hcju/isabelleps huggingface dataset:
intros_36: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and es :: "exp0 list" assumes "evaluate ck env s e (s', Rerr err)" shows "evaluate_list ck env s (e # es) (s', Rerr err)"
face_cycle_succ_def: shows "face_cycle_succ \<equiv> edge_succ M \<circ> edge_rev M"
cmp_def: fixes \<mu>\<nu> :: "'b \<times> 'b" shows "GoH.cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then G (H (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) \<cdot>\<^sub>D G (\<Phi>\<^sub>H (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>G (H (B.dom (fst \<mu>\<nu>)), H (B.dom (snd \<mu>\<nu>))) else D.null)"
ssubst_SubstFormP: fixes w :: "name"   and V :: "name set"   and V' :: "name set"   and A :: "fm" assumes "w \<in> Vs - V"   and "V \<subseteq> Vs"   and "V' = p \<bullet> V"   and "supp A \<subseteq> atom ` Vs" shows "insert (ConstP (F w)) {ConstP (F i) |i. i \<in> V} \<turnstile> SubstFormP \<guillemotleft>Var w\<guillemotright> (F w) (ssubst \<lfloor>A\<rfloor>V V F) (ssubst \<lfloor>A\<rfloor>(insert w V) (insert w V) F)"
get_global: fixes s :: "s"   and vs :: "v list"   and j :: "nat"   and i :: "nat" shows "\<lparr>s;vs;[$Get_global j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C sglob_val s i j]\<rparr>"
bex_Un: fixes A :: "'c set"   and B :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(\<exists>x\<in>A \<union> B. P x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x))"
mult: fixes A :: "'c mat"   and B :: "'c mat"   and nc :: "nat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "A * B \<in> carrier_mat n nc"
zlfm_I: fixes p :: "fm"   and i :: "int"   and bs :: "real list" assumes "qfree p" shows "Ifm (real_of_int i # bs) (zlfm p) = Ifm (real_of_int i # bs) p \<and> iszlfm (zlfm p) (real_of_int i # bs)"
set_intros_3: fixes x1 :: "'a" shows "x1 \<in> fv (Var x1)"
excess_non_negative: shows "\<forall>v\<in>V - {s, t}. (0::'a) \<le> excess f' v"
inf_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
chain_map_eq: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "singular_chain p X c"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "chain_map p f c = chain_map p g c"
A: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "le y x" shows "f x = f y"
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto><Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>S T x. (S, T) \<in> Rel' \<Longrightarrow> (<\<nu>x>S, <\<nu>x>T) \<in> Rel'" shows "P \<parallel> R \<leadsto><Rel'> Q \<parallel> R"
AE_return: fixes x :: "'b"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> space M"   and "Measurable.pred M P" shows "almost_everywhere (return M x) P = P x"
i_le_j_events_neq: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "i < j"   and "j < card X" shows "f i \<noteq> f j"
square_free_int_imp_square_free_mod_ring: fixes f :: "int poly" assumes "square_free f"   and "separable_bound f < int CARD('a)" shows "square_free (of_int_poly f)"
arr_dom: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.arr (V.dom f)"
nsum_mem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier M" shows "\<Sigma>\<^sub>e M f n \<in> carrier M"
permutation_of_list_id: fixes x :: "'b"   and xs :: "('b \<times> 'b) list" assumes "x \<notin> set (map fst xs)" shows "permutation_of_list xs x = x"
opposite_ocircline_def: shows "opposite_ocircline \<equiv> map_fun rep_ocircline abs_ocircline opposite_ocircline_clmat"
inject_1: fixes x11 :: "linear_poly"   and x12 :: "'a"   and y11 :: "linear_poly"   and y12 :: "'a" shows "(LEQ_ns x11 x12 = LEQ_ns y11 y12) = (x11 = y11 \<and> x12 = y12)"
awlast_in_verts: fixes u :: "'a"   and p :: "'b list" assumes "u \<in> verts G"   and "set p \<subseteq> arcs G" shows "awlast u p \<in> verts G"
rev_bl_order_rev_simp: fixes xs :: "bool list"   and ys :: "bool list"   and F :: "bool"   and x :: "bool"   and y :: "bool" assumes "length xs = length ys" shows "rev_bl_order F (xs @ [x]) (ys @ [y]) = (y \<and> \<not> x \<or> (y \<or> \<not> x) \<and> rev_bl_order F xs ys)"
ord_le_eq_trans: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
2: shows "(f j + g j) ^ m j = f j ^ m j + d * g j"
coeff_in_ring2: fixes x :: "'e"   and A :: "'e set"   and a :: "'e \<Rightarrow> 'a" assumes "x \<in> A"   and "a \<in> A \<rightarrow> carrier K" shows "a x \<in> carrier K"
no_roots_inbetween_imp_same_sign: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> poly p x \<noteq> 0" shows "sgn (poly p a) = sgn (poly p b)"
keys_rep_nat_pp_pp: fixes t :: "('b, 'c) pp" shows "keys_pp (rep_nat_pp t) = rep_nat ` keys_pp t"
psubst_cnj: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "psubst (cnj \<phi>1 \<phi>2) txs = cnj (psubst \<phi>1 txs) (psubst \<phi>2 txs)"
mu_fusion_equal_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_least_prefixpoint less_eq g"   and "order.has_least_fixpoint less_eq h"   and "less_eq (l (g (u (order.the_least_fixpoint less_eq h)))) (h (l (u (order.the_least_fixpoint less_eq h))))"   and "l (g (order.the_least_prefixpoint less_eq g)) = h (l (order.the_least_prefixpoint less_eq g))" shows "order.the_least_fixpoint less_eq h = l (order.the_least_prefixpoint less_eq g) \<and> order.the_least_fixpoint less_eq h = l (order.the_least_fixpoint less_eq g)"
free_vars_cexpr_comp_aux: fixes x :: "nat"   and e :: "cexpr"   and e' :: "cexpr" shows "free_vars (cexpr_comp_aux x e e') \<subseteq> free_vars e' - {x} \<union> free_vars e"
indaut_morph: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexEndomorphism X (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
degree_monom_eq: fixes a :: "'b"   and n :: "nat" assumes "a \<noteq> (0::'b)" shows "degree (monom a n) = n"
induct: fixes P :: "('a \<times> 'b) set \<Rightarrow> bool"   and a0 :: "('a \<times> 'b) set" assumes "\<And>KP. \<lbrakk>\<And>x xa xb. \<lbrakk>\<not> (infinite KP \<or> KP = {}); x = arg_min_on snd KP; xa = {p \<in> KP. fst p < fst x}; xb = {p \<in> KP. fst x < fst p}\<rbrakk> \<Longrightarrow> P xa; \<And>x xa xb. \<lbrakk>\<not> (infinite KP \<or> KP = {}); x = arg_min_on snd KP; xa = {p \<in> KP. fst p < fst x}; xb = {p \<in> KP. fst x < fst p}\<rbrakk> \<Longrightarrow> P xb\<rbrakk> \<Longrightarrow> P KP" shows "P a0"
moebius_translation_plus: fixes v1 :: "complex"   and v2 :: "complex" shows "moebius_translation v1 + moebius_translation v2 = moebius_translation (v1 + v2)"
prod_one_func: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "prod_one I A \<in> carr_prodag I A"
assms_2: assumes "\<not> colossless_gpv \<I> gpv" shows "plossless_gpv \<I> gpv'"
a2c_simp: fixes a :: "Arity\<^sub>\<bottom>" shows "a2c\<cdot>a = (if a \<sqsubseteq> \<bottom> then none else many)"
execn_to_execn_strip_guards_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and f :: "'f"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t = Fault f"   and "f \<notin> F" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> =n\<Rightarrow> Fault f"
that: fixes k :: "nat"   and r :: "'b list"   and p :: "'b" assumes "s = [Suc k..<Suc k + length r] || r"   and "v = (k, p)" shows "thesis"
conseq_2: fixes s :: "char list \<Rightarrow> int" shows "P s \<le> P' s"
finite_multisets: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "finite {M. set_mset M \<subseteq> A \<and> size M = k}"
iTILL_inter: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "[\<dots>n] \<inter> [\<dots>n'] = [\<dots>n]"
simple_packet_unext_def: fixes p :: "('b, 'c) simple_packet_scheme" shows "simple_packet_unext p \<equiv> \<lparr>p_iiface = p_iiface p, p_oiface = p_oiface p, p_src = p_src p, p_dst = p_dst p, p_proto = p_proto p, p_sport = p_sport p, p_dport = p_dport p, p_tcp_flags = p_tcp_flags p, p_payload = p_payload p\<rparr>"
distr_density_distr: fixes T :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and T' :: "'c \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ennreal" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "T' \<in> M' \<rightarrow>\<^sub>M M"   and "\<forall>x\<in>space M. T' (T x) = x"   and "f \<in> borel_measurable M'" shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)"
set_sel_1: fixes a :: "'b hp" shows "un_Hp1 a \<in> set_hp a"
distinct_quicksort_acc: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ac :: "'b list"   and xs :: "'b list" shows "distinct (ord.quicksort_acc less ac xs) = distinct (ac @ xs)"
dels_unsatisfied: fixes P :: "ast_problem"   and f :: "object atom formula"   and d :: "object atom formula list"   and a :: "object atom formula list"   and M :: "object atom formula set" assumes "wf_ast_problem P"   and "f \<in> set d"   and "f \<notin> set a" shows "f \<notin> apply_effect (Effect a d) M"
assms_3: shows "a \<in> maximalStrictAllocations N (set \<Omega>) b"
minus_other_side: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier K"   and "b \<in> carrier K" shows "(a \<ominus>\<^bsub>K\<^esub> b = \<zero>\<^bsub>K\<^esub>) = (a = b)"
binomial_altdef_nat: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "n choose k = fact n div (fact k * fact (n - k))"
SdlHPPB_restricted_on_PP_is_SdlHPPB: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and DA :: "'b \<Rightarrow> 'e"   and lH :: "('e \<times> 'a \<times> nat) set"   and d :: "'e"   and PP :: "nat set"   and R' :: "(('a, 'b) MWLsCom list \<times> ('a, 'b) MWLsCom list) set"   and V :: "('a, 'b) MWLsCom list"   and R :: "(('a, 'b) MWLsCom list \<times> ('a, 'b) MWLsCom list) set" assumes "WHATWHERE.Strong_dlHPP_Bisimulation (MWLs_semantics.MWLsSteps_det E BMap) E pp DA lH d PP R'"   and "(V, V) \<in> R'"   and "R = {(V', V''). (V', V'') \<in> R' \<and> set (PPV V') \<subseteq> set (PPV V) \<and> set (PPV V'') \<subseteq> set (PPV V)}" shows "WHATWHERE.Strong_dlHPP_Bisimulation (MWLs_semantics.MWLsSteps_det E BMap) E pp DA lH d PP R"
csa2: shows "set (characterize_consistent_signs_at_roots p qs2) \<subseteq> set signs2"
sgn_mult: fixes x :: "'b"   and y :: "'b" shows "sgn (x * y) = sgn x * sgn y"
known_ptr_new_shadow_root_ptr: fixes ptr :: "(_) object_ptr" assumes "ShadowRootClass.known_ptr ptr"   and "\<not> DocumentClass.known_ptr ptr" shows "is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"
ag_eq_diffzero: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (a \<plusminus> -\<^sub>a b = \<zero>)"
in_arcs: shows "in_arcs (with_proj G) w = {(u, w), (v, w)}"
simps: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and stack :: "'a \<Rightarrow> 'b list"   and a1 :: "'a"   and a2 :: "'a list"   and a3 :: "'a" assumes "balance_trace step stack" shows "balance_trace.bal step stack a1 a2 a3 = (\<exists>c T c'. a1 = c \<and> a2 = T \<and> a3 = c' \<and> traces.trace step c T c' \<and> (\<forall>c'\<in>set T. stack c \<lesssim> stack c') \<and> stack c' = stack c)"
HHH: shows "analz (AgentSet \<union> G \<union> (AgentSet \<union> H)) \<subseteq> synth (analz (G \<union> H))"
ThinChamberComplexFolding_axioms: shows "ThinChamberComplexFolding X f"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs = xs \<down> i @ xs ! i # xs \<up> Suc i"
PiE_UNIV_domain: fixes T :: "'a \<Rightarrow> 'b set" shows "Pi\<^sub>E UNIV T = Pi UNIV T"
char_eq_0_iff: fixes a :: "'a" shows "(principal_char (G\<lparr>carrier := H'\<rparr>) a = 0) = (a \<notin> carrier (G\<lparr>carrier := H'\<rparr>))"
IH: fixes xa :: "nat" assumes "xa \<in> {..<k}"   and "x\<^sub>0 \<le> (ts ! xa) x" shows "f ((ts ! xa) x) \<le> gc1 * f' ((ts ! xa) x)"
lt__nle: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "\<not> C D Le A B"
subintervalI: fixes t0 :: "real"   and t :: "real" assumes "t0 \<in> T"   and "t \<in> T" shows "{t0--t} \<subseteq> T"
M_def: shows "M = Max ({\<bar>fs_init ! i $ j\<bar> |i j. i < m \<and> j < n} \<union> {0})"
l_unit_add_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "\<zero>\<^bsub>rec_rng_of_frac\<^esub> \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = (r |\<^bsub>rel\<^esub> s)"
hyps: assumes "distinct (map fst xs)" shows "lookup_dflt (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_dflt xs k))"
a4: shows "\<forall>x\<in>set Xi. case x of (x, s) \<Rightarrow> invara x \<and> invar s \<and> \<alpha>' x \<subseteq> a (\<alpha> s)"
not_updated_aux: fixes t2 :: "(nat \<times> vname aexp) list" assumes "filter (\<lambda>(r', uu_). r' = r) t2 = []" shows "apply_updates t2 s s' $ r = s' $ r"
mult_of_nat_commute: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "nat"   and y :: "'c" assumes "class.semiring_1 one times plus zero" shows "times (semiring_1.of_nat one plus zero x) y = times y (semiring_1.of_nat one plus zero x)"
l2_inv1D: fixes x :: "l2_state"   and A :: "agent"   and B :: "agent" assumes "can_signal x A B"   and "x \<in> l2_inv1" shows "A \<notin> bad x \<and> B \<notin> bad x"
assms: shows "run_one_step d i (s, vs, ves, $Unreachable) = (s', vs', res)"
lincomb_cong: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "A = B"   and "A \<subseteq> carrier V"   and "\<And>v. v \<in> A \<Longrightarrow> a v = b v"   and "b \<in> B \<rightarrow> carrier K" shows "lincomb a A = lincomb b B"
o2: shows "t2 = s \<or> (s, t2) \<in> trm_ord"
oproj_in: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and a' :: "'c" assumes "oproj r s f"   and "(a, a') \<in> r" shows "(f a, f a') \<in> s"
rep_eq: fixes x :: "'a program"   and xa :: "'a program" shows "(x \<le> xa) = rep_program x \<sqsubseteq> rep_program xa"
IH: shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P \<parallel> Q''"
UNIV_nat_eq: shows "UNIV = insert 0 (range Suc)"
SUPR_progression: fixes X :: "'c \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> bool"   and Y :: "'d \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> bool" assumes "\<forall>n. \<exists>m. X n \<rightarrow> Y m" shows "Sup (range X) \<rightarrow> Sup (range Y)"
le_fininf_empty: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a stream" assumes "traces ind" shows "traces.le_fininf ind [] w"
item_end_finished_item: fixes \<alpha> :: "symbol list" shows "item_end (finished_item \<alpha>) = length Doc"
min_PInf2: fixes x :: "ereal" shows "min x \<infinity> = x"
spec_ops_id_inc: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and deps :: "'d \<Rightarrow> 'c set"   and x :: "'c" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "x \<in> set (map fst xs)" shows "x < oid"
assertion_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.assertion one times inf less_eq dual top = {x. less_eq x one \<and> inf (times x top) (dual x) = x}"
convergent_series_Cauchy: fixes a :: "nat \<Rightarrow> real"   and \<phi> :: "nat \<Rightarrow> 'c" assumes "\<exists>M. \<forall>n. sum a {0..n} \<le> M"   and "\<And>n. dist (\<phi> (Suc n)) (\<phi> n) \<le> a n" shows "Cauchy \<phi>"
simps_3: fixes \<sigma> :: "nat \<Rightarrow> val"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_sem \<sigma> <e1, e2>\<^sub>c = <|cexpr_sem \<sigma> e1, cexpr_sem \<sigma> e2|>"
ordering_mem: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "OrdinalVeblen.ordering A x \<in> A"
composition_invertible: fixes x :: "'a"   and y :: "'a" assumes "invertible x"   and "invertible y"   and "x \<in> M"   and "y \<in> M" shows "invertible (x \<cdot> y)"
face_distance_def: fixes X :: "'b set set"   and F :: "'b set"   and D :: "'b set" assumes "ChamberComplex X" shows "ChamberComplex.face_distance X F D \<equiv> ChamberComplex.chamber_distance X (ChamberComplex.closest_supchamber X F D) D"
Abs_inverse: fixes y :: "atom" assumes "y \<in> {a. sort_of a \<in> range sort_fun}" shows "Rep (Abs y) = y"
flush_all_until_volatile_write_unchanged_addresses: fixes a :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat" assumes "a \<notin> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ls. outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))" shows "flush_all_until_volatile_write ls m a = m a"
in_ivl_affine_of_ivlE: fixes k :: "'b"   and l :: "'b"   and u :: "'b" assumes "k \<in> {l..u}" obtains e :: "nat \<Rightarrow> real" where "e \<in> UNIV \<rightarrow> {- 1..1}" and "k = aform_val e (aform_of_ivl l u)"
span_base: fixes a :: "'b poly"   and S :: "'b poly set" assumes "a \<in> S" shows "a \<in> poly_vs.span S"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<cdot> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot>\<^sub>D \<lbrace>u\<rbrace>"
ff_bad: shows "inf_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (worst_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (\<lambda>ys xs. length xs < length ys))"
in_rel: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g list"   and b :: "'h list" shows "list_all2 R a b = (\<exists>z. z \<in> {x. set x \<subseteq> {(x, y). R x y}} \<and> map fst z = a \<and> map snd z = b)"
convergence: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs = set ys"   and "concurrent_ops_commute xs"   and "concurrent_ops_commute ys"   and "distinct xs"   and "distinct ys"   and "hb_consistent xs"   and "hb_consistent ys" shows "apply_operations xs = apply_operations ys"
key: shows "set_mset (J -s r \<down>m J) \<subseteq> set_mset (J - J \<inter># K)"
\<Gamma>_map_\<Gamma>: fixes f :: "'c set \<Rightarrow> 'b set"   and s :: "'c trace"   and i :: "nat" shows "\<Gamma> (map_\<Gamma> f s) i = f (\<Gamma> s i)"
src_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>C (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
iT_Div_def: fixes I :: "nat set"   and k :: "nat" shows "I \<oslash> k \<equiv> (\<lambda>n. n div k) ` I"
derived_S5_rules_2_a: fixes \<chi> :: "\<o>"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"   and "[\<^bold>\<box>\<chi> in v]" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
0: shows "Fvars \<chi> = Fvars \<phi> - snd ` set txs \<union> \<Union> {if x \<in> Fvars \<phi> then {u} else {} |u x. u \<in> var \<and> (Var u, x) \<in> set (zip (map Var us) (map snd txs))}"
Stirling_fps_aux: fixes m :: "nat"   and n :: "nat" shows "(fps_exp 1 - 1) ^ m $ n * fact n = fact m * real (Stirling n m)"
HFun_Sigma_fresh_iff: fixes a :: "atom"   and r :: "tm" shows "a \<sharp> HFun_Sigma r = a \<sharp> r"
length_transpose: fixes xs :: "'a list list" shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
path_appendI: fixes v :: "'d"   and v' :: "'d"   and pth1 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list"   and v'' :: "'d"   and pth2 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "path v v' pth1"   and "path v' v'' pth2" shows "path v v'' (pth1 @ pth2)"
proj5_eq_dhops: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>5 (the (rt dip)) = the (dhops rt dip)"
gFreshGVarRev_def: fixes MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'g, 'h, 'i) Recursion.model_scheme" shows "gFreshGVarRev MOD \<equiv> \<forall>xs y x. gFresh MOD xs y (Var xs x) (gVar MOD xs x) \<longrightarrow> y \<noteq> x"
STRUCT_EQ_tag_def: fixes x :: "'a"   and y :: "'a" shows "STRUCT_EQ_tag x y \<equiv> x = y"
bounded_iff: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
cofactorM_def: fixes A :: "(('a, 'b) vec, 'b) vec" shows "cofactorM A = (\<chi>i j. cofactor A i j)"
special_cosets_below_in: fixes w :: "'w"   and T :: "'w set" assumes "w \<in> W"   and "T \<in> Pow S" shows "dual_order.below_in \<P> (w +o \<langle>T\<rangle>) = (\<Union>R\<in>dual_order.below_in (Pow S) T. {w +o \<langle>R\<rangle>})"
no_defection_preserved: fixes s :: "v_state"   and r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and r' :: "nat"   and s' :: "v_state" assumes "s \<in> Vinv1"   and "r = next_round s"   and "no_defection s v_f r"   and "no_defection s (votes s r') r'"   and "votes s' = (votes s)(r := v_f)" shows "no_defection s' (votes s' r') r'"
formulaEntailsClauseAppend: fixes formula :: "Literal list list"   and clause :: "Literal list"   and formula' :: "Literal list list" assumes "formulaEntailsClause formula clause" shows "formulaEntailsClause (formula @ formula') clause"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case None of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
p_def: fixes i :: "'a"   and n :: "nat" shows "p i n = (P n)\<^sub>F i"
kolmogorov_0_1_law: fixes A :: "nat \<Rightarrow> 'a set set"   and X :: "'a set" assumes "\<And>i. sigma_algebra (space M) (A i)"   and "indep_sets A UNIV"   and "X \<in> tail_events A" shows "prob X = 0 \<or> prob X = 1"
eq_app_right_class_eq: shows "UNIV // eq_app_right language = (\<lambda>q. eq_app_right language `` {path_to q}) ` accessible"
free_tv_app_subst_type_scheme: fixes S :: "nat \<Rightarrow> typ"   and sch :: "type_scheme" shows "free_tv ($ S sch) \<subseteq> cod S \<union> free_tv sch"
simpfm_lin: fixes p :: "fm" assumes "qfree p" shows "islin (simpfm p)"