hcju/isabelleps huggingface dataset:
induct: fixes P :: "'b expr' \<Rightarrow> 'c expr' \<Rightarrow> bool"   and a0 :: "'b expr'"   and a1 :: "'c expr'" assumes "\<And>v v'. P (Var' v) (Var' v')"   and "\<And>i i'. P (Lit' i) (Lit' i')"   and "\<And>e\<^sub>1 e\<^sub>2 e\<^sub>1' e\<^sub>2'. \<lbrakk>P e\<^sub>1 e\<^sub>1'; P e\<^sub>2 e\<^sub>2'\<rbrakk> \<Longrightarrow> P (Add' e\<^sub>1 e\<^sub>2) (Add' e\<^sub>1' e\<^sub>2')"   and "\<And>v va. P (Lit' v) (Var' va)"   and "\<And>v va vb. P (Lit' v) (Add' va vb)"   and "\<And>v va vb. P (Add' v va) (Var' vb)"   and "\<And>v va vb. P (Add' v va) (Lit' vb)"   and "\<And>va v. P (Var' va) (Lit' v)"   and "\<And>va vb v. P (Add' va vb) (Lit' v)"   and "\<And>vb v va. P (Var' vb) (Add' v va)"   and "\<And>vb v va. P (Lit' vb) (Add' v va)" shows "P a0 a1"
v\<^sub>m_def_2: shows "N = \<parallel>v\<^sub>m\<parallel>\<^sup>2"
id_vwb_lens: shows "vwb_lens 1\<^sub>L"
take_rev: fixes n :: "nat"   and xs :: "'a list" shows "take n (rev xs) = rev (drop (length xs - n) xs)"
trans: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "y .=\<^bsub>S\<^esub> z"   and "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "x .=\<^bsub>S\<^esub> z"
distinct_3: fixes x1 :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" shows "\<Up>x1 \<noteq> x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34"
OF_match_eq: fixes ft :: "('a, 'b) flow_entry_match list"   and \<gamma> :: "'a set \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c" assumes "sorted_descending (map ofe_prio ft)"   and "check_no_overlap \<gamma> ft" shows "OF_same_priority_match2 \<gamma> ft p = OF_match_linear \<gamma> ft p"
r_cancel_one': fixes x :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "a \<in> carrier Zp" shows "(x = a \<oplus> x) = (a = \<zero>)"
check_ipresIGSubstAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGSubstAbs check check (errMOD MOD) MOD"
ed_ou_def: fixes X :: "'b set" shows "ed_ou X = (I \<circ> K = K \<circ> I)"
REC_unfold: fixes B :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "trimono B" shows "REC B = B (REC B)"
from_hma\<^sub>m_diff: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and b :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a - from_hma\<^sub>m b = from_hma\<^sub>m (a - b)"
simps_2: fixes t :: "'a tree"   and ts :: "'a tree list"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "splitLeaf\<^sub>F (t # ts) w\<^sub>a a w\<^sub>b b = splitLeaf t w\<^sub>a a w\<^sub>b b # splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b"
H_def: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_kat.H p x q = (rel_dioid_tests.t p ; x \<subseteq> x ; rel_dioid_tests.t q)"
finite_nempty_ex_degernate_in_lotteries: fixes out :: "'b set" assumes "out \<noteq> {}"   and "finite out" shows "\<exists>e\<in>lotteries_on out. \<exists>x\<in>out. pmf e x = 1"
cadd_0_right: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> \<oplus> 0 = \<kappa>"
f_def: shows "f \<equiv> \<lambda>x. THE_default (d x) (G x)"
pair_return_pmf2: fixes x :: "'a pmf"   and y :: "'b" shows "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
linorder_rank_strict_mono: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set"   and y :: "'b"   and x :: "'b" assumes "linorder_on B R"   and "finite A"   and "A \<subseteq> B"   and "y \<in> A"   and "(y, x) \<in> R"   and "x \<noteq> y" shows "linorder_rank R A y < linorder_rank R A x"
decisive_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "decisive_on A f = (\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> f B \<noteq> {})"
set_map_of_compr: fixes xs :: "('a \<times> 'b) list" assumes "distinct (map fst xs)" shows "set xs = {(k, v). map_of xs k = Some v}"
equal_interval_def: fixes a :: "'a interval"   and b :: "'a interval" shows "equal_class.equal a b \<equiv> lower a = lower b \<and> upper a = upper b"
valid_def: fixes \<phi> :: "'a" shows "\<tturnstile> \<phi> \<equiv> {} \<turnstile> \<phi>"
poly_eval_extend_indets: fixes a :: "'d option \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (extend_indets p) = poly_eval (a \<circ> Some) p"
brackets_unlift_vars_swap: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and x :: "nat itrm"   and n :: "nat"   and vs :: "nat list"   and x' :: "nat itrm" assumes "lifted_bracket term_bracket extra_rule itrm_bracket"   and "set (opaque x) \<subseteq> {0..<n}"   and "set vs \<subseteq> {0..<n}"   and "itrm_abstraction.itrm_brackets itrm_bracket vs x = Some x'" shows "bracket_abstraction.term_brackets term_bracket vs (unlift_vars n x) = Some (unlift_vars n x')"
simps_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ts :: "('c btree \<times> 'c) list"   and t :: "'c btree" assumes "split split" shows "nonempty_lasttreebal (Node ts t) = ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
refine: fixes D :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and Mi :: "nat"   and M :: "nat" assumes "DIM_precond TYPE('b) D"   and "(Xi, X) \<in> lvivl_rel"   and "(Mi, M) \<in> nat_rel" shows "(RETURN (local.sctnbounds_of_ivl_impl D Mi Xi), sctnbounds_of_ivl $ M $ X) \<in> \<langle>sctns_rel\<rangle>nres_rel"
commute: fixes a :: "'b list"   and b :: "'b list" shows "gcp a b = gcp b a"
independent_imp_trivial_combine: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and Ks :: "'a list" assumes "ring R"   and "subfield K R"   and "ring.independent R K Us"   and "set Ks \<subseteq> K"   and "ring.combine R Ks Us = \<zero>\<^bsub>R\<^esub>" shows "set (take (length Us) Ks) \<subseteq> {\<zero>\<^bsub>R\<^esub>}"
balance_right_inv1l: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b" assumes "inv1 lt"   and "inv1l rt" shows "inv1l (balance_right lt k x rt)"
U_intT_def: shows "U_intT \<equiv> InfModel.intTF any"
generate_H_def: fixes EM :: "bit list"   and emBits :: "nat"   and hLen :: "nat" shows "generate_H EM emBits hLen = take hLen (drop (roundup emBits 8 * 8 - hLen - 8) EM)"
adiv_eq: fixes z :: "int"   and a :: "ant"   and b :: "ant" assumes "z \<noteq> 0"   and "a * ant z = b * ant z" shows "a = b"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4, a5) \<in> pr_algo_lts'" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = f__" and "a2 = l__" and "a3 = NONSAT_PUSH'" and "a4 = push_effect f__ e__" and "a5 = l__" and "Height_Bounded_Labeling c s t f__ l__" and "nonsat_push_precond f__ l__ e__"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fa__" and "a2 = la__" and "a3 = SAT_PUSH' e" and "a4 = push_effect fa__ e" and "a5 = la__" and "Height_Bounded_Labeling c s t fa__ la__" and "sat_push_precond fa__ la__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = relabel_effect f l u" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
pcompose_0: fixes q :: "'b poly" shows "0 \<circ>\<^sub>p q = 0"
Transset_Vfrom: fixes A :: "V"   and i :: "V" assumes "Transset A" shows "Transset (Vfrom A i)"
prime_square_sum_nat_decomp_code_aux: fixes p :: "nat" assumes "prime p"   and "p = 2 \<or> [p = 1] (mod 4)" shows "prime_square_sum_nat_decomp p = the_elem (Set.filter (\<lambda>(x, y). x\<^sup>2 + y\<^sup>2 = p) (SIGMA x:{0..p}. {x..p}))"
fs_i_sumlist_\<kappa>: fixes i :: "nat"   and l :: "nat"   and j :: "nat" assumes "i < length fs"   and "l \<le> i"   and "j < l" shows "(fs ! i + M.sumlist (map (\<lambda>j. \<kappa> i l j \<cdot>\<^sub>v fs ! j) [0..<l])) \<bullet> fs ! j = (0::'a)"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "D.ide (V a)"
arr_Rel_comp_Rel: fixes \<alpha> :: "V"   and S :: "V"   and T :: "V" assumes "arr_Rel \<alpha> S"   and "arr_Rel \<alpha> T" shows "arr_Rel \<alpha> (S \<circ>\<^sub>R\<^sub>e\<^sub>l T)"
f2: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) \<and> j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
in_hhomE: fixes \<mu> :: "'c"   and a :: "'c"   and b :: "'c" assumes "C.in_hhom \<mu> a b" obtains "C.arr \<mu>" and "C.obj a" and "C.obj b" and "src\<^sub>C \<mu> = a" and "trg\<^sub>C \<mu> = b"
cong_int_iff: fixes m :: "nat"   and q :: "nat"   and n :: "nat" shows "[int m = int q] (mod int n) = [m = q] (mod n)"
maintainedD: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" assumes "maintained (A, B) G"   and "graph_homomorphism A G f" shows "extensible (A, B) G f"
SND_RKMO: shows "snd (r (k - 1)) = n"
prod_zero_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_zero I A \<in> carr_prodag I A"
SA_arrow: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "ty"   and S\<^sub>1 :: "ty"   and S\<^sub>2 :: "ty"   and T\<^sub>2 :: "ty" assumes "\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1"   and "\<Gamma>\<turnstile>S\<^sub>2<:T\<^sub>2" shows "\<Gamma>\<turnstile>S\<^sub>1 \<rightarrow> S\<^sub>2<:T\<^sub>1 \<rightarrow> T\<^sub>2"
multiset_inter_assoc: fixes A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset" shows "A \<inter># (B \<inter># C) = A \<inter># B \<inter># C"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VVV.seq g f = (VVV.arr f \<and> VVV.arr g \<and> CCC.seq g f)"
Card_iff_initial: fixes \<kappa> :: "V" shows "Card \<kappa> = (Ord \<kappa> \<and> (\<forall>\<alpha>. Ord \<alpha> \<and> \<alpha> < \<kappa> \<longrightarrow> \<not> elts \<alpha> \<approx> elts \<kappa>))"
fold_set_fold_remdups: fixes y :: "'a list"   and xs :: "'a list" shows "Finite_Set.fold insort y (set xs) = fold insort (remdups xs) y"
Bw: shows "B *\<^sub>v w = 0\<^sub>v n"
kernel_assoc_elem: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "h x = h y" obtains z :: "'a" where "x = y \<otimes> z" and "z \<in> kernel G H h"
ctr_transfer_1: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "list_all2 R [] []"
Gauss_Jordan_iarrays_PA_def: fixes A :: "'b iarray iarray" shows "Gauss_Jordan_iarrays_PA A = Gauss_Jordan_upt_k_iarrays_PA A (ncols_iarray A - 1)"
Henstock_lemma_part1: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and e :: "real"   and d :: "'a \<Rightarrow> 'a set"   and p :: "('a \<times> 'a set) set" assumes "f integrable_on cbox a b"   and "0 < e"   and "gauge d"   and "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - integral (cbox a b) f) < e"   and "p tagged_partial_division_of cbox a b"   and "d fine p" shows "norm (\<Sum>(x, K)\<in>p. content K *\<^sub>R f x - integral K f) \<le> e"
prems_2: shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
in_kD_case: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and en :: "'a"   and es :: "nat \<times> k \<times> f \<times> nat \<times> nat \<Rightarrow> 'a" assumes "dip \<in> kD rt" shows "(case rt dip of None \<Rightarrow> en | Some r \<Rightarrow> es r) = es (the (rt dip))"
comp_converter_parallel2: fixes conv1l :: "('h, 'j, 'p, 'r) converter"   and conv1r :: "('i, 'k, 'q, 's) converter"   and conv2l :: "('p, 'r, 'l, 'n) converter"   and conv2r :: "('q, 's, 'm, 'o) converter" shows "(conv1l |\<^sub>= conv1r) \<odot> (conv2l |\<^sub>= conv2r) = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r"
comm_semiring_1_cancel_ow_axioms_def: fixes U :: "'a set"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "comm_semiring_1_cancel_ow_axioms U minus times \<equiv> \<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times a (minus b c) = minus (times a b) (times a c)"
segment_inj: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "(a = b) = (segment D a = segment D b)"
Loss_lengthOut1: fixes a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and x :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list"   and t :: "nat" assumes "\<forall>t. length (a t) \<le> Suc 0"   and "Loss lose a i2 x i" shows "length (x t) \<le> Suc 0"
interior_Ici: fixes b :: "'b"   and x :: "'b" assumes "b < x" shows "interior {x..} = {x<..}"
Upper_antimono: fixes A :: "'a set"   and B :: "'a set"   and L :: "('a, 'b) gorder_scheme" assumes "A \<subseteq> B" shows "Upper L B \<subseteq> Upper L A"
prod_list_zero_iff: fixes xs :: "'c list" shows "(prod_list xs = (0::'c)) = ((0::'c) \<in> set xs)"
unit_disc_fix_iff: fixes M :: "moebius" assumes "unit_disc_fix M" shows "moebius_pt M ` unit_disc = unit_disc"
induct: fixes P :: "('c, 'd) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('c, 'd) FingerTreeStruc" assumes "P Empty"   and "\<And>a. P (Single a)"   and "\<And>uu_ a b m sf. P (Deep uu_ (Two a b) m sf)"   and "\<And>uv_ a b c m sf. P (Deep uv_ (Three a b c) m sf)"   and "\<And>uw_ a b c d m sf. P (Deep uw_ (Four a b c d) m sf)"   and "\<And>ux_ a m sf. P m \<Longrightarrow> P (Deep ux_ (One a) m sf)" shows "P a0"
summable_iff_convergent': fixes f :: "nat \<Rightarrow> 'a" shows "summable f = convergent (\<lambda>n. sum f {..n})"
lbound_ofI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "ordering less_eq less"   and "less_eq b x"   and "less_eq b y" shows "ordering.lbound_of less_eq x y b"
reflexive_axioms: shows "reflexive A (sympartp (\<sqsubseteq>))"
fmdrop_idle: fixes x :: "'c"   and B :: "('c, 'd) fmap" assumes "x |\<notin>| fmdom B" shows "fmdrop x B = B"
cong_midpoint__image: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p" assumes "Cong A X A Y"   and "B Midpoint X Y" shows "Y X Reflect A B"
simps_5: fixes t :: "'a local.term" shows "\<^bold>\<lfloor>\<^bold>\<l>\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
majorities_intersect: shows "\<forall>S T. IsMajority S \<and> IsMajority T \<longrightarrow> S \<inter> T \<noteq> {}"
simps: fixes a :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and p :: "nat \<times> 'c" shows "(a \<in> greachable A w V p) = (a = p \<or> (\<exists>q aa. a = aa \<and> q \<in> greachable A w V p \<and> (case q of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> aa \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V)))"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_eq: fixes A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
iso_is_retraction: fixes f :: "'d" assumes "D.iso f" shows "D.retraction f"
assm: shows "0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>x. \<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True))"
gen_wf_\<alpha>n_def: shows "gen_wf_\<alpha>n \<equiv> map_fun Rep_gen_cfg_wf id gen_\<alpha>n"
Rep_viewres_inverse: fixes x :: "('a, 'b) viewres" shows "Abs_viewres (Rep_viewres x) = x"
reindex_bij_witness: fixes S :: "'d set"   and i :: "'e \<Rightarrow> 'd"   and j :: "'d \<Rightarrow> 'e"   and T :: "'e set"   and h :: "'e \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "F g S = F h T"
assms: shows "\<forall>x1 x2. \<exists>y1 y2. P x1 x2 y1 y2"
iso_chain_group_sum: fixes \<U> :: "'b set set"   and X :: "'b topology"   and p :: "nat" assumes "disjoint \<U>"   and "\<Union> \<U> = topspace X"   and "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; \<not> disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T" shows "(\<lambda>f. sum' f \<U>) \<in> Group.iso (sum_group \<U> (\<lambda>S. chain_group p (subtopology X S))) (chain_group p X)"
psi_ubound_aux: fixes x :: "real"   and y :: "real" assumes "2 \<le> x"   and "x \<le> y" shows "(4 * ln y + 3) / (ln 2 * y) \<le> (4 * ln x + 3) / (ln 2 * x)"
freeword: fixes x :: "'a freeword" shows "freeword x \<in> {as. proper_signed_list as}"
extend_set_empty: fixes h :: "'e \<times> 'f \<Rightarrow> 'd" shows "extend_set h {} = {}"
unsigned_eq_0_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and w :: "'l word" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.unsigned one plus zero w = zero) = (w = 0)"
list_def: shows "list \<equiv> \<lambda>A. {x. listp (\<lambda>x. x \<in> A) x}"
pred_stream_cfg_on: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and xs :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" assumes "cfg \<in> MDP.valid_cfg"   and "MDP.MC.enabled cfg xs" shows "pred_stream (\<lambda>cfg. state cfg \<in> S \<and> cfg \<in> MDP.cfg_on (state cfg)) xs"
FunctorCompDef: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f"   and g :: "'f" assumes "Functor F"   and "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f \<approx>>\<^bsub>CatCod F\<^esub> F ## g"
1: fixes b :: "'a" assumes "b \<in> B" shows "g b \<in> A"
minus_pCons: fixes a :: "'a"   and p :: "'a Polynomial.poly" shows "- pCons a p = pCons (- a) (- p)"
inject: fixes x1 :: "'a extended"   and x2 :: "'a"   and x3 :: "'a extended"   and x4 :: "'b"   and y1 :: "'a extended"   and y2 :: "'a"   and y3 :: "'a extended"   and y4 :: "'b" shows "(InsertMessage x1 x2 x3 x4 = InsertMessage y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
bt_augment_simp: fixes t :: "('d, 'c) bintree"   and q :: "('c, 'd) pq" shows "bt_augment t q = PQ.push (val t) (priority t) (fold bt_augment (children t) q)"
is_done_oimpl_def: fixes v :: "'c"   and I :: "'c \<Rightarrow> node_state option" shows "is_done_oimpl v I \<equiv> case I v of None \<Rightarrow> False | Some (STACK nat) \<Rightarrow> False | Some DONE \<Rightarrow> True"
p_shunting_swap: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (x \<sqinter> z \<le> - y)"
retract_of_space_def: fixes S :: "'b set"   and X :: "'b topology" shows "S retract_of_space X \<equiv> S \<subseteq> topspace X \<and> (\<exists>r. continuous_map X (subtopology X S) r \<and> (\<forall>x\<in>S. r x = x))"
terminalE: fixes b :: "'a"   and a :: "'a" assumes "terminal b"   and "ide a" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<And>f'. \<guillemotleft>f' : a \<rightarrow> b\<guillemotright> \<Longrightarrow> f' = f"
B: shows "n \<notin> bad_shakey \<inter> bad_prikey"
invar1_finish: fixes A :: "('v \<times> 'v) set" assumes "prim_invar1 A"   and "edges g \<inter> S A \<times> - S A = {}" shows "is_MST w rg (graph {r} A)"
le_Inf_iff: fixes A :: "'b set"   and b :: "'b" shows "(\<Squnion> A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
Synchronized1Red2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and V :: "nat"   and a :: "'addr" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) e,s\<rangle> -ta\<rightarrow> \<langle>insync\<^bsub>V\<^esub> (a) e',s'\<rangle>"
uvw_arcs2_2: shows "(v, w) \<in> parcs G"
hyps_3: shows "X' \<sharp> (D' @ TVarB X Q # G)"
etp_axioms: shows "etp I domain range F F\<^sub>i\<^sub>n\<^sub>v"
preserves_assoc_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "F \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F h] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, h))) \<cdot>\<^sub>D D.inv (\<Phi> (f, g \<star>\<^sub>C h))"
normF_xx: fixes F :: "'a set"   and \<epsilon> :: "real"   and n :: "nat"   and m :: "nat" assumes "finite F"   and "0 < \<epsilon>"   and "I \<epsilon> \<le> n"   and "I \<epsilon> \<le> m" shows "normF F (\<lambda>a. x n a - x m a) \<le> \<epsilon>"
deleteMin_def: fixes bq :: "('c, 'd) BinomialTree list" shows "deleteMin bq \<equiv> let min = getMinTree bq in meld (rev (children min)) (remove1Prio (prio min) bq)"
col_transitivity_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "P \<noteq> Q"   and "Tarski_neutral_dimensionless.Col Bet P Q A"   and "Tarski_neutral_dimensionless.Col Bet P Q B" shows "Tarski_neutral_dimensionless.Col Bet P A B"
semidecidable_iff_domain: fixes X :: "nat set" shows "semidecidable X = (\<exists>f. recfn 1 f \<and> (\<forall>x. (eval f [x] \<down>) = (x \<in> X)))"
simps: fixes M :: "('a, 'b, 'c) FSM"   and f :: "'c \<Rightarrow> 'a list" shows "is_det_state_cover_ass M f = (f (initial M) = [] \<and> (\<forall>s\<in>d_reachable M (initial M). d_reaches M (initial M) (f s) s))"
cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) shadow_root_ptr" shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = document_ptr.Ext (Inl ptr)"
summable_ereal_pos: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "summable f"