hcju/isabelleps huggingface dataset:
mult_chain_group: fixes p :: "nat"   and X :: "'b topology" shows "(\<otimes>\<^bsub>chain_group p X\<^esub>) = (+)"
det_Gauss_Jordan_det_P: fixes A :: "(('b, 'c) vec, 'c) vec" shows "fst (Gauss_Jordan_det_P A) * det A = det (snd (Gauss_Jordan_det_P A))"
match_1: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and h :: "'c"   and g :: "'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C (C h g)"   and "partial_magma.arr C (C (C h g) f)" shows "partial_magma.arr C (C g f)"
extractFrameResChain: fixes xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "extractFrame (\<lparr>\<nu>*xvec\<rparr>P) = \<lparr>\<nu>*xvec\<rparr>extractFrame P"
WhileE: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and ct :: "com \<times> (char list \<Rightarrow> int)" assumes "(WHILE b DO c, s) \<rightarrow> ct" obtains "ct = (IF b THEN c;; WHILE b DO c ELSE SKIP, s)"
HcompaT_le_compHaT: fixes H :: "'a"   and a :: "'a" assumes "equivalence H"   and "injective (a * top)" shows "- H * a * top \<le> - (H * a * top)"
pareto_losers_def: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "pareto_losers R = {x. \<exists>y. x \<prec>[Pareto R] y}"
thin1: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B" shows "insert A H \<turnstile> B"
lists_def2: fixes A :: "'b set" shows "lists A = {l. set l \<subseteq> A}"
minus_web_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "\<Gamma> \<ominus> f = \<Gamma>\<lparr>weight := \<lambda>x. if x \<in> A \<Gamma> then weight \<Gamma> x - d_OUT f x else weight \<Gamma> x + d_OUT f x - d_IN f x\<rparr>"
Repl: fixes b :: "ZF"   and A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "Elem b (Repl A f) = (\<exists>a. Elem a A \<and> b = f a)"
simps_1: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr> = ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some [], Set = ipurge_ref_aux I D U [] X\<rparr>"
Mprefix_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "Mprefix A P \<equiv> Abs_process ({(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}, {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))})"
assms: shows "\<exists>s1'. (if \<mu>2 s2' s2 then trsys1.silent_moves else trsys1.silent_movet) s1 s1' \<and> s1' \<approx> s2'"
natUnion_decompose: fixes p :: "'a"   and S :: "nat \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool" assumes "p \<in> natUnion S"   and "\<And>n p. p \<in> S n \<Longrightarrow> P p" shows "P p"
awalk_decomp_verts: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "wf_digraph G"   and "pre_digraph.cas G u p v"   and "pre_digraph.awalk_verts G u p = xs @ y # ys" obtains q :: "'d list" and r :: "'d list" where "pre_digraph.cas G u q y" and "pre_digraph.cas G y r v" and "p = q @ r" and "pre_digraph.awalk_verts G u q = xs @ [y]" and "pre_digraph.awalk_verts G y r = y # ys"
runit_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b"   and f :: "'b" assumes "right_hom_with_unit V H \<a> \<iota> a" shows "right_hom_with_unit.runit V H \<a> \<iota> a f \<equiv> THE \<mu>. partial_magma.in_hom (subcategory.comp V (weak_composition.right V H a)) \<mu> (weak_composition.H\<^sub>R H a f) f \<and> weak_composition.H\<^sub>R H a \<mu> = subcategory.comp V (weak_composition.right V H a) (H f \<iota>) (\<a> f a a)"
rbt_lookup_rbtreeify: fixes kvs :: "('a \<times> 'c) list" assumes "sorted (map fst kvs)"   and "distinct (map fst kvs)" shows "rbt_lookup (rbtreeify kvs) = map_of kvs"
seqp_choiceTE: fixes \<xi> :: "'e"   and p1 :: "('e, 'f, 'g, 'h) seqp"   and p2 :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, p1 \<oplus> p2), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "((\<xi>, p1), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   | "((\<xi>, p2), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
c2: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
assms_1: shows "\<forall>t. (M \<turnstile> t) = (M' \<turnstile>\<^sub>c t)"
pick_in_set_le: fixes n :: "nat"   and S :: "nat set" assumes "n < card S" shows "pick S n \<in> S"
inject: fixes x :: "fol fm"   and ya :: "fol fm" shows "(Thm x = Thm ya) = (x = ya)"
simps_1: fixes x21 :: "'c"   and x22 :: "'c"   and y21 :: "'c"   and y22 :: "'c" shows "(Point x21 x22 = Point y21 y22) = (x21 = y21 \<and> x22 = y22)"
ide_leg0: shows "ide g\<^sub>0h\<^sub>1.p\<^sub>0"
UNIV_cenum: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "cEnum = Some (enum, enum_all, enum_ex)" shows "UNIV = set enum"
a: shows "\<forall>Q R. ((\<lambda>s. Q s \<and> R s) \<and>* P) = (\<lambda>s. (Q \<and>* P) s \<and> (R \<and>* P) s)"
simps_1: fixes t' :: "transition"   and t :: "transition"   and r :: "nat" shows "stored_reused_aux t' t r 0 = stored_reused_aux_per_reg t' t r 0"
chamber_in_opp_fold_A_im: fixes F :: "'a set" assumes "chamber F"   and "F \<in> opp_fold_A \<turnstile> A" shows "F \<in> opp_fold_A \<turnstile> ChamberComplex.chamber_system A"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> TargetTerm TQ"
fv_subst_subset: fixes x :: "'d"   and t :: "('e, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('e, 'f) Term.term" assumes "x \<in> fv t" shows "fv (\<theta> x) \<subseteq> fv (t \<cdot> \<theta>)"
Top_fun_def: shows "\<top> = (\<lambda>x. \<top>)"
step_3_c_dims_2: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and i :: "(nat \<times> nat) list"   and A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.step_3_c x l k i A) = dim_col A"
inj_on_id: fixes A :: "'b set" shows "inj_on id A"
nontriv_def: shows "class.nontriv TYPE('b) \<equiv> 1 < CARD('b)"
polyfun_diff: fixes n :: "nat"   and a :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "1 \<le> n" shows "(\<Sum>i\<le>n. a i * x ^ i) - (\<Sum>i\<le>n. a i * y ^ i) = (x - y) * (\<Sum>j<n. (\<Sum>i = Suc j..n. a i * y ^ (i - j - 1)) * x ^ j)"
l0_f1f1': shows "(f1' * c + 1) / ((f1' - f1) * c) \<le> real l0"
maj: shows "IsMajority {d \<in> disksWritten s p. \<forall>q\<in>UNIV - {p}. hasRead s p d q}"
in_cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> o[F](h)) = (g \<in> o[F](h))"
elims_2: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "modality"   and xa :: "safety"   and xb :: "'b regex" assumes "Regex.safe_regex fv safe x xa xb" obtains m :: "modality" and uu_ :: "safety" and n :: "nat" where "x = m__" and "xa = uu_" and "xb = Skip n"   | m :: "modality" and g :: "safety" and \<phi> :: "'b" where "x = ma__" and "xa = g__" and "xb = Test \<phi>" and "safe g__ \<phi>"   | m :: "modality" and g :: "safety" and r :: "'b regex" and s :: "'b regex" where "x = mb__" and "xa = ga__" and "xb = Plus rb__ s__" and "(ga__ = Lax \<or> fv_regex fv rb__ = fv_regex fv s__) \<and> Regex.safe_regex fv safe mb__ ga__ rb__ \<and> Regex.safe_regex fv safe mb__ ga__ s__"   | g :: "safety" and r :: "'b regex" and s :: "'b regex" where "x = Futu" and "xa = gb__" and "xb = Times rc__ sa__" and "(gb__ = Lax \<or> fv_regex fv rc__ \<subseteq> fv_regex fv sa__) \<and> Regex.safe_regex fv safe Futu gb__ sa__ \<and> Regex.safe_regex fv safe Futu Lax rc__"   | g :: "safety" and r :: "'b regex" and s :: "'b regex" where "x = Past" and "xa = gc__" and "xb = Times rd__ s" and "(gc__ = Lax \<or> fv_regex fv s \<subseteq> fv_regex fv rd__) \<and> Regex.safe_regex fv safe Past gc__ rd__ \<and> Regex.safe_regex fv safe Past Lax s"   | m :: "modality" and g :: "safety" and r :: "'b regex" where "x = m" and "xa = g" and "xb = Star r" and "(g = Lax \<or> fv_regex fv r = {}) \<and> Regex.safe_regex fv safe m g r"
wf_subset: fixes r :: "('k \<times> 'k) set"   and p :: "('k \<times> 'k) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto> a<x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"
simps_1: fixes uu :: "('c, 'd) tm" shows "position_of uu [] = True"
rep_eq: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b fset" shows "fset (ffilter x xa) = Set.filter x (fset xa)"
fermat_theorem_mod_ring2: fixes a :: "'b mod_ring"   and n :: "nat" shows "a ^ CARD('b) ^ n = a"
cycle_free_diag_equiv: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs) = cycle_free M n"
rcons_def: fixes t :: "('c, 'd) FingerTree"   and a :: "'c \<times> 'd" shows "rcons t a \<equiv> Abs_FingerTree (FingerTreeStruc.rcons (Rep_FingerTree t) a)"
logic_aux1: fixes i :: "'a"   and x :: "'b"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and j :: "'c" shows "(\<exists>k. (i = k \<longrightarrow> x = f i j) \<and> (i \<noteq> k \<longrightarrow> x = (0::'b))) = ((\<exists>k. i = k \<and> x = f i j) \<or> (\<exists>k. i \<noteq> k \<and> x = (0::'b)))"
\<B>_4: fixes S :: "'a set" assumes "open S"   and "x \<in> S" shows "\<exists>B\<in>\<B>. B \<subseteq> S"
intros_1: fixes A :: "'b \<Rightarrow> bool" shows "alllstsp A LNil"
C_eq_id: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy1_strong p" shows "Cp (list2FWpolicy (insertDeny p)) = Cp (list2FWpolicy p)"
jkbpT_tLength_inv: fixes t :: "'s Trace"   and T :: "'s Trace set"   and n :: "nat" assumes "t \<in> jkbpT T"   and "tLength t = n" shows "t \<in> jkbpT\<^bsub>n\<^esub> T"
additive_increasing: fixes M :: "'b measure"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive (sets M) f"   and "Measure_Space.additive (sets M) f" shows "increasing (sets M) f"
add_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(b = plus b a) = (a = zero)"
Cassoc: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e"   and h :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g"   and "g \<approx>>\<^bsub>C\<^esub> h" shows "f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)"
discrim_def: shows "discrim \<equiv> (rep_msg ---> id) freediscrim"
sig_of_spair_commute: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_of_pair (Inl (p, q)) = sig_of_pair (Inl (q, p))"
supp_6: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "supp ([x3\<noteq>x2]x1) = supp x3 \<union> (supp x2 \<union> supp x1)"
eventually_at_top_dense: fixes P :: "'c \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n>N. P n)"
awalk_verts_append: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" assumes "awalk u (p @ q) v" shows "awalk_verts u (p @ q) = awalk_verts u p @ tl (awalk_verts (awlast u p) q)"
transitive: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and p :: "'b"   and q :: "'b"   and r :: "'b"   and s :: "'b"   and t :: "'b" assumes "ccw_system1235 ccw S"   and "distinct5 p q r s t"   and "in5 S p q r s t"   and "ccw t s p"   and "ccw t s q"   and "ccw t s r"   and "ccw t p q"   and "ccw t q r" shows "ccw t p r"
\<b>_zero_gr: fixes X :: "'x set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "finite X"   and "(h, U) \<in> set ps" shows "poly_deg h < \<b> ps 0"
ICons_3: assumes "s \<noteq> full_sort"   and "sort_ex cs s" shows "normalize_sort cs s \<noteq> full_sort"
positive_minus: fixes x :: "rat" assumes "\<not> positive x"   and "x \<noteq> 0" shows "positive (- x)"
alpha_1: fixes a :: "tyvrs"   and x :: "'a"   and b :: "tyvrs"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
prod_emb_PiE: fixes J :: "'b set"   and I :: "'b set"   and E :: "'b \<Rightarrow> 'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
simps_1: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre SKIP Q = Q"
A_def: fixes n :: "nat" shows "A n = fst ((bisect ^^ n) (a, b))"
mult_inj_eq: fixes x :: "nat"   and q :: "nat"   and y :: "nat"   and y' :: "nat" assumes "coprime x q"   and "y < q"   and "y' < q"   and "x * y mod q = x * y' mod q" shows "y = y'"
PO_m1_refines_a0ii_ri: shows "refines (R_a0iim1_ri \<inter> a0i_inv1_iagree \<times> (m1_inv1r_cache \<inter> m1_inv0_fin)) med_a0iim1_ri a0i m1"
expand: fixes envT :: "('d, 'e) envT"   and envT' :: "('d, 'e) envT" assumes "run_env envT = run_env envT'" shows "envT = envT'"
mset_subset_eq_single: fixes a :: "'b"   and B :: "'b multiset" assumes "a \<in># B" shows "\<LM> a  \<RM> \<subseteq># B"
sign_r_pos_pderiv: fixes p :: "real poly"   and x :: "real" assumes "poly p x = 0"   and "p \<noteq> 0" shows "sign_r_pos (pderiv p * p) x"
lQuots_append: fixes v :: "'b list"   and w :: "'b list"   and A :: "'b list set" shows "(v \<in> lQuots w A) = (w @ v \<in> A)"
wf_mbufn_map_tl: fixes i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and buf :: "event_data option list set list list" assumes "wf_mbufn i js Ps buf"   and "[] \<notin> set buf" shows "wf_mbufn (Suc i) js Ps (map tl buf)"
resolve_def: fixes clause1 :: "Literal list"   and clause2 :: "Literal list"   and literal :: "Literal" shows "resolve clause1 clause2 literal \<equiv> removeAll literal clause1 @ removeAll (opposite literal) clause2"
valid_path_uminus_comp: fixes g :: "real \<Rightarrow> 'a" shows "valid_path (uminus \<circ> g) = valid_path g"
arr_iff_in_hom: fixes f :: "'a \<times> 'a" shows "VoV.arr f = VoV.in_hom f (VoV.dom f) (VoV.cod f)"
HSem_below: fixes h :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c"   and r :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<notin> domA h \<Longrightarrow> \<rho> x \<sqsubseteq> r x"   and "\<And>x. x \<in> domA h \<Longrightarrow> \<lbrakk> the (map_of h x) \<rbrakk>\<^bsub>r\<^esub> \<sqsubseteq> r x" shows "\<lbrace> h \<rbrace>\<rho> \<sqsubseteq> r"
homeomorphism_symD: fixes S :: "'a set"   and t :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphism S t f g" shows "homeomorphism t S g f"
iso_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "C.iso \<a>\<^sub>C[a, b, c]"
assms: shows "length (trace_parallel_plan_strips I \<pi>) < length \<pi> + 1"
entails_trans: fixes N1 :: "'f set"   and N2 :: "'f set"   and N3 :: "'f set" assumes "N1 \<Turnstile>Q N2"   and "N2 \<Turnstile>Q N3" shows "N1 \<Turnstile>Q N3"
is_sup_uniq: fixes x :: "'c"   and y :: "'c"   and sup :: "'c"   and sup' :: "'c" assumes "is_sup x y sup"   and "is_sup x y sup'" shows "sup = sup'"
bigtheta_refl: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<in> \<Theta>[F](f)"
PR_CONST_def: fixes x :: "'d" shows "PR_CONST x \<equiv> x"
nonneg_coeffs_aux: fixes n :: "nat" assumes "0 < n" shows "fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0"
P_def: shows "P = Pascal_Property.inter (line A B) (line D E)"
bounds_1: shows "\<B>\<^sub>l (l ! (j - 1)) = \<B>\<^sub>l (l ! i)"
weak_upper_semilattice_axioms: shows "weak_upper_semilattice (fpl \<P> \<H>)"
swap_inverse: shows "swap\<^sub>C \<odot> swap\<^sub>C = 1\<^sub>C"
that: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "homeomorphism (cbox a b) (cbox a b) f g"   and "f a = a"   and "f b = b"   and "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v" shows "thesis"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Cod \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = Cod t \<^bold>\<star> Cod u \<^bold>\<star> Cod v"
semiring_normalization_rules_25: fixes a :: "'a"   and c :: "'a"   and d :: "'a" shows "a + (c + d) = a + c + d"
Units_bijective: shows "left.transformation.Sym = {\<alpha> \<in> G \<rightarrow>\<^sub>E G. bij_betw \<alpha> G G}"
elims_3: fixes x :: "'a list" assumes "\<not> decr x" obtains xa :: "'a" and y :: "'a" and xs :: "'a list" where "x = xa # y # xs" and "\<not> (y \<le> xa \<and> decr (y # xs))"
funpow_dist1_prop: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "(f ^^ funpow_dist1 f x y) x = y"
emb_step_at_chop: fixes p :: "dir list"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "emb_step_at p dir.Right t = s"   and "position_of t (p @ [dir.Right])"   and "list_all (\<lambda>x. x = dir.Left) p" shows "chop t = s \<or> chop t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s"
count_rev: fixes xs :: "'a list"   and x :: "'a" shows "count_list (rev xs) x = count_list xs x"
F: fixes x :: "'b" assumes "x \<in> T" shows "openin (top_of_set T') (F x) \<and> x \<in> F x \<and> S \<times> F x \<subseteq> U"
set_8_atLeast0: shows "{0..<8} = {0, 1, 2, 3, 4, 5, 6, 7}"
wf_mdecl_bodyD: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "methd" assumes "wf_mdecl G C (sig, m)" shows "\<exists>T. \<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile>Body C (stmt (mbody m))\<Colon>-T \<and> G\<turnstile>T\<preceq>resTy m"
map_comp: fixes f :: "'s \<Rightarrow> 'r"   and g :: "'t \<Rightarrow> 's"   and p :: "'t spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
section_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.section C f \<equiv> \<exists>g. partial_magma.ide C (C g f)"
Gmult_closed: fixes g :: "'g"   and v :: "'v" assumes "g \<in> G"   and "v \<in> V" shows "g *\<cdot> v \<in> V"
directed_prop: fixes X :: "'a set" assumes "X \<noteq> {}"   and "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y" shows "ord.directed (\<lambda>x y. y \<le> x) X"
vars_iff_subterm_or_eq: fixes x :: "'a"   and t :: "('b, 'a) Term.term" shows "(x \<in> fv t) = (Var x \<sqsubset> t \<or> Var x = t)"
sums_vec_nth: fixes f :: "nat \<Rightarrow> ('c, 'd) vec"   and a :: "('c, 'd) vec"   and i :: "'d" assumes "f sums a" shows "(\<lambda>x. f x $ i) sums a $ i"
iterate_nth: fixes k :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "k < n - m" shows "iterate m n f x ! k = (f ^^ (m + k)) x"
Restrp_Restrp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" shows "r \<restriction> A \<restriction> B \<equiv> r \<restriction> A \<inter> B"
id: shows "[:f, e, 0, 1:] = [:- x1, 1:] * [:- x2, 1:] * [:- x3, 1:]"
permutes_in_image: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "p permutes S" shows "(p x \<in> S) = (x \<in> S)"
prm_input_def: fixes pi :: "('d \<times> 'd) list"   and x :: "('a, 'b, 'c) input" shows "pi \<bullet> x \<equiv> Abs_input (pi \<bullet> Rep_input x)"
simps_6: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (Le (CN 0 c e)) = (0 < c \<and> numbound0 e)"
existence_ivl_initial_time_iff: fixes x0 :: "'a" shows "(0 \<in> existence_ivl0 x0) = (0 \<in> UNIV \<and> x0 \<in> X)"
inject_1: fixes x1 :: "'e"   and y1 :: "'e" shows "(Normal x1 = Normal y1) = (x1 = y1)"
inject_1: fixes x1 :: "Preliminaries.var"   and y1 :: "Preliminaries.var" shows "(Var x1 = Var y1) = (x1 = y1)"
can_select_def: fixes P :: "'g \<Rightarrow> bool"   and A :: "'g set" shows "can_select P A = (\<exists>!x. x \<in> A \<and> P x)"
vsubsetD: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<le> b"   and "c \<in> elts a" shows "c \<in> elts b"
hyps_1: shows "(ftv T1 - ftv \<Gamma>) \<sharp>* \<theta>"
estep_Pr3: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Pr n f g, xs, ls) # fs, rv)"   and "ls \<noteq> []"   and "length ls = Suc (hd xs)"   and "recfn (length xs) (Pr n f g)" shows "estep_Pr (encode_config c) = encode_config (step c)"
rep_meridian_reflect_def: shows "rep_meridian_reflect \<equiv> vector [vector [- 1, 0, 0], vector [0, 1, 0], vector [0, 0, 1]]"
distinct_288: fixes x151 :: "floatarith"   and x152 :: "nat"   and x14 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Ln x14"
DC_def: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "a \<Sqinter> b \<equiv> \<lambda>ab Q s. min (a ab Q s) (b ab Q s)"
subst_mgt_comp': fixes \<sigma> :: "'d \<Rightarrow> ('b, 'c) Term.term"   and \<theta> :: "'d \<Rightarrow> ('b, 'e) Term.term"   and \<delta> :: "'e \<Rightarrow> ('b, 'a) Term.term" assumes "\<theta> \<circ>\<^sub>s \<delta> \<preceq>\<^sub>\<circ> \<sigma>" shows "\<theta> \<preceq>\<^sub>\<circ> \<sigma>"
\<Psi>_sel_4: shows "sink \<Psi> = vertex.SINK"
bij_imp_permutes': fixes p :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "bij p"   and "\<forall>x. x \<notin> A \<longrightarrow> p x = x" shows "p permutes A"
noFaultI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and f :: "'e" assumes "\<And>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<Longrightarrow> t \<noteq> Fault f" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Fault f}"
pm_of_idx_pm_minus: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd"   and g :: "nat \<Rightarrow>\<^sub>0 'd" shows "pm_of_idx_pm xs (f - g) = pm_of_idx_pm xs f - pm_of_idx_pm xs g"
covector_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "top \<sqinter> x = x" shows "top \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
c_drop_at_Suc: fixes y :: "nat"   and x :: "nat" shows "c_drop (Suc y) x = c_tl (c_drop y x)"
coherent_Assoc'_Ide: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a"   and "Ide b"   and "Ide c" shows "coherent \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[a, b, c\<^bold>]"
return_def: fixes a :: "'d" shows "return a = State (Pair a)"
simps_12: fixes f :: "'b \<Rightarrow> 'a" shows "lmap f LNil = LNil"
ide_char': fixes a :: "(bool, 'arr list) concrete_category.arr" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
closure_of_eq_diff: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S = topspace X - \<Union> {T. openin X T \<and> disjnt S T}"