hcju/isabelleps huggingface dataset:
assms: shows "h \<turnstile> get_ancestors_di ptr \<rightarrow>\<^sub>r ancestors"
d3: fixes x :: "'a"   and y :: "'a" shows "d (x \<sqinter> d y) \<le> d (x \<sqinter> y)"
in_q11_code: fixes x :: "nat" shows "(x mod 11 \<in> q11) = q11_array !! (x mod 11)"
C_eq_All_untilSorted: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = Cp p"
q1_def: shows "q1 = fcompose p [:(z0 + complex_of_real r) * \<i>, z0 - complex_of_real r:] [:\<i>, 1:]"
orderE: fixes b :: "'a::semilattice_sup"   and a :: "'a::semilattice_sup" assumes "(b::'a::semilattice_sup) \<le> (a::'a::semilattice_sup)" obtains "(a::'a::semilattice_sup) = a \<squnion> (b::'a::semilattice_sup)"
Sup_dual_upset_var: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and X :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "\<Inter> (ord.upset less_eq ` X) = ord.upset less_eq (Sup X)"
fps_unit_factor_fls_base_factor: fixes f :: "'a fls" shows "unit_factor (fls_base_factor_to_fps f) = fls_base_factor_to_fps f"
bounded_linear_const_mult: fixes g :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "bounded_linear g" shows "bounded_linear (\<lambda>xa. x * g xa)"
HF_Ord: fixes i :: "hf" assumes "Ord i" shows "Goedel_I.HF i = W i"
ivl_subset: fixes i :: "'b"   and j :: "'b"   and m :: "'b"   and n :: "'b" shows "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> n)"
of_circline_unit_circle: shows "of_circline unit_circle = ounit_circle"
summ_zf: fixes A :: "nat set" shows "summable (zero_on f A)"
simps_4: fixes ao :: "'a option"   and n :: "nat"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_interval_set_Compl_aux ao n (x # xs) (y # ys) = (if x < y then proper_interval ao (Some x) \<or> proper_interval_set_Compl_aux (Some x) (n + 1) xs (y # ys) else if y < x then proper_interval ao (Some y) \<or> proper_interval_set_Compl_aux (Some y) (n + 1) (x # xs) ys else proper_interval ao (Some x) \<and> (let m = CARD('a) - n in m - length ys \<noteq> 2 \<or> m - length xs \<noteq> 2))"
odd_degree_imp_real_root: fixes p :: "real poly" assumes "odd (degree p)" shows "\<exists>x. poly p x = 0"
poly_minimum_modulus: fixes p :: "complex poly" shows "\<exists>z. \<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
S\<^sub>i\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i" shows "(E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E x \<^bold>\<leftarrow> E (dom x)) \<^bold>\<and> (E y \<^bold>\<leftarrow> E (cod y))"
simps: fixes x :: "'b extC"   and y :: "'b extC" shows "x * y = x \<sqinter> y"
ev: fixes v :: "'a"   and e :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T" shows "e * v = bot"
da_elim_cases_44: fixes Env :: "env"   and B :: "lname set"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (c ? e1 : e2)\<guillemotright> A" obtains C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C__" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1__" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2__" and "nrm A = B \<union> assigns_if True (c ? e1 : e2) \<inter> assigns_if False (c ? e1 : e2)" and "brk A = (\<lambda>l. UNIV)"   | C :: "assigned" and E1 :: "assigned" and E2 :: "assigned" where "\<not> Env\<turnstile>\<langle>c ? e1 : e2\<rangle>\<Colon>Inl (PrimT Boolean)" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2" and "nrm A = nrm E1 \<inter> nrm E2" and "brk A = (\<lambda>l. UNIV)"
numeral_Bit1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
word_1_0: fixes a :: "'b word"   and b :: "'b word"   and x :: "nat" assumes "a + 1 \<le> b"   and "a < word_of_nat x" shows "a < b"
\<iota>_in_hom': shows "\<guillemotleft>\<iota>\<^sub>D : D.tensor (D.cod \<iota>\<^sub>D) (D.cod \<iota>\<^sub>D) \<rightarrow>\<^sub>D D.cod \<iota>\<^sub>D\<guillemotright>"
divide_eq_minus_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = uminus one) = (b \<noteq> zero \<and> a = uminus b)"
elims: fixes x :: "'c list"   and xa :: "'c list"   and y :: "'c list" assumes "splice x xa = y" obtains ys :: "'c list" where "x = []" and "xa = ys__" and "y = ys__"   | xb :: "'c" and xs :: "'c list" and ys :: "'c list" where "x = xb # xs" and "xa = ys" and "y = xb # splice ys xs"
us: shows "us = getFrN (map snd txs) (Var x # map fst txs) [] (length txs)"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'a list" shows "rotate n xs = rotate (n mod |xs|) xs"
iff: shows "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a <\<^bsub>P' i\<^esub> b)"
posaux4: fixes a :: "('b, 'c) Combinators"   and b :: "('b, 'c) Combinators"   and l :: "'b set list"   and x :: "'b set"   and y :: "'b set" assumes "singleCombinators [a]"   and "a \<noteq> DenyAll"   and "b \<noteq> DenyAll"   and "in_list a l"   and "in_list b l"   and "smaller a b l"   and "x = bothNet a"   and "y = bothNet b" shows "position x l \<le> position y l"
simps_1: fixes c :: "'form"   and uu :: "nat" shows "in_port_at (c, []) uu = plain_ant c"
bin_nth_0_BIT: fixes w :: "int"   and b :: "bool" shows "bit (w BIT b) 0 = b"
S: shows "S \<in> carrier_mat CARD('nr mod_ring) CARD('nc mod_ring)"
drop_Suc: fixes n :: "nat"   and xs :: "'d list" shows "drop (Suc n) xs = drop n (tl xs)"
dr: shows "dim_row (submatrix A I J) = dim_row (submatrix (submatrix A I UNIV) UNIV J)"
FSwSbImorph_iff: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'h, 'j, 'm) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'i, 'k, 'n) model_scheme" shows "FSwSbImorph h hA MOD MOD' = (FSbImorph h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD')"
S_verts_def: fixes p :: "real"   and n :: "nat" assumes "edge_space p" shows "edge_space.S_verts n \<equiv> {1..n}"
wf0_FV0_LESS: fixes idx :: "'n"   and a :: "'a"   and v :: "nat"   and k :: "'k" assumes "wf0 idx a"   and "v \<in> FV0 k a" shows "LESS k v idx"
set_mset_single: fixes b :: "'b" shows "set_mset (\<LM> b  \<RM>) = {b}"
elims_2: fixes x :: "char list"   and xa :: "char list" assumes "lleq_charlist x xa" obtains "xa__ = []" and "xa = []"   | v :: "char" and va :: "char list" where "xa__ = []" and "xa = v # va"   | x :: "char" and xs :: "char list" and y :: "char" and ys :: "char list" where "xa__ = x # xs" and "xa = y # ys" and "if x = y then lleq_charlist xs ys else x < y"
less_eq_list_not_le: fixes xs :: "'a list"   and ys :: "'a list" shows "local.less_list xs ys = (local.less_eq_list xs ys \<and> \<not> local.less_eq_list ys xs)"
induct: fixes P2 :: "('c, 'd) bintree list \<Rightarrow> bool"   and P1 :: "('c, 'd) bintree \<Rightarrow> bool"   and bintree_list :: "('c, 'd) bintree list" assumes "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (Node x1 x2 x3)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 bintree_list"
node_anycast_msg_zhops: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(uu_, a, uu_). castmsg msg_zhops a)"
conj_def: fixes p :: "fm"   and q :: "fm" shows "MIR.conj p q \<equiv> if p = F \<or> q = F then F else if p = T then q else if q = T then p else if p = q then p else And p q"
lformula_nFAll: fixes b :: "'k"   and \<phi> :: "('a, 'k) aformula" assumes "lformula (FAll b \<phi>)" shows "lformula (nFAll b \<phi>)"
index_mult_mat_2: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_row (A * B) = dim_row A"
conv: fixes x :: "complex" shows "(poly (complex_of_int_poly p) x = 0) = (poly (map_poly complex_of_real q) x = 0)"
RES_sng_eq_RETURN: fixes x :: "'a" shows "RES {x} = RETURN x"
not_HLD: fixes X :: "'c set" shows "(\<lambda>xs. \<not> HLD X xs) = HLD (- X)"
Let_1: shows "atom ` domA as \<sharp>* \<Gamma>"
first_tail_second: fixes s :: "nat \<Rightarrow> 'a" shows "first (tail s) = second s"
exi: fixes j :: "name"   and i :: "name"   and x :: "trm"   and A :: "fmla" assumes "atom j \<sharp> (i, x)" shows "(exi j A)(i::=x) = exi j (A(i::=x))"
identity_functor_axioms: shows "identity_functor (\<cdot>)"
proots_count_monom: fixes A :: "'a set"   and d :: "nat" assumes "(0::'a) \<notin> A" shows "proots_count (monom (1::'a) d) A = 0"
assms_3: shows "has_greatest_postfixpoint (\<lambda>x. \<nu> f x)"
balanced_def: fixes kdt :: "'a kdt" shows "balanced kdt = (height kdt - min_height kdt \<le> 1)"
stable_no_blocking_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Pds'.stable_no_blocking_on ds X = (\<forall>h X'. \<not> Pds'.blocking_on ds X h X')"
interleave_filter: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<in> filter P ys \<otimes> filter P zs" obtains xs' :: "'b list" where "xs' \<in> ys \<otimes> zs" and "xs = filter P xs'"
g: shows "g \<equiv> \<lambda>xs. sum_list (map h xs)"
sup_loc_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and LT :: "ty err list" shows "P \<turnstile> LT [\<le>\<^sub>\<top>] LT"
cet_minus_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : REAL" shows "\<Gamma> \<turnstile>\<^sub>c -\<^sub>c e : REAL"
hcomp_arr_obj_1: fixes \<mu> :: "'a"   and a :: "'a" assumes "B.arr \<mu>"   and "B.obj a"   and "src\<^sub>B \<mu> = a" shows "\<mu> \<star>\<^sub>B a = B.runit' (B.cod \<mu>) \<cdot>\<^sub>B \<mu> \<cdot>\<^sub>B B.runit (B.dom \<mu>)"
supp_atm: fixes x :: "name" shows "supp x = {x}"
irreflp_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "irreflp_on P A = (\<forall>a\<in>A. \<not> P a a)"
distinct_9: fixes x1 :: "('e, 'f) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "send\<langle>x1\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
noStuckn_startD: 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" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Normal t" shows "s \<noteq> Stuck"
induct_list012: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y zs. \<lbrakk>P zs; P (y # zs)\<rbrakk> \<Longrightarrow> P (x # y # zs)" shows "P xs"
5: shows "length w\<^sub>2 = length (w\<^sub>2 @ v\<^sub>2)"
simps_17: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and f3 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and f4 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and x2 :: "nat" shows "(case TVar x2 of TUnit \<Rightarrow> f1 | TVar x \<Rightarrow> f2 x | TArr x xa \<Rightarrow> f3 x xa | TPair x xa \<Rightarrow> f4 x xa) = f2 x2"
lincomb_smult: fixes A :: "'a vec set"   and c :: "'a"   and a :: "'a vec \<Rightarrow> 'a" assumes "A \<subseteq> carrier_vec n" shows "lincomb (\<lambda>w. c * a w) A = c \<cdot>\<^sub>v lincomb a A"
ax5_0: fixes x :: "'a" shows "d (x \<cdot> (0::'a)) \<cdot> U = x \<cdot> (0::'a) \<parallel> U"
Cd_single: fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (Pd d)" shows "x \<in> Cd d X"
gather_squares: fixes a :: "real"   and b :: "real" shows "a * (a * b) = a\<^sup>2 * b"
predict_imp_adapt: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "predict_stoch_proc F X N" shows "adapt_stoch_proc F X N"
keys_extend_indets: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "keys (extend_indets p) = (\<lambda>t. \<Sum>y\<in>keys t. monomial (lookup t y) (Some y)) ` keys p"
encode_frame_def: fixes s :: "recf \<times> nat list \<times> nat list" shows "encode_frame s \<equiv> triple_encode (encode (fst s)) (Partial_Recursive.list_encode (fst (snd s))) (Partial_Recursive.list_encode (snd (snd s)))"
idiv_infinity_if: fixes n :: "enat" shows "n div \<infinity> = (if n = \<infinity> then \<infinity> else 0)"
has_integral_cong: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and i :: "'d" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(f has_integral i) s = (g has_integral i) s"
elims_2: fixes x :: "'a st"   and xa :: "char list set" assumes "top_on\<^sub>s x xa" obtains S :: "'a st" and X :: "char list set" where "x = S" and "xa = X" and "fun S = \<lambda>x. \<top> on X"
one_step: fixes s :: "'a"   and t :: "'a"   and tt :: "SN_rel_ext_type set" assumes "M s"   and "(s, t) \<in> \<Union> (SN_rel_ext_step P Pw R Rw ` tt) \<inter> M'" shows "\<exists>typ\<in>tt. (s, t) \<in> SN_rel_ext_step P Pw R Rw typ \<and> M s \<and> M t"
assms_1: fixes n :: "nat" assumes "0 < n" shows "f n = f' n"
min_eq_1: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "min a b = a"
simps_1: fixes I :: "('a, 'b, 'c) interp"   and x :: "'c"   and v :: "(real, 'c) vec" shows "sterm_sem I (Var x) v = v $ x"
ideal_generated_pair_UNIV: fixes s :: "'a"   and a :: "'a"   and t :: "'a"   and b :: "'a"   and u :: "'a" assumes "s * a + t * b = u"   and "u dvd (1::'a)" shows "ideal_generated {a, b} = UNIV"
simp_Plus_simps: fixes p1 :: "'b rexp \<times> ('c val \<Rightarrow> 'c val)"   and p2 :: "'b rexp \<times> ('c val \<Rightarrow> 'c val)" shows "simp_Plus p1 p2 = (if fst p1 = Zero then (fst p2, F_RIGHT (snd p2)) else if fst p2 = Zero then (fst p1, F_LEFT (snd p1)) else (Plus (fst p1) (fst p2), F_Plus (snd p1) (snd p2)))"
base: assumes "a = b"   and "b \<in> F" shows "thesis"
assms_2: fixes t1 :: "('a, unit) RBT_Impl.rbt"   and t2 :: "('a, unit) RBT_Impl.rbt"   and color :: "color"   and a :: "'a"   and b :: "unit" assumes "P t1"   and "P t2"   and "t2 = rbt.Empty" shows "P (Branch color t1 a b t2)"
induct: fixes P :: "('b word \<times> 'b word) negation_type list \<Rightarrow> bool"   and a0 :: "('b word \<times> 'b word) negation_type list" assumes "P []"   and "\<And>s e ls. P ls \<Longrightarrow> P (Pos (s, e) # ls)"   and "\<And>s e ls. P ls \<Longrightarrow> P (Neg (s, e) # ls)" shows "P a0"
nfoldli_transfer_dres: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd dres"   and f' :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and l :: "'c list"   and c :: "'d \<Rightarrow> bool"   and s :: "'d" assumes "\<And>x s. nres_of (f x s) \<le> f' x s" shows "nres_of (foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f x) (dRETURN s)) \<le> nfoldli l c f' s"
hcomp_simps_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
prebicategory_with_homs_and_units_axioms: shows "prebicategory_with_homs_and_units (\<cdot>) (\<star>) \<a> \<i>' src trg"
continuous_trivial_limit: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "net = bot" shows "continuous net f"
inducts: fixes P :: "'b list \<Rightarrow> bool"   and list :: "'b list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
V_iff_non_\<phi>: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" shows "(V (trn # tr) = V tr) = (\<not> \<phi> trn)"
unique: fixes x :: "'a"   and y :: "'a" assumes "Domainp A x"   and "Domainp A y"   and "f x"   and "f y" shows "f' x = f' y"
primitive_extractor_correct_1: fixes m :: "'a match_expr"   and disc :: "'a \<Rightarrow> bool"   and sel :: "'a \<Rightarrow> 'b"   and C :: "'b \<Rightarrow> 'a"   and as :: "'b negation_type list"   and ms :: "'a match_expr"   and \<gamma> :: "('a \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "(matches \<gamma> (alist_and (NegPos_map C as)) a p \<and> matches \<gamma> ms a p) = matches \<gamma> m a p"
e'_2: shows "(\<lambda>x. f x - eval C x * b x powr e) \<in> o(\<lambda>x. b x powr e')"
wb_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "wb_lens x \<equiv> weak_lens x \<and> wb_lens_axioms x"
commute: fixes a :: "'b"   and b :: "'b" shows "gcd a b = gcd b a"
mono_\<pi>: fixes k :: "nat"   and T :: "(symbol \<times> character list) set" shows "mono (\<pi> k T)"
assms: shows "is_path_undir (delete_node v G) x p y"
Frame_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a \<Rightarrow> bool" shows "Frame P Q F \<equiv> \<forall>s. P s \<longrightarrow> (Q \<and>* F) s"
rwp_on_trans: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "wca_on A f"   and "f_range_on A f"   and "decisive_on A f" shows "trans (rwp_on A f)"
row_def: fixes i :: "'f"   and A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "row i A = (\<chi>j. A $h i $h j)"
scast_scast_id_1: fixes x :: "'b word" shows "SCAST('b signed \<rightarrow> 'b) (SCAST('b \<rightarrow> 'b signed) x) = x"
F: shows "range F \<subseteq> {A \<times> B |A B. A \<in> sets S \<and> B \<in> sets T} \<and> incseq F \<and> \<Union> (range F) = space S \<times> space T \<and> (\<forall>i. emeasure (S \<Otimes>\<^sub>M T) (F i) \<noteq> \<infinity>)"
up_filter_dist_sup: fixes x :: "'c"   and y :: "'c" shows "up_filter (x \<squnion> y) = up_filter x \<sqinter> up_filter y"
real_of_inj: fixes x :: "real_alg"   and y :: "real_alg" shows "(real_of x = real_of y) = (x = y)"
x_2: shows "t = (\<theta>1 \<circ>\<^sub>s \<theta>2) x"
frontier_complement: fixes S :: "'c set" shows "frontier (- S) = frontier S"
extend_act_D: fixes z :: "'c"   and z' :: "'c"   and act :: "('a \<times> 'a) set" assumes "(z, z') \<in> extend_act h act" shows "(f z, f z') \<in> act"
swap_def: fixes p :: "'d \<times> 'b" shows "prod.swap p = (snd p, fst p)"
in_minimal_antichain: fixes x :: "'a"   and A :: "'a set" shows "(x \<in> minimal_antichain A) = (x \<in> A \<and> \<not> (\<exists>y\<in>A. y < x))"
interval_integral_cong: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "\<And>x. x \<in> einterval (min a b) (max a b) \<Longrightarrow> f x = g x" shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
ANR_imp_absolute_neighbourhood_retract: fixes S :: "'a set"   and S' :: "'b set"   and U :: "'b set" assumes "ANR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" obtains V :: "'b set" where "openin (top_of_set U) V" and "S' retract_of V"
seqE: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.seq g f" obtains "J_C.arr f" and "J_C.arr g" and "J_C.dom g = J_C.cod f"
Card_lt_csucc_iff: fixes \<kappa>' :: "V"   and \<kappa> :: "V" assumes "Card \<kappa>'"   and "Card \<kappa>" shows "(\<kappa>' \<subset>\<^sub>\<circ> csucc \<kappa>) = (\<kappa>' \<subseteq>\<^sub>\<circ> \<kappa>)"
fun_cong: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l"   and x :: "'k" assumes "f = g" shows "f x = g x"
assms_2: shows "A *\<^sub>m\<^sub>m B = C"
central_binomial_odd: fixes n :: "nat" assumes "odd n" shows "n choose Suc (n div 2) = n choose n div 2"
AlphaNegNeg: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<turnstile> A # G" shows "\<turnstile> Neg (Neg A) # G"
interp_alt_factorization_reflect: fixes fctrs :: "'a \<times> ('a \<times> nat) list" assumes "(0::'a) \<notin> fst ` set (snd fctrs)" shows "reflect_poly (interp_alt_factorization fctrs) = interp_factorization fctrs"
annotateI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and anno :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P anno Q,A"   and "c = anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
block_size_lt_v: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card bl) \<le> \<v>"
isometry_on_injective: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "inj_on f X"
Iod_carr_segment: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D" shows "carrier (Iod D (segment D a)) = segment D a"
simps_3: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp" shows "vars (Plus a\<^sub>1 a\<^sub>2) = vars a\<^sub>1 \<union> vars a\<^sub>2"
pr_var_mwb_lens: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "mwb_lens (&x)"
IH_2: fixes t :: "'a bro" assumes "t \<in> U h" shows "ins x t \<in> T h"
induct: fixes x1 :: "'c"   and x2 :: "'d list"   and x3 :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set"   and P :: "'c \<Rightarrow> 'd list \<Rightarrow> 'c \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> trcl t"   and "\<And>c. P c [] c"   and "\<And>c a c' w c''. \<lbrakk>(c, a, c') \<in> t; (c', w, c'') \<in> trcl t; P c' w c''\<rbrakk> \<Longrightarrow> P c (a # w) c''" shows "P x1 x2 x3"
Sup_bot_conv_2: fixes A :: "'a set" shows "(\<bottom> = Sup A) = (\<forall>x\<in>A. x = \<bottom>)"
map_add_None: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b" shows "((n \<Oplus> m) k = \<bottom>) = (n k = \<bottom> \<and> m k = \<bottom>)"
distinct_butlast: fixes xs :: "'b list" assumes "distinct xs" shows "distinct (butlast xs)"
monom_rep_X_pow: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a n = a \<odot>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> n"
Rep_status: fixes x :: "('d, 'e, 'f) status" shows "Rep_status x \<in> status"
agree_UNIV_snd: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "Vagree \<nu> \<omega> (range Inr)" shows "snd \<nu> = snd \<omega>"
simple_path_rectpath: fixes a1 :: "complex"   and a3 :: "complex" assumes "Re a1 \<noteq> Re a3"   and "Im a1 \<noteq> Im a3" shows "simple_path (rectpath a1 a3)"
frameIntIdentity: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" shows "\<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<one>\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"