hcju/isabelleps huggingface dataset:
mult_lc_monic: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "local.punit.monom_mult (lcf p) 0 (punit.monic p) = p"
nonempty_listE: fixes xs :: "nat list"   and m :: "nat" assumes "Suc 0 \<le> length xs" shows "map ((\<lambda>a. rec_exec a (m # xs)) \<circ> recf.id (Suc (length xs))) [Suc 0..<length xs] @ [(m # xs) ! length xs] = xs"
wt_subst_trm'': fixes \<sigma> :: "(('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>" shows "\<Gamma> t = \<Gamma> (t \<cdot> \<sigma>)"
minorM_eq: fixes M :: "(('c, 'd) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "minorM M i j = to_vec (minor (from_vec M) i j)"
all_imp_alw: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<And>xs. \<phi> xs" shows "alw \<phi> xs"
up_ucast_inj: fixes x :: "'c word"   and y :: "'c word" assumes "ucast x = ucast y"   and "LENGTH('c) \<le> LENGTH('b)" shows "x = y"
sym_bis: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set" assumes "Bisim.mono Retr" shows "sym (bis Retr)"
a_get_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "CD.a_get_disconnected_nodes_locs document_ptr = {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)}"
head_\<omega>_def: fixes M :: "hmultiset" shows "head_\<omega> M = (if M = 0 then 0 else \<omega>\<^sub>h^ (Max_mset (hmsetmset M)))"
mult_neg_numeral_simps_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = numeral.numeral one plus (m * n)"
hyps_6: fixes b :: "ty" assumes "\<Gamma>\<turnstile>b<:Q" shows "D @ TVarB X b # \<Gamma> \<turnstile> t1 : (\<forall>X'<:T11. T12)"
rep_eq: fixes x :: "('b, 'c) mapping" shows "keys x = dom (rep x)"
normalize_rules_dnf_def2: shows "normalize_rules_dnf = normalize_rules normalize_match"
subst_bvs1'_step_extend_lower_level: fixes a :: "Core.term"   and args :: "Core.term list"   and t :: "Core.term"   and lev :: "nat" assumes "\<forall>x\<in>set (a # args). is_closed x" shows "subst_bv2 (subst_bvs1' t (Suc lev) args) lev a = subst_bvs1' t lev (a # args)"
vals_pos_nonempty: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "{x \<in> v ` carrier K. 0 < x} \<noteq> {}"
abs_range_map_Some: fixes v :: "nat" assumes "v \<in> set abs_ast_variable_section" shows "abs_range_map v = Some [0..<length (snd (snd (astDom ! v)))]"
record_simps_27: fixes f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "refs (update_defined_types f x) = refs x"
par_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "B A Par D C"
cfg_on_div_def: fixes st :: "'s \<times> ('c \<Rightarrow> real)" shows "cfg_on_div st \<equiv> MDP.cfg_on st \<inter> Collect div_cfg"
af_G_letter_opt_sat_core_lifted: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl_prop_equiv_quotient"   and \<nu> :: "'b set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P Rep \<phi>" shows "\<G> \<Turnstile>\<^sub>P Rep (\<up>af\<^sub>G\<^sub>\<UU> \<phi> \<nu>)"
smod_int_compares_6: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "b < 0" shows "0 \<le> a smod b"
match_ex_table_pc_length_compEs2: fixes P :: "'a prog"   and a :: "String.literal"   and pc :: "nat"   and es :: "'b expr1 list"   and pc' :: "nat"   and d :: "nat"   and pcd :: "nat \<times> nat" assumes "match_ex_table P a pc (compxEs2 es pc' d) = \<lfloor>pcd\<rfloor>" shows "pc' \<le> pc \<and> pc < length (compEs2 es) + pc'"
mp_weakly_fair: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "weakly_fair enabled taken \<sigma>"   and "(\<box>enabled) \<sigma>" shows "(\<diamond>taken) \<sigma>"
filter_leD: assumes "F \<le> F'"   and "\<forall>\<^sub>F x in F'. \<not> P x" shows "\<forall>\<^sub>F x in F. \<not> P x"
cols_length: fixes A :: "'a mat" shows "length (cols A) = dim_col A"
ufa_invarD: fixes l :: "nat list"   and i :: "nat" assumes "ufa_invar l"   and "i < length l" shows "rep_of_dom (l, i) \<and> l ! i < length l"
sign_denormal_of_Float: fixes x :: "Float.float" shows "sign (denormal_of_Float x) = (if 0 \<le> x then 0 else 1)"
ord_isom_mem: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "f a \<in> carrier E"
iso_def: fixes "from" :: "'c \<Rightarrow> 'd"   and to :: "'d \<Rightarrow> 'c" shows "local.iso from to = ((\<forall>a. to (from a) = a) \<and> (\<forall>b. from (to b) = b))"
assms_1: shows "are_perspective_from_point A B C A' B' C' R"
append_sublist: fixes l1 :: "'a list"   and l2 :: "'a list"   and l3 :: "'a list"   and l :: "'a list" assumes "subseq (l1 @ l2 @ l3) l" shows "subseq (l1 @ l3) l"
inner_prod_eq: fixes i :: "'d" shows "(\<lambda>v. v \<bullet> axis i 1) = (\<lambda>v. v $ i)"
Nset_img: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and B :: "'e set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> B" shows "insert (f (Suc n)) (f ` {j. j \<le> n}) = f ` {j. j \<le> Suc n}"
finfun_exhaust_disj: fixes x :: "('b, 'a) finfun" shows "(\<exists>b. x = finfun_const b) \<or> (\<exists>f a b. x = finfun_update f a b)"
that_1: assumes "ns = [n]"   and "m = n" shows "thesis"
fls_inverse_base_factor': fixes f :: "'a fls" assumes "inverse (fls_nth f (fls_subdegree f)) \<noteq> (0::'a)" shows "inverse (fls_base_factor f) = fls_base_factor (inverse f)"
getIJProperties_2: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" assumes "2 \<le> card V"   and "(I, J) = getIJ Q_pos Q_neg V" shows "1 \<le> card J"
finite_linorder_induct: fixes A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite A"   and "P {}"   and "\<And>a A. \<lbrakk>finite A; A \<subseteq> {..<a}; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P A"
P_def: shows "P \<equiv> \<lambda>a. not (StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x a)"
abs_eq_iff: fixes x :: "('e, 'f, 'g) Tree"   and y :: "('e, 'f, 'g) Tree" shows "(abs_Tree\<^sub>p x = abs_Tree\<^sub>p y) = hull_relp x y"
valid_prefix_def: fixes pf :: "'b prefix_match" shows "valid_prefix pf = (pfxm_mask pf && pfxm_prefix pf = 0)"
seq_compact_eq_compact: fixes U :: "'a set" shows "seq_compact U = compact U"
nat_term_compare_cases: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "y \<in> {cmp. comparator cmp \<and> nat_term_comp cmp}" obtains x :: "'a nat_term_order" where "y = nat_term_compare x"
rep_eq: fixes x :: "'b permutation"   and xa :: "'b permutation" shows "(\<rightarrow>) (x - xa) = (\<rightarrow>) x \<circ> the_inv ((\<rightarrow>) xa)"
independent_columns_fst_QR_decomposition: fixes A :: "((real, 'b) vec, 'a) vec" assumes "vec.independent (columns A)"   and "card (columns A) = ncols A" shows "vec.independent (columns (fst (QR_decomposition A))) \<and> card (columns (fst (QR_decomposition A))) = ncols A"
are_complementary_test: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "are_complementary x y" shows "test y"
normalizePr_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "normalizePr p = removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) (Nets_List p))))"
rconf_In1: fixes G :: "prog"   and L :: "lenv"   and s :: "st"   and ec :: "expr + stmt"   and v :: "val"   and T :: "ty" shows "G,L,s\<turnstile>In1 ec\<succ>In1 v\<Colon>\<preceq>Inl T = G,s\<turnstile>v\<Colon>\<preceq>T"
fls_base_factor_to_fps_delta: fixes a :: "int"   and c :: "'a" shows "fls_base_factor_to_fps (Abs_fls (\<lambda>n. if n = a then c else (0::'a))) = fps_const c"
dotprod_unit: fixes a :: "nat list"   and n :: "nat"   and k :: "nat"   and zk :: "nat" assumes "length a = n"   and "k < n" shows "a \<bullet> (zeroes n)[k := zk] = a ! k * zk"
37513: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) \<sqinter> - (y \<squnion> x) = bot"
coincides_stocks_cls_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt2 :: "('c, 'd) discrete_market" assumes "stock_portfolio Mkt pf"   and "coincides_on Mkt Mkt2 (stocks Mkt)" shows "\<forall>n w. cls_val_process Mkt pf n w = cls_val_process Mkt2 pf n w"
chamber_retraction1: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>" shows "fixespointwise f C"
lift_resumption_DONE: fixes x :: "'d" shows "lift_resumption (DONE x) = Generative_Probabilistic_Value.Done x"
Max_image_pair: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite S"   and "finite T"   and "T \<noteq> {}" shows "(MAX s\<in>S. Max (f s ` T)) = (MAX (s, t)\<in>S \<times> T. f s t)"
wordinterval_union_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_union r1 r2) = wordinterval_to_set r1 \<union> wordinterval_to_set r2"
all_self_eq_1: fixes v :: "i" shows "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<alpha> \<^bold>= \<alpha>) in v]"
Dyz: shows "hausdorff_distance (f ` {y--z}) {f y--f z} \<le> D"
ext_inject: fixes obj_mark :: "bool"   and obj_fields :: "'a \<Rightarrow> 'c option"   and obj_payload :: "'a \<Rightarrow> 'b option"   and more :: "'d"   and obj_mark' :: "bool"   and obj_fields' :: "'a \<Rightarrow> 'c option"   and obj_payload' :: "'a \<Rightarrow> 'b option"   and more' :: "'d" shows "(\<lparr>obj_mark = obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr> = \<lparr>obj_mark = obj_mark', obj_fields = obj_fields', obj_payload = obj_payload', \<dots> = more'\<rparr>) = (obj_mark = obj_mark' \<and> obj_fields = obj_fields' \<and> obj_payload = obj_payload' \<and> more = more')"
Limsup_const_add: fixes F :: "'b filter"   and c :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "F \<noteq> bot" shows "Limsup F (\<lambda>x. c + f x) = c + Limsup F f"
funpow_Suc_right: fixes n :: "nat"   and f :: "'g \<Rightarrow> 'g" shows "f ^^ Suc n = f ^^ n \<circ> f"
iMODb_subset_mod_0_conv: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and r :: "nat"   and c :: "nat" shows "([ r', mod m', c' ] \<subseteq> [ r, mod 0, c ]) = (r' = r \<and> (m' = 0 \<or> c' = 0))"
Lambert_W'_neg: fixes x :: "real" shows "Lambert_W' x < 0"
initial_r_def: fixes prog :: "lambda" shows "initial_r prog = Discr (Inl (the_elem (\<aA> (L prog) Map.empty {}.), [{AStop}], {}., \<abinit>))"
\<tau>external'_red_external_TA_empty: fixes P :: "'a prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "\<tau>external' P h a M" shows "ta = \<lbrace>\<rbrace>"
execute_parallel_operator_sas_plus_def: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list" shows "execute_parallel_operator_sas_plus s ops \<equiv> foldl (++) s (map (map_of \<circ> effect_of) ops)"
content_primitive_part_fract: fixes p :: "'a fract poly" assumes "p \<noteq> 0" shows "content (primitive_part_fract p) = (1::'a)"
markedElementsToLastMarkedAreAllMarkedElements: fixes M :: "('b \<times> bool) list" assumes "uniq (elements M)"   and "markedElements M \<noteq> []" shows "markedElementsTo (lastMarked M) M = markedElements M"
eq_foldr: fixes xs :: "'a list" shows "F xs = foldr (\<^bold>*) xs \<^bold>1"
tab_in_hom_2: shows "\<guillemotleft>D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> : F g \<Rightarrow>\<^sub>D F r \<star>\<^sub>D F f\<guillemotright>"
common_subdivision_imp_eq_line_integral_1: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "common_boundary_sudivision_exists one_chain1 one_chain2"   and "boundary_chain one_chain1"   and "boundary_chain one_chain2"   and "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"   and "finite one_chain1"   and "finite one_chain2"   and "finite basis" shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
inf_sup_ord_4: fixes inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and y :: "'k"   and x :: "'k" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
WT_gpv_left_gpv: fixes \<I>1 :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and \<I>2 :: "('k, 'l) \<I>" assumes "\<I>1 \<turnstile>g gpv \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>g left_gpv gpv \<surd>"
coeff_pCons_Suc: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "coeff (pCons a p) (Suc n) = coeff p n"
supp_Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "supp (Not\<^sub>\<alpha> t\<^sub>\<alpha>) = supp t\<^sub>\<alpha>"
strict_implies_not_eq: fixes a :: "'t"   and b :: "'t" assumes "a \<prec>\<^sub>t b" shows "a \<noteq> b"
simps_4: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "Cp (x \<oplus> y) = Cp y \<Oplus> Cp x"
simps_3: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and T :: "('b, 'a) process_term" assumes "isACT sys T" shows "prefOf (solution sys T) = PREF sys T"
impr_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a r\<rightarrow> b \<equiv> Abs_N ((- Rep_N a + Rep_N b) \<sqinter> (0::'a))"
le_inf_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.le_inf ind w\<^sub>1 w\<^sub>2 \<equiv> \<forall>v\<^sub>1. v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 \<longrightarrow> traces.le_fininf ind v\<^sub>1 w\<^sub>2"
negative_flip_kauff_mat: shows "kauff_mat left_under = kauff_mat right_under"
Inl_ST_iff: fixes q :: "hf"   and u :: "'a list" shows "(HF.Inl q \<in> ST.nextl {HF.Inl (dfa.init MS)} u) = (q = dfa.nextl MS (dfa.init MS) u)"
inj_on_imageI2: fixes f' :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "inj_on (f' \<circ> f) A" shows "inj_on f A"
bind_return_pmf: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a pmf" shows "bind_pmf (return_pmf x) f = f x"
h2: shows "\<forall>k<n. nSend k t = []"
bounded_semilattice_sup_bot_ow_def: fixes U :: "'a set"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a" shows "bounded_semilattice_sup_bot_ow U sup le ls bot \<equiv> semilattice_sup_ow U sup le ls \<and> order_bot_ow U bot le ls"
simps_2: fixes ts :: "t list"   and ts' :: "t list" shows "produce (Type ts) (Type ts') = Type (ts @ ts')"
elem_of_symcl_1: fixes a :: "'b"   and b :: "'b"   and Rel :: "('b \<times> 'b) set" assumes "(a, b) \<in> Rel" shows "(a, b) \<in> symcl Rel"
rel_flip: fixes R :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b H"   and b :: "'d H" shows "rel_H R\<inverse>\<inverse> a b = rel_H R b a"
diagonal_to_Smith_PQ_JNF: fixes A :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and S :: "'b mat"   and Q :: "'b mat" assumes "isDiagonal_mat A"   and "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "(P, S, Q) = diagonal_to_Smith_PQ_JNF A bezout"   and "1 < n"   and "1 < m" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
RG_aezdelta0fun_closed: fixes G :: "'a set"   and r :: "'b" assumes "Group G" shows "r \<delta>\<delta> (0::'a) \<in> aezfun_setspan G"
iwlsFSbSwTR_iterSTR_ipresCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "ipresCons (iterSTR MOD) (iterAbsSTR MOD) MOD"
narrow_Boolean2: fixes G :: "prog"   and S :: "ty" assumes "G\<turnstile>S\<succ>PrimT Boolean" shows "S = PrimT Boolean"
set_intros_18: fixes yp :: "'b"   and x141 :: "'b ltlc"   and x142 :: "'b ltlc" assumes "yp \<in> atoms_ltlc x141" shows "yp \<in> atoms_ltlc (x141 M\<^sub>c x142)"
step: fixes x :: "'a" assumes "P x" shows "P (iterate k\<cdot>f\<cdot>x)"
r: shows "{x. poly p x = 0} = (\<Union>p\<in>fst ` set pis. {x. poly p x = 0})"
sup_corr_eq_chain: fixes M :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>(A, B)\<in>M. f A = g B" shows "Sup {f A |A. A \<in> fst_proj M} = Sup {g B |B. B \<in> snd_proj M}"
card_equiv_class: fixes A :: "'a set"   and R :: "('a \<times> 'a) set" assumes "finite A"   and "equiv A R" shows "card A = sum card (A // R)"
cubic_root_radical_sqrt_steplemma_sqrt: fixes P :: "real set"   and a :: "real"   and b :: "real"   and c :: "real"   and z :: "real"   and u :: "real"   and v :: "real"   and s :: "real" assumes "\<nat> \<subseteq> P"   and "\<forall>x\<in>P. - x \<in> P"   and "\<forall>x\<in>P. x \<noteq> 0 \<longrightarrow> 1 / x \<in> P"   and "\<forall>x\<in>P. \<forall>y\<in>P. x + y \<in> P"   and "\<forall>x\<in>P. \<forall>y\<in>P. x * y \<in> P"   and "a \<in> P"   and "b \<in> P"   and "c \<in> P"   and "z ^ 3 + a * z\<^sup>2 + b * z + c = 0"   and "u \<in> P"   and "v \<in> P"   and "s \<in> P"   and "0 \<le> s"   and "z = u + v * sqrt s" shows "\<exists>w\<in>P. w ^ 3 + a * w\<^sup>2 + b * w + c = 0"
subst_subst: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and \<rho> :: "'c \<Rightarrow> 'd"   and e :: "('a, 'd) uexpr" shows "\<sigma> \<dagger> \<rho> \<dagger> e = (\<rho> \<circ> \<sigma>) \<dagger> e"
vec_of_poly_rev_shifted_def: fixes p :: "'b poly"   and n :: "nat"   and j :: "nat" shows "vec_of_poly_rev_shifted p n j \<equiv> vec n (\<lambda>i. if i \<le> j \<and> j \<le> degree p + i then coeff p (degree p + i - j) else (0::'b))"
residual_l_set_def: fixes Z :: "'a set"   and Y :: "'a set" shows "Z \<leftarrow> Y = \<Union> {X. X \<cdot> Y \<subseteq> Z}"
case_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x7 :: "'b ltln" shows "(case X\<^sub>n x7 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f7 x7"
limit_ordinal_oLimitI: fixes f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. f n < oLimit f" shows "limit_ordinal (oLimit f)"
crename_OrR1: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = OrR1 <c>.N d"   and "c \<sharp> (R, a, b)"   and "d \<sharp> (a, b)" shows "\<exists>N'. R = OrR1 <c>.N' d \<and> N'[a\<turnstile>c>b] = N"
lfilter_lmap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "lfilter P (lmap f xs) = lmap f (lfilter (P \<circ> f) xs)"
pmf_q: fixes x :: "'alt" shows "pmf q x = pmf p x + \<epsilon> * (pmf q' x - pmf p' x)"
simps_2: fixes n :: "nat"   and x :: "'b" shows "x \<up> Suc n = x # x \<up> n"
list_ex1_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "list_ex1 P xs = (\<exists>!x. x \<in> set xs \<and> P x)"
homotopic_paths_subset: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and t :: "'b set" assumes "homotopic_paths s p q"   and "s \<subseteq> t" shows "homotopic_paths t p q"
echelon_form_upt_k_if_equal: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and B :: "(('d, 'e) vec, 'f) vec" assumes "echelon_form_upt_k A k"   and "\<forall>a b. b < mod_type_class.from_nat k \<longrightarrow> A $ a $ b = B $ a $ b"   and "k < ncols A" shows "echelon_form_upt_k B k"
dom_one: fixes domain_op :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.domain_monoid domain_op times one" shows "domain_op one = one"
tSplice_envAction: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tLength t = tLength t'" shows "envAction (tLast (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t')) = envAction (tLast t)"
progress: fixes s :: "char list \<Rightarrow> int option"   and c :: "com"   and A' :: "char list set" assumes "D (dom s) c A'"   and "c \<noteq> SKIP" shows "\<exists>cs'. (c, s) \<rightarrow> cs'"
icd: shows "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
strict_trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "strict_prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
unless_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(P \<W> Q) = (\<lambda>s. (P \<U> Q) s \<or> (\<box>P) s)"
orderE: fixes b :: "'h"   and a :: "'h" assumes "b \<le> a" obtains "a = max a b"
minus_natural_def: shows "(-) \<equiv> map_fun nat_of_natural (map_fun nat_of_natural natural_of_nat) (-)"
or_eq_not_not_and: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "or a b = not (and (not a) (not b))"
exhaust: fixes y :: "pheap" obtains x1 :: "heap" and x2 :: "nat set" where "y = pHeap x1 x2"