hcju/isabelleps huggingface dataset:
lprefix_expand: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<Longrightarrow> \<not> lnull ys \<and> lhd xs = lhd ys \<and> lprefix (ltl xs) (ltl ys)" shows "lprefix xs ys"
perfect_map_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "perfect_map X Y f \<equiv> continuous_map X Y f \<and> proper_map X Y f \<and> f ` topspace X = topspace Y"
\<phi>: shows "\<guillemotleft>\<phi> : w \<Rightarrow> w'\<guillemotright> \<and> local.iso \<phi>"
zero_closed: fixes G :: "'a set" assumes "AbGroup G" shows "(0::'a) \<in> G"
simps: fixes a :: "'c set"   and sp :: "'c set"   and A :: "'c set set" shows "(a \<in> sigma_sets sp A) = ((\<exists>aa. a = aa \<and> aa \<in> A) \<or> a = {} \<or> (\<exists>aa. a = sp - aa \<and> aa \<in> sigma_sets sp A) \<or> (\<exists>aa. a = \<Union> (range aa) \<and> (\<forall>x. aa x \<in> sigma_sets sp A)))"
sorted_lower: shows "sorted_sublist_map R h xs' lo (p - Suc 0)"
d_diff_def: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_diff g x = d_OUT g x - d_IN g x"
hoare_triple_if: shows "\<turnstile>\<^sub>p {tensor_P post (1\<^sub>m K)} Measure_P vars1 N testN (replicate N SKIP) {tensor_P post (1\<^sub>m K)}"
assms_2: shows "(v, a) \<in> set (precondition_of op) \<or> (v, a) \<in> set (effect_of op)"
Positive_def: shows "Positive \<equiv> Direction (<) \<B>\<^sub>i\<^sub>l \<B>\<^sub>i\<^sub>u \<B>\<^sub>l \<B>\<^sub>u \<I>\<^sub>l \<I>\<^sub>u \<B>\<^sub>i\<^sub>u_update Leq Geq (\<le>)"
e': fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space M" shows "expr_sem \<sigma> e' = return_val (expr_sem_rf \<sigma> e')"
strict_trans2: fixes b :: "'c word"   and a :: "'c word"   and c :: "'c word" assumes "b <s a"   and "c \<le>s b" shows "c <s a"
prems_2: shows "vs all_in\<^sub>i range_tms e VS"
map_upd_triv: fixes t :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and x :: "'d" assumes "t k = Some x" shows "t(k \<mapsto> x) = t"
assms_1: shows "(x, y) \<in> Validity.alpha_Tree_rel"
fv_transform: fixes a :: "Arity"   and e :: "exp" shows "fv (ccTransform a e) \<subseteq> fv e"
hs_ins_impl: shows "imp_set_ins is_hashset hs_ins"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "idom_hom hom" shows "map_poly_idom_hom hom"
I2: assumes "Can u" shows "\<lbrace>Inv u\<rbrace> = local.inv \<lbrace>u\<rbrace>"
A: shows "((\<lambda>t. complex_of_real t powr (complex_of_real x - 1) / complex_of_real (exp t)) has_integral complex_of_real (Gamma x)) {0..}"
subst_sym: fixes s1 :: "('b \<times> 'b trm) list"   and s2 :: "('b \<times> 'b trm) list" assumes "s1 \<doteq> s2" shows "s2 \<doteq> s1"
exp_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "exp x \<in>\<^sub>r exp_float_interval p X"
finite_extension_finite_dimension_1: fixes K :: "'a set"   and xs :: "'a list" assumes "subfield K R"   and "set xs \<subseteq> carrier R"   and "\<And>x. x \<in> set xs \<Longrightarrow> (algebraic over K) x" shows "finite_dimension K (finite_extension K xs)"
card_eq_1_iff: fixes A :: "'a set" shows "(card A = 1) = (\<exists>x. A = {x})"
finprod_reindex_bij_betw: fixes h :: "'e \<Rightarrow> 'f"   and S :: "'e set"   and T :: "'f set"   and g :: "'f \<Rightarrow> 'a" assumes "bij_betw h S T"   and "g \<in> h ` S \<rightarrow> carrier G" shows "(\<Otimes>x\<in>S. g (h x)) = finprod G g T"
i_set_cont_FROM: fixes n :: "nat" shows "i_set_cont_minp [n\<dots>]"
sums_iff_shift: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "'a" shows "(\<lambda>i. f (i + n)) sums s = f sums (s + sum f {..<n})"
t_wise_balance_axioms: shows "t_wise_balance \<V> \<B> 2 \<Lambda>"
vec_of_list_index: fixes xs :: "'a list"   and j :: "nat" shows "vec_of_list xs $ j = xs ! j"
trivial_limit_within: fixes a :: "'a"   and S :: "'a set" shows "(at a within S = bot) = (\<not> a islimpt S)"
apath_nonempty_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "apath u p v"   and "p \<noteq> []" shows "u \<noteq> v"
transp_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less_eq"
Suc_pos: fixes k :: "nat"   and r :: "nat" assumes "Suc k \<le> r" shows "0 < r"
simps_1: shows "consequent andI = [Fun and [X, Y]]"
2: shows "\<forall>i. (mtop \<odot> f) (i, j) \<noteq> bot \<and> (\<forall>k. (mtop \<odot> f) (i, j) = (mtop \<odot> f) (k, j)) \<and> (\<forall>k. j \<noteq> k \<longrightarrow> (mtop \<odot> f) (i, k) = bot)"
bij_lcoset_perm: fixes e :: "'g set"   and g :: "'g set" assumes "Group e \<and> order.greater_eq G e"   and "normal e"   and "g \<in> G // LCoset_rel e" shows "bij (restrict1 ((+) g) (G // LCoset_rel e))"
fact_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 2 = numeral.numeral one plus (num.Bit0 num.One)"
closure_is_superset: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set" assumes "decomp_closure M M'" shows "M \<subseteq> M'"
approx_eq_imp: fixes a :: "'a star"   and b :: "'a star" assumes "a = b" shows "a \<approx> b"
field_Taylor: fixes S :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "real"   and w :: "'a"   and z :: "'a" assumes "convex S"   and "\<And>i x. \<lbrakk>x \<in> S; i \<le> n\<rbrakk> \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f (Suc n) x) \<le> B"   and "w \<in> S"   and "z \<in> S" shows "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / fact i)) \<le> B * norm (z - w) ^ Suc n / fact n"
assms_1: shows "[int y * z * x = y' * x] (mod int (order \<G>))"
inj_on_imageI: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "inj_on (g \<circ> f) A" shows "inj_on g (f ` A)"
zip_with_altdef: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" shows "zip_with f xs ys = map2 f xs ys"
square_free_poly_def: fixes Gcd :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> 'c poly"   and p :: "'c poly" shows "square_free_poly Gcd p = (if p = 0 then 0 else yun_gcd.square_free_monic_poly Gcd (smult (inverse (lead_coeff p)) p))"
simps_1: fixes y :: "Vars.var"   and z :: "Vars.var" shows "[][y::h=z] = []"
hb_broadcast_exists2: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and m2 :: "'c" assumes "network history msg_id"   and "network.hb history m1 m2" shows "\<exists>i. Broadcast m2 \<in> set (history i)"
weakenStatImp_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q \<equiv> \<exists>Q'. \<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q' \<and> insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi> \<and> (\<Psi>, P, Q') \<in> Rel"
spn_hcomp: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<tau> :: "'b"   and \<mu> :: "'b"   and g :: "'b"   and f :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V (V \<tau> \<mu>)"   and "category.isomorphic V g (bicategory_of_spans.spn V H \<a> \<i> src trg \<tau>)"   and "category.isomorphic V f (bicategory_of_spans.spn V H \<a> \<i> src trg \<mu>)" shows "category.isomorphic V (bicategory_of_spans.spn V H \<a> \<i> src trg (V \<tau> \<mu>)) (H g f)"
ImpL_inv': fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n \<and> G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
Dp_Mp_eq: fixes f :: "int poly"   and m :: "int" shows "f = poly_mod.Mp m f + Polynomial.smult m (poly_mod.Dp m f)"
ideal_explicit: fixes 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 S :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "ring.ideal_generated plus zero uminus times S = {y. \<exists>f U. finite U \<and> U \<subseteq> S \<and> comm_monoid_add.sum plus zero (\<lambda>i. times (f i) i) U = y}"
AF_def: shows "AF \<equiv> \<lambda>k i. enum (nxt N (Max (DF k i))) ` {..<\<Sqinter> (DF k i)}"
bind_pure_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> pure (g x) h" shows "pure (Heap_Error_Monad.bind f g) h"
complex_Re_of_nat: fixes n :: "nat" shows "Re (of_nat n) = real n"
abs_of_nonpos: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a zero" shows "abs a = uminus a"
lossless_eval: fixes c_o :: "'c_o"   and c_a :: "'c_a"   and s :: "'s"   and s' :: "'s"   and "guess" :: "'guess" assumes "(c_o, c_a, s) \<in> set_spmf init" shows "lossless_spmf (eval c_o c_a s' guess)"
unit_l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "unit x" shows "(x ** y = x ** z) = (y = z)"
wtl_inst_def: fixes cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and pc :: "nat"   and s :: "'c" shows "wtl_inst cert f r T step pc s = merge cert f r T pc (step pc s) (cert ! (pc + 1))"
con3: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and err :: "v error_result" assumes "do_con_check (c env) cn (length es)"   and "list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs"   and "sequence_result rs = Rerr err" shows "cupcake_evaluate_single env (Con cn es) (Rerr err)"
remove_for_all_empty_children: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> forall_M Shadow_DOM.remove children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r []"
finfun_curry_const: fixes c :: "'c" shows "finfun_curry (finfun_const c) = finfun_const (finfun_const c)"
certificate_characterization: shows "(K\<^bsub>3,3\<^esub> (with_proj (gen_contr_graph (with_proj G) (verts3 (with_proj G)))) \<or> K\<^bsub>5\<^esub> (with_proj (gen_contr_graph (with_proj G) (verts3 (with_proj G))))) = (\<exists>H. (K\<^bsub>3,3\<^esub> (with_proj H) \<or> K\<^bsub>5\<^esub> (with_proj H)) \<and> subdivision_pair H slim \<and> verts3 (with_proj G) = verts3 (with_proj slim))"
igSubstIGOpSTR_imp_igSubstIGOp: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstIGOpSTR Delta MOD" shows "FixSyn.igSubstIGOp Delta MOD"
a_export': fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> y) = d x + ad y"
bex_Un: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>A \<union> B. P x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x))"
autoref_monadicI2_1: fixes b :: "'a nres"   and a :: "'b nres"   and R :: "('a \<times> 'b) set"   and c :: "'a dres" assumes "(b, a) \<in> \<langle>R\<rangle>nres_rel"   and "nres_of c \<le> b" shows "(nres_of c, a) \<in> \<langle>R\<rangle>nres_rel"
hyps_1: shows "c \<noteq> 0 \<or> cs \<noteq> 0"
diff_cancel: fixes A :: "'c multiset" shows "A - A = {#}"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (R f)"
preal_add_assoc: fixes x :: "preal"   and y :: "preal"   and z :: "preal" shows "x + y + z = x + (y + z)"
UINF_mem_UNIV: fixes P :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> x \<in> UNIV \<bullet> P x) = (\<Sqinter> x \<bullet> P x)"
ta_bisim_empty: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool" shows "ta_bisim bisim \<lbrace>\<rbrace> \<lbrace>\<rbrace>"
ind_is_open_def: fixes U :: "'a set" shows "it.ind_is_open U \<equiv> U \<subseteq> U \<and> (\<exists>V\<subseteq>S. is_open V \<and> U = U \<inter> V)"
subst_def: fixes x :: "'c"   and t :: "('d, 'c) Term.term" shows "subst x t = Var(x := t)"
trace_snoc: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c'' :: "('a, 'b, 'c) configuration" assumes "trace c t c'"   and "c' \<turnstile> ev \<mapsto> c''" shows "trace c (t @ [ev]) c''"
arrI: fixes f :: "'t"   and a :: "'t"   and b :: "'t" assumes "\<guillemotleft>f : a \<rightarrow>' b\<guillemotright>" shows "S'.arr f"
distinct_5: fixes x1 :: "('e, 'f, 'g) Tree set['e]"   and x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree" shows "tConj x1 \<noteq> tAct x41 x42"
igSubstIGVar1STR_imp_igSubstIGVar1: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSubstIGVar1STR MOD" shows "igSubstIGVar1 MOD"
trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "prefix b a"   and "prefix c b" shows "prefix c a"
VDM_validn_def: fixes n :: "nat"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows " \<Turnstile>\<^sub>n c : A  = (\<forall>m\<le>n. \<forall>s t.  s , c \<rightarrow>\<^sub>m  t  \<longrightarrow> A s t)"
edges_conv_Un_Edges: fixes f :: "face"   and x :: "nat"   and y :: "nat" assumes "distinct (vertices f)"   and "x \<in> \<V> f"   and "y \<in> \<V> f"   and "x \<noteq> y" shows "\<E> f = Edges (x # between (vertices f) x y @ [y]) \<union> Edges (y # between (vertices f) y x @ [x])"
insert_union: fixes x :: "'a"   and X :: "'a set" shows "insert x X = X \<union> {x}"
split_indicator_asm: fixes P :: "'c \<Rightarrow> bool"   and S :: "'d set"   and x :: "'d" shows "P (indicator S x) = (\<not> (x \<in> S \<and> \<not> P (1::'c) \<or> x \<notin> S \<and> \<not> P (0::'c)))"
suppa_distincts: 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 D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D E F" shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F"
pass12_merge_pairs: fixes hs :: "'b hp list" shows "pass\<^sub>2 (pass\<^sub>1 hs) = merge_pairs hs"
basis_reduction_add_rows_impl_1: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and state' :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and fs' :: "int vec list" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "LLL_Impl.basis_reduction_add_rows n upw i state = state'"   and "i < m"   and "fs' = fs_state state'" shows "LLL_impl_inv state' i fs'"
simps_1: fixes x :: "'c"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "Var x \<cdot> \<sigma> = \<sigma> x"
Axiom_of_Unions: fixes \<alpha> :: "V"   and a :: "V" assumes "\<Z> \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> a \<in>\<^sub>\<circ> Vset \<alpha>"
ge_l: fixes a :: "MIR.num" shows "isrlfm (rsplit ge a)"
is_sig_red_sing_reg_cases: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_sig_red (\<preceq>\<^sub>t) top_tail F p = (is_sig_red (=) top_tail F p \<or> is_sig_red (\<prec>\<^sub>t) top_tail F p)"
in_imp_not_empty: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "A \<noteq> {}"
ordered_insert_pairs_map: fixes x :: "'c"   and v :: "'d"   and ys :: "('c \<times> 'd) list" shows "map_of_alist (ordered_insert_pairs x v ys) =  map_of_alist ys { x \<rightarrow> v }"
SAStates_notmem_Chi: fixes S :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs A" shows "T \<notin> Chi A S"
ContractsWithSubstitutes_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithSubstitutes Xd Xh Pd Ch \<equiv> Contracts Xd Xh Pd Ch \<and> ContractsWithSubstitutes_axioms Ch"
plus_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "Mp f + g =m f + g"
is_none_simps_2: fixes x :: "'c" shows "\<not> Option.is_none (Some x)"
strip_comb_def: fixes u :: "Core.term" shows "strip_comb u = strip_comb_imp (u, [])"
subgroup_generated_group_carrier: shows "subgroup_generated C (carrier C) = C"
take_Cons': fixes n :: "nat"   and x :: "'e"   and xs :: "'e list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
in_matched_in_HRB_slice: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "matched n ns n'"   and "n'' \<in> set ns"   and "n' \<in> S" shows "n'' \<in> HRB_slice S"
mu_refine_intro: fixes F :: "'a upred \<Rightarrow> 'a upred"   and C :: "'a upred"   and S :: "'a upred" assumes "(C \<Rightarrow> S) \<sqsubseteq> F (C \<Rightarrow> S)"   and "(C \<and> \<mu> F) = (C \<and> \<nu> F)" shows "(C \<Rightarrow> S) \<sqsubseteq> \<mu> F"
resolvable_def: fixes A :: "'a"   and D :: "'a literal list"   and CA :: "'a literal list"   and Ls :: "'a literal list" shows "resolvable A D CA Ls = (let \<sigma> = mgu {insert A (atms_of (mset Ls))} in \<sigma> \<noteq> None \<and> Ls \<noteq> [] \<and> maximal_wrt (A \<cdot>a the \<sigma>) (add_mset (Neg A) (mset D) \<cdot> the \<sigma>) \<and> strictly_maximal_wrt (A \<cdot>a the \<sigma>) ((mset CA - mset Ls) \<cdot> the \<sigma>) \<and> (\<forall>L\<in>set Ls. is_pos L))"
qe_eps_def: shows "qe_eps = DLO.lift_nnf_qe qe_eps\<^sub>1"
PDer: shows "derivative P' ([(x, y)] \<bullet> Q') a y Rel"
induces_pseudofunctor_between_opposites: shows "pseudofunctor (\<cdot>\<^sub>C) (\<lambda>f g. g \<star>\<^sub>C f) (op_bicategory.\<a> (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C src\<^sub>C trg\<^sub>C) \<i>\<^sub>C trg\<^sub>C src\<^sub>C (\<cdot>\<^sub>D) (\<lambda>f g. g \<star>\<^sub>D f) (op_bicategory.\<a> (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D src\<^sub>D trg\<^sub>D) \<i>\<^sub>D trg\<^sub>D src\<^sub>D F (\<lambda>f. \<Phi> (snd f, fst f))"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> contained.Partition" shows "contained.Class a = A"
sharing_consis_axioms: shows "sharing_consis \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
Int_insert_left: fixes a :: "'c"   and B :: "'c set"   and C :: "'c set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
that: fixes a :: "real" assumes "A \<le> a"   and "f \<in> \<Omega>(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a (real x)))" shows "thesis"
flip_altsublist_adjacent_length: fixes ss :: "'w list"   and ts :: "'w list" assumes "flip_altsublist_adjacent ss ts" shows "length ts = length ss"
X: shows "X \<in>\<^sub>\<circ> vproduct I A"
Input: fixes x :: "name"   and a :: "name"   and Q' :: "pi" assumes "x \<sharp> C"   and "Q \<longmapsto> a<x> \<prec> Q'" shows "\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"
ran_m_mapsto_upd_notin: fixes C :: "'c"   and N :: "('c, 'd) fmap"   and C' :: "'d" assumes "C \<notin># dom_m N" shows "ran_m (fmupd C C' N) = add_mset C' (ran_m N)"
poly_deg_times: fixes p :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'c"   and q :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "poly_deg (p * q) = poly_deg p + poly_deg q"
assms_2: fixes x :: "'a" assumes "x \<in> C" shows "x \<in> B"
nsqn_rt_fresh_asI: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt' :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt"   and "dip \<in> kD rt'"   and "nsqn rt dip = nsqn rt' dip"   and "\<pi>\<^sub>5 (the (rt dip)) = \<pi>\<^sub>5 (the (rt' dip))" shows "rt \<approx>\<^bsub>dip\<^esub> rt'"
bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst: fixes A :: "('g strand_label \<times> ('f, 'e) stateful_strand_step) list"   and \<delta> :: "'e \<Rightarrow> ('f, 'e) Term.term" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
nstd_case2: fixes b :: "cell list"   and a :: "nat"   and c :: "cell list" assumes "\<forall>m. b \<noteq> Bk \<up> m" shows "NSTD (trpl_code (a, b, c))"
assms: shows "let N = SMP0 Ana \<Gamma> M in set M \<subseteq> set N \<and> comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> N"
module_on_axioms: shows "module_on U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>2)"
induct: fixes x :: "'b list"   and G :: "'b list set"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> \<langle>G\<rangle>"   and "P \<epsilon>"   and "\<And>w1 w2. \<lbrakk>w1 \<in> G; w2 \<in> \<langle>G\<rangle>; P w2\<rbrakk> \<Longrightarrow> P (w1 \<cdot> w2)" shows "P x"
False: shows "\<nexists>n. \<forall>m\<ge>n. f m = f n"
4: shows "Domain a \<inter> Domain ({seller} \<times> ({\<Omega> - \<Union> (Range a)} - {{}})) = {}"
rel_adv_eq_dis_log_adv: fixes \<G> :: "'a cyclic_group"   and \<A> :: "'a \<Rightarrow> nat spmf" assumes "schnorr_sigma_protocol \<G>" shows "\<Sigma>_protocols_to_commitments.rel_advantage (schnorr_base.R_DL \<G>) (schnorr_base.G \<G>) \<A> = dis_log.advantage \<G> \<A>"
simps_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
chain_boundary_sum: fixes p :: "nat"   and g :: "'d \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and I :: "'d set" shows "chain_boundary p (sum g I) = sum (chain_boundary p \<circ> g) I"
pt_list_set_supp: fixes xs :: "'a list" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)" shows "supp (set xs) = supp xs"
simps_3: fixes \<nu> :: "'a"   and u :: "'a local.term" shows "Nml (\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> u) = (arr \<nu> \<and> Nml u \<and> \<not> is_Prim\<^sub>0 u \<and> \<^bold>\<langle>src \<nu>\<^bold>\<rangle>\<^sub>0 = Trg u)"