hcju/isabelleps huggingface dataset:
not_shadows_vconsts_alt_def_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "not_shadows_vconsts (Vrecabs css name \<Gamma>) = (not_shadows_vconsts_env \<Gamma> \<and> pred_fmap (list_all (\<lambda>(pat, t). fdisjnt all_consts (frees pat) \<and> \<not> shadows_consts t)) css \<and> name |\<in>| fmdom css \<and> True)"
mset_nempty_set_nempty: fixes A :: "'b multiset" shows "(A \<noteq> {#}) = (set_mset A \<noteq> {})"
add_block_alt: fixes bl :: "'a set"   and b :: "'a set" shows "del_block_sys.add_block bl b = add_mset b (add_point_sys.del_block bl)"
list_of_vec_map: fixes xs :: "'b Matrix.vec" shows "list_of_vec xs = map (($v) xs) [0..<dim_vec xs]"
integral_f: fixes k :: "nat" shows "integral {0..c} (f k) = F k"
set_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and node_ptrs :: "(_) node_ptr list" assumes "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok set_disconnected_nodes document_ptr node_ptrs"
exp_def: fixes b :: "'a"   and c :: "'a" shows "exp b c \<equiv> SOME x. ide x \<and> (\<exists>e. \<guillemotleft>e : local.prod x b \<rightarrow> c\<guillemotright> \<and> (\<forall>a g. ide a \<and> \<guillemotleft>g : local.prod a b \<rightarrow> c\<guillemotright> \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> x\<guillemotright> \<and> g = C e (local.prod f b))))"
rev_gext_objD: fixes s :: "st"   and r :: "oref"   and obj :: "obj"   and s' :: "st" assumes "globs s r = Some obj"   and "s\<le>|s'" shows "\<exists>obj'. globs s' r = Some obj' \<and> tag obj' = tag obj"
eq_neg_iff_add_eq_0: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (plus a b = zero)"
a1: shows "snd (dispatch_instruction instr s) = False \<and> s' = snd (fst (dispatch_instruction instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
src_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src\<^sub>B (trg\<^sub>B \<mu>) = trg\<^sub>B \<mu>"
simps_5: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree" shows "is_heap (T v (T va vb vc) (T vd ve vf)) = (val (T vd ve vf) \<le> v \<and> is_heap (T vd ve vf) \<and> val (T va vb vc) \<le> v \<and> is_heap (T va vb vc))"
bfs_spec_def: fixes src :: "nat"   and dst :: "nat"   and r :: "(nat \<times> (nat \<Rightarrow> nat option)) option" shows "bfs_spec src dst r \<equiv> case r of None \<Rightarrow> \<not> local.connected src dst | Some (d, PRED) \<Rightarrow> local.connected src dst \<and> min_dist src dst = d \<and> valid_PRED c src PRED \<and> dst \<in> dom PRED"
set_attribute_writes: fixes element_ptr :: "(_) element_ptr"   and k :: "char list"   and v :: "char list option"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_attribute_locs element_ptr) (set_attribute element_ptr k v) h h'"
eexp_lt_eexp_iff_lt: fixes x :: "ereal"   and y :: "ereal" shows "(eexp x < eexp y) = (x < y)"
wf_eq_fv: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term"   and S' :: "('b, 'a) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ \<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')" shows "fv t' \<subseteq> wfvarsoccs\<^sub>s\<^sub>t S \<union> V"
o_id: fixes f :: "'b \<Rightarrow> 'c" shows "f \<circ> id = f"
current_next_state: shows "\<forall>s execs. current (int.next_state s execs) = current s"
del_unEdge_connectivity: fixes v :: "'v"   and e :: "'w"   and v' :: "'v" assumes "local.connected"   and "\<exists>ps. valid_graph.is_path (del_unEdge v e v' G) v ps v'" shows "valid_unMultigraph.connected (del_unEdge v e v' G)"
Union_rel_def: fixes L :: "('c \<times> 'e set) set"   and S :: "('e \<times> 'd set) set" shows "\<langle>L, S\<rangle>Union_rel = L O \<langle>S\<rangle>set_rel O br \<Union> top"
poly_mod_minus_left: fixes x :: "'b poly"   and y :: "'b poly" shows "- x mod y = - (x mod y)"
indeps_6: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" assumes "myss2 x y a z" shows "a \<bowtie> z"
iterative_HSem': fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value"   and e :: "exp" assumes "x \<notin> domA \<Gamma>" shows "(\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<lbrace>\<Gamma>\<rbrace>xa))(x := \<lbrakk> e \<rbrakk>\<^bsub>xa\<^esub>)) = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<lbrace>\<Gamma>\<rbrace>xa))(x := \<lbrakk> e \<rbrakk>\<^bsub>\<lbrace>\<Gamma>\<rbrace>xa\<^esub>))"
satisfies_nFOr: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile> nFOr \<phi> \<psi>) = (\<AA> \<Turnstile> FOr \<phi> \<psi>)"
factor_bound: fixes f :: "int poly"   and g :: "int poly"   and n :: "nat"   and k :: "nat" assumes "f \<noteq> 0"   and "g dvd f"   and "degree g \<le> n" shows "\<bar>coeff g k\<bar> \<le> factor_bound f n"
cmp_simps_1: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.arr (cmp (f, g))"
assms: fixes i :: "nat" assumes "i < max_Var_form f" shows "xs ! i = ys ! i"
yy_2: shows "y \<notin> Fvars \<phi>"
convex_hull_Int_subset: fixes S :: "'a set"   and T :: "'a set" assumes "\<not> affine_dependent (S \<union> T)" shows "convex hull S \<inter> convex hull T \<subseteq> convex hull (S \<inter> T)"
inv_comp_right_2: fixes g :: "'b"   and f :: "'b" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso f" shows "D.iso g"
simp_Times_simps: fixes p1 :: "'b rexp \<times> ('c val \<Rightarrow> 'c val)"   and p2 :: "'b rexp \<times> ('c val \<Rightarrow> 'c val)" shows "simp_Times p1 p2 = (if fst p1 = One then (fst p2, F_Times1 (snd p1) (snd p2)) else if fst p2 = One then (fst p1, F_Times2 (snd p1) (snd p2)) else (Times (fst p1) (fst p2), F_Times (snd p1) (snd p2)))"
assms_6: fixes p :: "nat" assumes "prime p" shows "f p \<noteq> 0 \<and> f p \<noteq> h p"
smult_eq_iff: fixes b :: "'c"   and a :: "'c"   and p :: "'c poly"   and q :: "'c poly" assumes "b \<noteq> (0::'c)" shows "(smult a p = smult b q) = (smult (a / b) p = q)"
set_2: fixes x2 :: "'a" shows "set_option (Some x2) = {x2}"
card_less_sym_Diff: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A < card B" shows "card (A - B) < card (B - A)"
invertible_inverse_closed: fixes u :: "'a" assumes "comm.additive.invertible u"   and "u \<in> R" shows "comm.additive.inverse u \<in> R"
epi_char: fixes f :: "'s" shows "epi f = (arr f \<and> (Img f = Cod f \<or> (\<forall>t t'. t \<in> Univ \<and> t' \<in> Univ \<longrightarrow> t = t')))"
simps: fixes i :: "Integer"   and j :: "Integer" shows "supto\<cdot>i\<cdot>j = If le\<cdot>i\<cdot>j then i :# supto\<cdot>(i + 1)\<cdot>j else [::]"
"2_3": fixes x :: "'t list"   and xa :: "'t \<Rightarrow>\<^sub>0 'b" assumes "x = new_syz_sigs ss bs p"   and "\<not> sig_crit bs x p"   and "xa = sig_trd bs (poly_of_pair p)"   and "rep_list xa = 0" shows "snd (snd (fst (rb_aux ((bs, lt xa # x, ps), Suc z)))) = []"
healthy_unitary: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "healthy t"   and "unitary P" shows "unitary (t P)"
sig_red_singleD8: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "t \<oplus> lt f \<preceq>\<^sub>t lt p"
eq: shows "h = fds (\<lambda>n. if n = 1 then l * g 1 else fds_nth f n * g n * mangoldt n /\<^sub>R ln (real n))"
ord_pp_def: fixes cmp_term :: "'b nat_term_order"   and s :: "'b"   and t :: "'b" shows "ord_pp_punit cmp_term s t = le_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))"
pmf_Pi': fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = dflt" shows "pmf (Pi_pmf A dflt p) f = (\<Prod>x\<in>A. pmf (p x) (f x))"
a_set_child_nodes_locs_def: fixes ptr :: "(_) object_ptr" shows "local.a_set_child_nodes_locs ptr \<equiv> (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes_update) else {}) \<union> (if is_document_ptr_kind ptr then all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^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_element_update) else {})"
dagger_prod_unfold: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "Id \<union> (x O (y O x)\<^sup>*) O y = (x O y)\<^sup>*"
h_plus_le_g: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_plus \<Delta> g i e \<le> g e"
lnull_wit: fixes xs :: "'a llist" shows "lnull (local.wit xs) = lnull xs"
isNotDSource_def: fixes i :: "AbstrLevelsID"   and S :: "CSet" shows "isNotDSource i S \<equiv> \<forall>x\<in>OUT S. \<forall>Z\<in>AbstrLevel i. x \<notin> IN Z"
ab_semigroup_mult_hom_axioms: shows "ab_semigroup_mult_hom (map_poly hom)"
antisym: fixes a :: "'al"   and b :: "'al" assumes "a \<in> U"   and "b \<in> U"   and "a \<le>\<^sub>o\<^sub>w b"   and "b \<le>\<^sub>o\<^sub>w a" shows "a = b"
rpd_fv_regex: fixes s :: "'d regex"   and test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and fv :: "'d \<Rightarrow> 'f set" assumes "s \<in> rpd test i r" shows "fv_regex fv s \<subseteq> fv_regex fv r"
sym_diff_inc: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" shows "A \<Delta> C \<subseteq> A \<Delta> B \<union> B \<Delta> C"
abs_eq: fixes x :: "'d" shows "bsingleton x = Abs_bset {x}"
not_arg_cong_Inr: fixes x :: "'g"   and y :: "'g" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
intro: fixes B :: "'e set"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "\<And>x y z. \<lbrakk>x \<in> B; y \<in> B; z \<in> D\<rbrakk> \<Longrightarrow> f x (f y z) = f y (f x z)"   and "\<And>x y. \<lbrakk>x \<in> B; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D" shows "LCD B D f"
sup_ge1: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<subseteq># x \<union># y"
aux: shows "X \<union> weakBisim = {(P, Q). (P, Q) \<in> X \<or> P \<approx> Q}"
elements_mat_h_def': fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "elements_mat_h A = {A $h i $h j |i j. True}"
rett_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "rett_instr instr \<equiv> let op_list = snd instr in do psr_val \<leftarrow> gets (cpu_reg_val PSR); curr_win \<leftarrow> gets (\<lambda>s. get_CWP (cpu_reg_val PSR s)); new_cwp \<leftarrow> gets (\<lambda>s. word_of_int ((uint curr_win + 1) mod NWINDOWS)); new_cwp_int \<leftarrow> gets (\<lambda>s. (uint curr_win + 1) mod NWINDOWS); addr \<leftarrow> gets (get_addr op_list); et_val \<leftarrow> gets (\<lambda>s. ucast (get_ET psr_val)); s_val \<leftarrow> gets (\<lambda>s. ucast (get_S psr_val)); ps_val \<leftarrow> gets (\<lambda>s. ucast (get_PS psr_val)); wim_val \<leftarrow> gets (cpu_reg_val WIM); npc_val \<leftarrow> gets (cpu_reg_val nPC); if et_val = 1 then if s_val = 0 then do raise_trap privileged_instruction; return () od else do raise_trap illegal_instruction; return () od else if s_val = 0 then do write_cpu_tt 3; set_exe_mode False; set_err_mode True; raise_trap privileged_instruction; fail () od else if get_WIM_bit (nat new_cwp_int) wim_val \<noteq> 0 then do write_cpu_tt 6; set_exe_mode False; set_err_mode True; raise_trap window_underflow; fail () od else if addr AND 3 \<noteq> 0 then do write_cpu_tt 7; set_exe_mode False; set_err_mode True; raise_trap mem_address_not_aligned; fail () od else do write_cpu npc_val PC; write_cpu addr nPC; new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_rett new_cwp 1 ps_val psr_val); write_cpu new_psr_val PSR; return () od od"
approximating_bigstep_to_decision1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and Y :: "final_decision"   and X :: "final_decision" assumes "\<gamma>,p\<turnstile> \<langle>rs, Decision Y\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision X" shows "Y = X"
Gr_incl: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" shows "(BNF_Def.Gr A f \<subseteq> A \<times> B) = (f ` A \<subseteq> B)"
list_update_le_cong: fixes i :: "nat"   and xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and x :: "'b"   and y :: "'b" assumes "i < length xs"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y" shows "xs[i := x] [\<sqsubseteq>\<^bsub>r\<^esub>] ys[i := y]"
le_cong: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> y"   and "z .=\<^bsub>L\<^esub> w"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L"   and "w \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> z) = (y \<sqsubseteq>\<^bsub>L\<^esub> w)"
pmf_of_list_remove_zeros_2: fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs" shows "pmf_of_list (filter (\<lambda>z. snd z \<noteq> 0) xs) = pmf_of_list xs"
order_iff_strict: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b) = (a \<subset># b \<or> a = b)"
quasi_inversesE: fixes f :: "'a"   and g :: "'a" assumes "quasi_inverses f g" obtains "ide f" and "ide g" and "src f \<cong> g \<star> f" and "f \<star> g \<cong> trg f"
smult_sum_distrib: fixes r :: "('f, 'g) aezfun"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'v" assumes "r \<in> FG"   and "finite A"   and "f ` A \<subseteq> V" shows "r \<cdot> sum f A = (\<Sum>a\<in>A. r \<cdot> f a)"
theFTOne_length_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'e) cmd list"   and dl :: "('f, 'g, 'h) cmd list"   and n :: "nat" assumes "PL_Indis cval indis"   and "length cl = length dl"   and "n \<in> theFTOne cl dl" shows "n < length cl"
hyps_2: fixes H :: "'a pair_pre_digraph" assumes "I = with_proj H"   and "rev_I = swap_in (parcs H)" shows "pair_sd G H"
simps: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" shows "((a1, a2) \<in> upRules) = (\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> ModalSequents.mset c = \<LM> Compound R Fs  \<RM> \<and> ps \<noteq> [])"
append_Nil: fixes ys :: "'c list" shows "[] @ ys = ys"
closure_approachable: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> closure S) = (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
surjective: fixes r :: "'b cbody_scheme" shows "r = \<lparr>access = access r, cfields = cfields r, methods = methods r, init = init r, \<dots> = cbody.more r\<rparr>"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>t s. a1 = t \<and> a2 = s \<and> wt t >\<^sub>p wt s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> \<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s \<and> num_args t = 1 \<and> (\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0) \<and> arg t \<ge>\<^sub>t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> head t >\<^sub>h\<^sub>d head s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> head t = head s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s))))"
pullback_commutes: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "commutative_square f g (prj1 f g) (prj0 f g)"
tcontra_unhalt: assumes "\<not> halts tcontra [code_tcontra]" shows "False"
l9_41_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and P :: "'a"   and R :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TSP Bet A B C P R"   and "Tarski_neutral_dimensionless.TSP Bet A B C Q R" shows "Tarski_neutral_dimensionless.OSP Bet A B C P Q"
euler_incr_fas_def: fixes ode_ops :: "ode_ops"   and X0 :: "floatarith list"   and h :: "floatarith"   and CX :: "floatarith list" shows "ode_operations.euler_incr_fas ode_ops X0 h CX = map (ode_operations.euler_incr_fas_nth ode_ops X0 h CX) [0..<length X0]"
hn_ctxt_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a :: "'c"   and c :: "'d" shows "hn_ctxt P a c \<equiv> P a c"
assoc: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
intro: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool" assumes "nominal_ts satisfies transition"   and "indexed_nominal_ts_axioms TYPE('d) TYPE('a)" shows "indexed_nominal_ts TYPE('d) satisfies transition"
assms_3: shows "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> z"
unat_minus_one_word: shows "unat (- 1) = 2 ^ LENGTH('b) - 1"
eps_def: shows "eps = (\<lambda>n. d2 * (1 / 2) ^ n)"
monom_a_inv: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
is_fmapI: fixes M :: "('a \<times> 'b) fset" assumes "\<And>a b\<^sub>1 b\<^sub>2. \<lbrakk>(a, b\<^sub>1) |\<in>| M; (a, b\<^sub>2) |\<in>| M\<rbrakk> \<Longrightarrow> b\<^sub>1 = b\<^sub>2" shows "is_fmap M"
division_segment_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment one = one"
jointfun_surj: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and A :: "'c set"   and g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and B :: "'c set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "surj_to f {j. j \<le> n} A"   and "g \<in> {j. j \<le> m} \<rightarrow> B"   and "surj_to g {j. j \<le> m} B" shows "surj_to (jointfun n f m g) {j. j \<le> Suc (n + m)} (A \<union> B)"
class_assoc_end_type_def: fixes "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" shows "class_assoc_end_type end \<equiv> \<langle>assoc_end_class end\<rangle>\<^sub>\<T>[1]"
assms: shows "l icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
coerce_Leaf: fixes x :: "'b" shows "COERCE('b\<cdot>btree, 'a\<cdot>btree)\<cdot>(Leaf\<cdot>x) = Leaf\<cdot>(COERCE('b, 'a)\<cdot>x)"
mzero_def: shows "mzero = COERCE(udom\<cdot>'c, 'b\<cdot>'c)\<cdot>zeroU"
resolution_not_empty: assumes "\<B> \<noteq> {#}" shows "\<P> \<noteq> {#}"
rsr2: fixes y :: "'a"   and x :: "'a" shows "ad (ad (y \<cdot> ad (ad x))) = ad (ad (y \<cdot> x))"
grid_multi_dimensional_specification: fixes dm :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" assumes "dm \<le> length b"   and "length p = length b"   and "\<And>d. d < dm \<Longrightarrow> odd (ix p d) \<and> lv b d \<le> lv p d \<and> ix p d < (ix b d + 1) * 2 ^ (lv p d - lv b d) \<and> (ix b d - 1) * 2 ^ (lv p d - lv b d) < ix p d"   and "\<And>d. \<lbrakk>dm \<le> d; d < length b\<rbrakk> \<Longrightarrow> p ! d = b ! d" shows "p \<in> grid b {0..<dm}"
inject: fixes x21 :: "'c"   and x22 :: "'c list"   and y21 :: "'c"   and y22 :: "'c list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
chine_simps_2: shows "C.dom f.chine = f.apex"
simps_1: fixes uu :: "predicate"   and uv :: "'a list" shows "is_STRIPS_fmla (Atom (predAtm uu uv)) = True"
k: shows "a - k *s b \<in> span (S - {b})"
hyps_1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
lexord_simps_1: fixes y :: "'b list"   and r :: "('b \<times> 'b) set" shows "(([], y) \<in> lexord r) = (\<exists>a x. y = a # x)"
dma_type_eq: shows "type_max (dma_type x) mem\<^sub>1 = type_max (dma_type x) mem\<^sub>2"
assms_3: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary (Suc p) (h c) + h' (chain_boundary p c) = f c - g c"
mult_neg_numeral_simps_3: 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 (numeral.numeral one plus m) (uminus (numeral.numeral one plus n)) = uminus (numeral.numeral one plus (m * n))"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c llist"   and y :: "'d llist"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "llist_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> lset x; yb \<in> lset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "llist_all2 Ra x y"
NatMultiples_def: shows "NatMultiples \<equiv> \<lambda>F. {x. NatMultiplesp (\<lambda>x. x \<in> F) x}"
poly_mod_type_simps_1: fixes m :: "int" shows "poly_mod_type TYPE('b) m = (m = int CARD('b))"
simps_11: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and y1 :: "'b" shows "rel_vertex R (Vertex x1) (Vertex y1) = R x1 y1"
derivs_Zero: fixes xs :: "'b list" shows "derivs xs Zero = Zero"
simps: fixes s1 :: "('a, 'b, 'c) SLin_state"   and s2 :: "('a, 'e, 'f, 'g) SLin_state_scheme" shows "P8 (s1, s2) = (\<forall>iv\<in>initVals s2. \<exists>rs\<in>pendingSeqs s1. iv = dstate s1 \<star> rs)"
eval_dioph_div_mod: fixes ks :: "int list"   and xs :: "nat list"   and l :: "int" shows "(eval_dioph ks xs = l) = (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and> eval_dioph ks (map (\<lambda>x. x div 2) xs) = (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"
enc: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"