows "nofail (step s)"
fmlookup_add: fixes m :: "('b, 'a) fmap"   and n :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup (m ++\<^sub>f n) x = (if x |\<in>| fmdom n then fmlookup n x else fmlookup m x)"
pid_system_automaton'_simps_1: shows "g_V pid_system_automaton' = UNIV"
analyzed_closed_mod_timpls_is_analyzed_closed_timpl_closure_set_aux2: fixes m :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term set"   and TI :: "('sets set \<times> 'sets set) set"   and t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "\<forall>s\<in>set (snd (Ana m)). timpl_closure_set M TI \<turnstile>\<^sub>c s"   and "m \<in> M"   and "t \<in> timpl_closure m TI"   and "s \<in> set (snd (Ana t))" shows "timpl_closure_set M TI \<turnstile>\<^sub>c s"